| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-muls 28134 | . . 3
⊢ 
·s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})))) | 
| 2 | 1 | norec2ov 27991 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (𝐴 ·s 𝐵) = (〈𝐴, 𝐵〉(𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})))( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})))) | 
| 3 |  | opex 5468 | . . . 4
⊢
〈𝐴, 𝐵〉 ∈ V | 
| 4 |  | mulsfn 28135 | . . . . . 6
⊢ 
·s Fn ( No  ×  No ) | 
| 5 |  | fnfun 6667 | . . . . . 6
⊢ (
·s Fn ( No  ×  No ) → Fun ·s ) | 
| 6 | 4, 5 | ax-mp 5 | . . . . 5
⊢ Fun
·s | 
| 7 |  | fvex 6918 | . . . . . . . . 9
⊢ ( L
‘𝐴) ∈
V | 
| 8 |  | fvex 6918 | . . . . . . . . 9
⊢ ( R
‘𝐴) ∈
V | 
| 9 | 7, 8 | unex 7765 | . . . . . . . 8
⊢ (( L
‘𝐴) ∪ ( R
‘𝐴)) ∈
V | 
| 10 |  | snex 5435 | . . . . . . . 8
⊢ {𝐴} ∈ V | 
| 11 | 9, 10 | unex 7765 | . . . . . . 7
⊢ ((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) ∈ V | 
| 12 |  | fvex 6918 | . . . . . . . . 9
⊢ ( L
‘𝐵) ∈
V | 
| 13 |  | fvex 6918 | . . . . . . . . 9
⊢ ( R
‘𝐵) ∈
V | 
| 14 | 12, 13 | unex 7765 | . . . . . . . 8
⊢ (( L
‘𝐵) ∪ ( R
‘𝐵)) ∈
V | 
| 15 |  | snex 5435 | . . . . . . . 8
⊢ {𝐵} ∈ V | 
| 16 | 14, 15 | unex 7765 | . . . . . . 7
⊢ ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵}) ∈ V | 
| 17 | 11, 16 | xpex 7774 | . . . . . 6
⊢ (((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∈ V | 
| 18 | 17 | difexi 5329 | . . . . 5
⊢ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}) ∈ V | 
| 19 |  | resfunexg 7236 | . . . . 5
⊢ ((Fun
·s ∧ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}) ∈ V) → (
·s ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) ∈ V) | 
| 20 | 6, 18, 19 | mp2an 692 | . . . 4
⊢ (
·s ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) ∈ V | 
| 21 |  | fveq2 6905 | . . . . . 6
⊢ (𝑧 = 〈𝐴, 𝐵〉 → (1st ‘𝑧) = (1st
‘〈𝐴, 𝐵〉)) | 
| 22 |  | fveq2 6905 | . . . . . . 7
⊢ (𝑧 = 〈𝐴, 𝐵〉 → (2nd ‘𝑧) = (2nd
‘〈𝐴, 𝐵〉)) | 
| 23 | 22 | csbeq1d 3902 | . . . . . 6
⊢ (𝑧 = 〈𝐴, 𝐵〉 → ⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})) = ⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))}))) | 
| 24 | 21, 23 | csbeq12dv 3907 | . . . . 5
⊢ (𝑧 = 〈𝐴, 𝐵〉 → ⦋(1st
‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})) = ⦋(1st
‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))}))) | 
| 25 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑝𝑚𝑦) = (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦)) | 
| 26 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑥𝑚𝑞) = (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) | 
| 27 | 25, 26 | oveq12d 7450 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → ((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) = ((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))) | 
| 28 |  | oveq 7438 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑝𝑚𝑞) = (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) | 
| 29 | 27, 28 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞)) = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))) | 
| 30 | 29 | eqeq2d 2747 | . . . . . . . . . . 11
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞)) ↔ 𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 31 | 30 | 2rexbidv 3221 | . . . . . . . . . 10
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 32 | 31 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} = {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))}) | 
| 33 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑟𝑚𝑦) = (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦)) | 
| 34 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑥𝑚𝑠) = (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) | 
| 35 | 33, 34 | oveq12d 7450 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → ((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) = ((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))) | 
| 36 |  | oveq 7438 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑟𝑚𝑠) = (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) | 
| 37 | 35, 36 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠)) = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))) | 
| 38 | 37 | eqeq2d 2747 | . . . . . . . . . . 11
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠)) ↔ 𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 39 | 38 | 2rexbidv 3221 | . . . . . . . . . 10
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠)) ↔ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 40 | 39 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))} = {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) | 
| 41 | 32, 40 | uneq12d 4168 | . . . . . . . 8
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) = ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))})) | 
| 42 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑡𝑚𝑦) = (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦)) | 
| 43 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑥𝑚𝑢) = (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) | 
| 44 | 42, 43 | oveq12d 7450 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → ((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) = ((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))) | 
| 45 |  | oveq 7438 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑡𝑚𝑢) = (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) | 
| 46 | 44, 45 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢)) = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))) | 
| 47 | 46 | eqeq2d 2747 | . . . . . . . . . . 11
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢)) ↔ 𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 48 | 47 | 2rexbidv 3221 | . . . . . . . . . 10
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢)) ↔ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 49 | 48 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} = {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))}) | 
| 50 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑣𝑚𝑦) = (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦)) | 
| 51 |  | oveq 7438 | . . . . . . . . . . . . . 14
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑥𝑚𝑤) = (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) | 
| 52 | 50, 51 | oveq12d 7450 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → ((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) = ((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))) | 
| 53 |  | oveq 7438 | . . . . . . . . . . . . 13
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑣𝑚𝑤) = (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) | 
| 54 | 52, 53 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤)) = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))) | 
| 55 | 54 | eqeq2d 2747 | . . . . . . . . . . 11
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤)) ↔ 𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 56 | 55 | 2rexbidv 3221 | . . . . . . . . . 10
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 57 | 56 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))} = {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}) | 
| 58 | 49, 57 | uneq12d 4168 | . . . . . . . 8
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))}) = ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) | 
| 59 | 41, 58 | oveq12d 7450 | . . . . . . 7
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 60 | 59 | csbeq2dv 3905 | . . . . . 6
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) →
⦋(2nd ‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})) = ⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 61 | 60 | csbeq2dv 3905 | . . . . 5
⊢ (𝑚 = ( ·s
↾ ((((( L ‘𝐴)
∪ ( R ‘𝐴)) ∪
{𝐴}) × ((( L
‘𝐵) ∪ ( R
‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) →
⦋(1st ‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})) = ⦋(1st
‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 62 |  | eqid 2736 | . . . . 5
⊢ (𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))}))) = (𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))}))) | 
| 63 |  | ovex 7465 | . . . . . . 7
⊢ (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) ∈ V | 
| 64 | 63 | csbex 5310 | . . . . . 6
⊢
⦋(2nd ‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) ∈ V | 
| 65 | 64 | csbex 5310 | . . . . 5
⊢
⦋(1st ‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) ∈ V | 
| 66 | 24, 61, 62, 65 | ovmpo 7594 | . . . 4
⊢
((〈𝐴, 𝐵〉 ∈ V ∧ (
·s ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) ∈ V) → (〈𝐴, 𝐵〉(𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})))( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))) = ⦋(1st
‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 67 | 3, 20, 66 | mp2an 692 | . . 3
⊢
(〈𝐴, 𝐵〉(𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})))( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))) = ⦋(1st
‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) | 
| 68 |  | op1stg 8027 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | 
| 69 | 68 | csbeq1d 3902 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋(1st
‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = ⦋𝐴 / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 70 |  | op2ndg 8028 | . . . . . 6
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | 
| 71 | 70 | csbeq1d 3902 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 72 | 71 | csbeq2dv 3905 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋𝐴 / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 73 |  | simpl 482 | . . . . . 6
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐴 ∈  No
) | 
| 74 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴)) | 
| 75 |  | oveq1 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐴 → (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞) = (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) | 
| 76 | 75 | oveq2d 7448 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝐴 → ((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) = ((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))) | 
| 77 | 76 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝐴 → (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))) | 
| 78 | 77 | eqeq2d 2747 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝐴 → (𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ 𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 79 | 78 | rexbidv 3178 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → (∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ ∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 80 | 74, 79 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 81 | 80 | abbidv 2807 | . . . . . . . . . 10
⊢ (𝑥 = 𝐴 → {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} = {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))}) | 
| 82 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴)) | 
| 83 |  | oveq1 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐴 → (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠) = (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) | 
| 84 | 83 | oveq2d 7448 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝐴 → ((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) = ((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))) | 
| 85 | 84 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝐴 → (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))) | 
| 86 | 85 | eqeq2d 2747 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝐴 → (𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ 𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 87 | 86 | rexbidv 3178 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → (∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ ∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 88 | 82, 87 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 89 | 88 | abbidv 2807 | . . . . . . . . . 10
⊢ (𝑥 = 𝐴 → {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))} = {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) | 
| 90 | 81, 89 | uneq12d 4168 | . . . . . . . . 9
⊢ (𝑥 = 𝐴 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) = ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))})) | 
| 91 |  | oveq1 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐴 → (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢) = (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) | 
| 92 | 91 | oveq2d 7448 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝐴 → ((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) = ((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))) | 
| 93 | 92 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝐴 → (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))) | 
| 94 | 93 | eqeq2d 2747 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝐴 → (𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ 𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 95 | 94 | rexbidv 3178 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → (∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ ∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 96 | 74, 95 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 97 | 96 | abbidv 2807 | . . . . . . . . . 10
⊢ (𝑥 = 𝐴 → {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} = {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))}) | 
| 98 |  | oveq1 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐴 → (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤) = (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) | 
| 99 | 98 | oveq2d 7448 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝐴 → ((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) = ((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))) | 
| 100 | 99 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝐴 → (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))) | 
| 101 | 100 | eqeq2d 2747 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝐴 → (𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ 𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 102 | 101 | rexbidv 3178 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → (∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ ∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 103 | 82, 102 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 104 | 103 | abbidv 2807 | . . . . . . . . . 10
⊢ (𝑥 = 𝐴 → {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))} = {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}) | 
| 105 | 97, 104 | uneq12d 4168 | . . . . . . . . 9
⊢ (𝑥 = 𝐴 → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}) = ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) | 
| 106 | 90, 105 | oveq12d 7450 | . . . . . . . 8
⊢ (𝑥 = 𝐴 → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 107 | 106 | csbeq2dv 3905 | . . . . . . 7
⊢ (𝑥 = 𝐴 → ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 108 | 107 | adantl 481 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 109 | 73, 108 | csbied 3934 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 110 |  | simpr 484 | . . . . . 6
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐵 ∈  No
) | 
| 111 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → ( L ‘𝑦) = ( L ‘𝐵)) | 
| 112 |  | oveq2 7440 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝐵 → (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) = (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵)) | 
| 113 | 112 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝐵 → ((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) = ((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))) | 
| 114 | 113 | oveq1d 7447 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝐵 → (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))) | 
| 115 | 114 | eqeq2d 2747 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → (𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ 𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 116 | 111, 115 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑦 = 𝐵 → (∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ ∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 117 | 116 | rexbidv 3178 | . . . . . . . . . 10
⊢ (𝑦 = 𝐵 → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)))) | 
| 118 | 117 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑦 = 𝐵 → {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} = {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))}) | 
| 119 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → ( R ‘𝑦) = ( R ‘𝐵)) | 
| 120 |  | oveq2 7440 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝐵 → (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) = (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵)) | 
| 121 | 120 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝐵 → ((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) = ((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))) | 
| 122 | 121 | oveq1d 7447 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝐵 → (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))) | 
| 123 | 122 | eqeq2d 2747 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → (𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ 𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 124 | 119, 123 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑦 = 𝐵 → (∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ ∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 125 | 124 | rexbidv 3178 | . . . . . . . . . 10
⊢ (𝑦 = 𝐵 → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)))) | 
| 126 | 125 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑦 = 𝐵 → {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))} = {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) | 
| 127 | 118, 126 | uneq12d 4168 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) = ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))})) | 
| 128 |  | oveq2 7440 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝐵 → (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) = (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵)) | 
| 129 | 128 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝐵 → ((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) = ((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))) | 
| 130 | 129 | oveq1d 7447 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝐵 → (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))) | 
| 131 | 130 | eqeq2d 2747 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → (𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ 𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 132 | 119, 131 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑦 = 𝐵 → (∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ ∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 133 | 132 | rexbidv 3178 | . . . . . . . . . 10
⊢ (𝑦 = 𝐵 → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)))) | 
| 134 | 133 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑦 = 𝐵 → {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} = {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))}) | 
| 135 |  | oveq2 7440 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝐵 → (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) = (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵)) | 
| 136 | 135 | oveq1d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝐵 → ((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) = ((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))) | 
| 137 | 136 | oveq1d 7447 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝐵 → (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))) | 
| 138 | 137 | eqeq2d 2747 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → (𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ 𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 139 | 111, 138 | rexeqbidv 3346 | . . . . . . . . . . 11
⊢ (𝑦 = 𝐵 → (∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ ∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 140 | 139 | rexbidv 3178 | . . . . . . . . . 10
⊢ (𝑦 = 𝐵 → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)))) | 
| 141 | 140 | abbidv 2807 | . . . . . . . . 9
⊢ (𝑦 = 𝐵 → {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))} = {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}) | 
| 142 | 134, 141 | uneq12d 4168 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}) = ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) | 
| 143 | 127, 142 | oveq12d 7450 | . . . . . . 7
⊢ (𝑦 = 𝐵 → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 144 | 143 | adantl 481 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ 𝑦 = 𝐵) → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 145 | 110, 144 | csbied 3934 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}))) | 
| 146 |  | elun1 4181 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 ∈ ( L ‘𝐴) → 𝑝 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 147 | 146 | ad2antrl 728 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 148 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑝 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 149 | 147, 148 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 150 |  | snidg 4659 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐵 ∈ 
No  → 𝐵 ∈
{𝐵}) | 
| 151 | 150 | adantl 481 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐵 ∈ {𝐵}) | 
| 152 |  | elun2 4182 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 153 | 151, 152 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 154 | 153 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 155 | 149, 154 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝑝, 𝐵〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 156 |  | leftirr 27930 | . . . . . . . . . . . . . . . . . . . . 21
⊢  ¬
𝐴 ∈ ( L ‘𝐴) | 
| 157 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = 𝐴 → (𝑝 ∈ ( L ‘𝐴) ↔ 𝐴 ∈ ( L ‘𝐴))) | 
| 158 | 156, 157 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = 𝐴 → ¬ 𝑝 ∈ ( L ‘𝐴)) | 
| 159 | 158 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 ∈ ( L ‘𝐴) → 𝑝 ≠ 𝐴) | 
| 160 | 159 | ad2antrl 728 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ≠ 𝐴) | 
| 161 | 160 | orcd 873 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)) | 
| 162 |  | vex 3483 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑝 ∈ V | 
| 163 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑝 ∈ V ∧ 𝐵 ∈ 
No ) → (〈𝑝, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 164 | 162, 163 | mpan 690 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ 
No  → (〈𝑝,
𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 165 | 164 | ad2antlr 727 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (〈𝑝, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 166 | 161, 165 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝑝, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 167 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝑝, 𝐵〉 ∈ V | 
| 168 | 167 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝑝, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑝, 𝐵〉 = 〈𝐴, 𝐵〉) | 
| 169 | 168 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝑝, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑝, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 170 | 166, 169 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ¬ 〈𝑝, 𝐵〉 ∈ {〈𝐴, 𝐵〉}) | 
| 171 | 155, 170 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝑝, 𝐵〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 172 | 171 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑝, 𝐵〉) = ( ·s
‘〈𝑝, 𝐵〉)) | 
| 173 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑝( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑝, 𝐵〉) | 
| 174 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑝 ·s 𝐵) = ( ·s
‘〈𝑝, 𝐵〉) | 
| 175 | 172, 173,
174 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (𝑝 ·s 𝐵)) | 
| 176 |  | snidg 4659 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ 
No  → 𝐴 ∈
{𝐴}) | 
| 177 | 176 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐴 ∈ {𝐴}) | 
| 178 |  | elun2 4182 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 179 | 177, 178 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 180 | 179 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 181 |  | elun1 4181 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑞 ∈ ( L ‘𝐵) → 𝑞 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 182 | 181 | ad2antll 729 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 183 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑞 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑞 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 184 | 182, 183 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 185 | 180, 184 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝐴, 𝑞〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 186 |  | leftirr 27930 | . . . . . . . . . . . . . . . . . . . . 21
⊢  ¬
𝐵 ∈ ( L ‘𝐵) | 
| 187 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑞 = 𝐵 → (𝑞 ∈ ( L ‘𝐵) ↔ 𝐵 ∈ ( L ‘𝐵))) | 
| 188 | 186, 187 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑞 = 𝐵 → ¬ 𝑞 ∈ ( L ‘𝐵)) | 
| 189 | 188 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑞 ∈ ( L ‘𝐵) → 𝑞 ≠ 𝐵) | 
| 190 | 189 | ad2antll 729 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ≠ 𝐵) | 
| 191 | 190 | olcd 874 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵)) | 
| 192 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ 
No  ∧ 𝑞 ∈
V) → (〈𝐴, 𝑞〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵))) | 
| 193 | 192 | elvd 3485 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 
No  → (〈𝐴,
𝑞〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵))) | 
| 194 | 193 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (〈𝐴, 𝑞〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵))) | 
| 195 | 191, 194 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝐴, 𝑞〉 ≠ 〈𝐴, 𝐵〉) | 
| 196 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝐴, 𝑞〉 ∈ V | 
| 197 | 196 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝐴, 𝑞〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑞〉 = 〈𝐴, 𝐵〉) | 
| 198 | 197 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝐴, 𝑞〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑞〉 ≠ 〈𝐴, 𝐵〉) | 
| 199 | 195, 198 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ¬ 〈𝐴, 𝑞〉 ∈ {〈𝐴, 𝐵〉}) | 
| 200 | 185, 199 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝐴, 𝑞〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 201 | 200 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑞〉) = ( ·s
‘〈𝐴, 𝑞〉)) | 
| 202 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑞〉) | 
| 203 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴 ·s 𝑞) = ( ·s
‘〈𝐴, 𝑞〉) | 
| 204 | 201, 202,
203 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞) = (𝐴 ·s 𝑞)) | 
| 205 | 175, 204 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) = ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞))) | 
| 206 | 149, 184 | opelxpd 5723 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝑝, 𝑞〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 207 | 190 | olcd 874 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵)) | 
| 208 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑞 ∈ V | 
| 209 | 162, 208 | opthne 5486 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑝, 𝑞〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑝 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵)) | 
| 210 | 207, 209 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝑝, 𝑞〉 ≠ 〈𝐴, 𝐵〉) | 
| 211 |  | opex 5468 | . . . . . . . . . . . . . . . . 17
⊢
〈𝑝, 𝑞〉 ∈ V | 
| 212 | 211 | elsn 4640 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑝, 𝑞〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑝, 𝑞〉 = 〈𝐴, 𝐵〉) | 
| 213 | 212 | necon3bbii 2987 | . . . . . . . . . . . . . . 15
⊢ (¬
〈𝑝, 𝑞〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑝, 𝑞〉 ≠ 〈𝐴, 𝐵〉) | 
| 214 | 210, 213 | sylibr 234 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ¬ 〈𝑝, 𝑞〉 ∈ {〈𝐴, 𝐵〉}) | 
| 215 | 206, 214 | eldifd 3961 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 〈𝑝, 𝑞〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 216 | 215 | fvresd 6925 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑝, 𝑞〉) = ( ·s
‘〈𝑝, 𝑞〉)) | 
| 217 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑝( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑝, 𝑞〉) | 
| 218 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑝 ·s 𝑞) = ( ·s
‘〈𝑝, 𝑞〉) | 
| 219 | 216, 217,
218 | 3eqtr4g 2801 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞) = (𝑝 ·s 𝑞)) | 
| 220 | 205, 219 | oveq12d 7450 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) | 
| 221 | 220 | eqeq2d 2747 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) | 
| 222 | 221 | 2rexbidva 3219 | . . . . . . . 8
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) | 
| 223 | 222 | abbidv 2807 | . . . . . . 7
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} = {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))}) | 
| 224 |  | elun2 4182 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑟 ∈ ( R ‘𝐴) → 𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 225 | 224 | ad2antrl 728 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 226 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 227 | 225, 226 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 228 | 153 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 229 | 227, 228 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝑟, 𝐵〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 230 |  | rightirr 27931 | . . . . . . . . . . . . . . . . . . . . 21
⊢  ¬
𝐴 ∈ ( R ‘𝐴) | 
| 231 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 = 𝐴 → (𝑟 ∈ ( R ‘𝐴) ↔ 𝐴 ∈ ( R ‘𝐴))) | 
| 232 | 230, 231 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 = 𝐴 → ¬ 𝑟 ∈ ( R ‘𝐴)) | 
| 233 | 232 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 ∈ ( R ‘𝐴) → 𝑟 ≠ 𝐴) | 
| 234 | 233 | ad2antrl 728 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ≠ 𝐴) | 
| 235 | 234 | orcd 873 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)) | 
| 236 |  | vex 3483 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑟 ∈ V | 
| 237 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑟 ∈ V ∧ 𝐵 ∈ 
No ) → (〈𝑟, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 238 | 236, 237 | mpan 690 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ 
No  → (〈𝑟,
𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 239 | 238 | ad2antlr 727 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (〈𝑟, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 240 | 235, 239 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝑟, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 241 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝑟, 𝐵〉 ∈ V | 
| 242 | 241 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝑟, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑟, 𝐵〉 = 〈𝐴, 𝐵〉) | 
| 243 | 242 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝑟, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑟, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 244 | 240, 243 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ¬ 〈𝑟, 𝐵〉 ∈ {〈𝐴, 𝐵〉}) | 
| 245 | 229, 244 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝑟, 𝐵〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 246 | 245 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑟, 𝐵〉) = ( ·s
‘〈𝑟, 𝐵〉)) | 
| 247 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑟( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑟, 𝐵〉) | 
| 248 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑟 ·s 𝐵) = ( ·s
‘〈𝑟, 𝐵〉) | 
| 249 | 246, 247,
248 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (𝑟 ·s 𝐵)) | 
| 250 | 179 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 251 |  | elun2 4182 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ ( R ‘𝐵) → 𝑠 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 252 | 251 | ad2antll 729 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 253 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑠 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 254 | 252, 253 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 255 | 250, 254 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝐴, 𝑠〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 256 |  | rightirr 27931 | . . . . . . . . . . . . . . . . . . . . 21
⊢  ¬
𝐵 ∈ ( R ‘𝐵) | 
| 257 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = 𝐵 → (𝑠 ∈ ( R ‘𝐵) ↔ 𝐵 ∈ ( R ‘𝐵))) | 
| 258 | 256, 257 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 𝐵 → ¬ 𝑠 ∈ ( R ‘𝐵)) | 
| 259 | 258 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ ( R ‘𝐵) → 𝑠 ≠ 𝐵) | 
| 260 | 259 | ad2antll 729 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ≠ 𝐵) | 
| 261 | 260 | olcd 874 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵)) | 
| 262 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ 
No  ∧ 𝑠 ∈
V) → (〈𝐴, 𝑠〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵))) | 
| 263 | 262 | elvd 3485 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 
No  → (〈𝐴,
𝑠〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵))) | 
| 264 | 263 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (〈𝐴, 𝑠〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵))) | 
| 265 | 261, 264 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝐴, 𝑠〉 ≠ 〈𝐴, 𝐵〉) | 
| 266 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝐴, 𝑠〉 ∈ V | 
| 267 | 266 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝐴, 𝑠〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑠〉 = 〈𝐴, 𝐵〉) | 
| 268 | 267 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝐴, 𝑠〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑠〉 ≠ 〈𝐴, 𝐵〉) | 
| 269 | 265, 268 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ¬ 〈𝐴, 𝑠〉 ∈ {〈𝐴, 𝐵〉}) | 
| 270 | 255, 269 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝐴, 𝑠〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 271 | 270 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑠〉) = ( ·s
‘〈𝐴, 𝑠〉)) | 
| 272 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑠〉) | 
| 273 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴 ·s 𝑠) = ( ·s
‘〈𝐴, 𝑠〉) | 
| 274 | 271, 272,
273 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠) = (𝐴 ·s 𝑠)) | 
| 275 | 249, 274 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) = ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠))) | 
| 276 | 227, 254 | opelxpd 5723 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝑟, 𝑠〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 277 | 260 | olcd 874 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵)) | 
| 278 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑠 ∈ V | 
| 279 | 236, 278 | opthne 5486 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑟, 𝑠〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑟 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵)) | 
| 280 | 277, 279 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝑟, 𝑠〉 ≠ 〈𝐴, 𝐵〉) | 
| 281 |  | opex 5468 | . . . . . . . . . . . . . . . . 17
⊢
〈𝑟, 𝑠〉 ∈ V | 
| 282 | 281 | elsn 4640 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑟, 𝑠〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑟, 𝑠〉 = 〈𝐴, 𝐵〉) | 
| 283 | 282 | necon3bbii 2987 | . . . . . . . . . . . . . . 15
⊢ (¬
〈𝑟, 𝑠〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑟, 𝑠〉 ≠ 〈𝐴, 𝐵〉) | 
| 284 | 280, 283 | sylibr 234 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ¬ 〈𝑟, 𝑠〉 ∈ {〈𝐴, 𝐵〉}) | 
| 285 | 276, 284 | eldifd 3961 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 〈𝑟, 𝑠〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 286 | 285 | fvresd 6925 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑟, 𝑠〉) = ( ·s
‘〈𝑟, 𝑠〉)) | 
| 287 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑟( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑟, 𝑠〉) | 
| 288 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑟 ·s 𝑠) = ( ·s
‘〈𝑟, 𝑠〉) | 
| 289 | 286, 287,
288 | 3eqtr4g 2801 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠) = (𝑟 ·s 𝑠)) | 
| 290 | 275, 289 | oveq12d 7450 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) | 
| 291 | 290 | eqeq2d 2747 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) | 
| 292 | 291 | 2rexbidva 3219 | . . . . . . . 8
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) | 
| 293 | 292 | abbidv 2807 | . . . . . . 7
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))} = {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) | 
| 294 | 223, 293 | uneq12d 4168 | . . . . . 6
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) = ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})) | 
| 295 |  | elun1 4181 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ ( L ‘𝐴) → 𝑡 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 296 | 295 | ad2antrl 728 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 297 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑡 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 298 | 296, 297 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 299 | 153 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 300 | 298, 299 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝑡, 𝐵〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 301 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝐴 → (𝑡 ∈ ( L ‘𝐴) ↔ 𝐴 ∈ ( L ‘𝐴))) | 
| 302 | 156, 301 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝐴 → ¬ 𝑡 ∈ ( L ‘𝐴)) | 
| 303 | 302 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 ∈ ( L ‘𝐴) → 𝑡 ≠ 𝐴) | 
| 304 | 303 | ad2antrl 728 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ≠ 𝐴) | 
| 305 | 304 | orcd 873 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)) | 
| 306 |  | vex 3483 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑡 ∈ V | 
| 307 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑡 ∈ V ∧ 𝐵 ∈ 
No ) → (〈𝑡, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 308 | 306, 307 | mpan 690 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ 
No  → (〈𝑡,
𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 309 | 308 | ad2antlr 727 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (〈𝑡, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 310 | 305, 309 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝑡, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 311 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝑡, 𝐵〉 ∈ V | 
| 312 | 311 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝑡, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑡, 𝐵〉 = 〈𝐴, 𝐵〉) | 
| 313 | 312 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝑡, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑡, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 314 | 310, 313 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ¬ 〈𝑡, 𝐵〉 ∈ {〈𝐴, 𝐵〉}) | 
| 315 | 300, 314 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝑡, 𝐵〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 316 | 315 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑡, 𝐵〉) = ( ·s
‘〈𝑡, 𝐵〉)) | 
| 317 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑡( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑡, 𝐵〉) | 
| 318 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑡 ·s 𝐵) = ( ·s
‘〈𝑡, 𝐵〉) | 
| 319 | 316, 317,
318 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (𝑡 ·s 𝐵)) | 
| 320 | 179 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 321 |  | elun2 4182 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑢 ∈ ( R ‘𝐵) → 𝑢 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 322 | 321 | ad2antll 729 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 323 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑢 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑢 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 324 | 322, 323 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 325 | 320, 324 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝐴, 𝑢〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 326 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 = 𝐵 → (𝑢 ∈ ( R ‘𝐵) ↔ 𝐵 ∈ ( R ‘𝐵))) | 
| 327 | 256, 326 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 𝐵 → ¬ 𝑢 ∈ ( R ‘𝐵)) | 
| 328 | 327 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 ∈ ( R ‘𝐵) → 𝑢 ≠ 𝐵) | 
| 329 | 328 | ad2antll 729 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ≠ 𝐵) | 
| 330 | 329 | olcd 874 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵)) | 
| 331 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ 
No  ∧ 𝑢 ∈
V) → (〈𝐴, 𝑢〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵))) | 
| 332 | 331 | elvd 3485 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 
No  → (〈𝐴,
𝑢〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵))) | 
| 333 | 332 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (〈𝐴, 𝑢〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵))) | 
| 334 | 330, 333 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝐴, 𝑢〉 ≠ 〈𝐴, 𝐵〉) | 
| 335 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝐴, 𝑢〉 ∈ V | 
| 336 | 335 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝐴, 𝑢〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑢〉 = 〈𝐴, 𝐵〉) | 
| 337 | 336 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝐴, 𝑢〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑢〉 ≠ 〈𝐴, 𝐵〉) | 
| 338 | 334, 337 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ¬ 〈𝐴, 𝑢〉 ∈ {〈𝐴, 𝐵〉}) | 
| 339 | 325, 338 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝐴, 𝑢〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 340 | 339 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑢〉) = ( ·s
‘〈𝐴, 𝑢〉)) | 
| 341 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑢〉) | 
| 342 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴 ·s 𝑢) = ( ·s
‘〈𝐴, 𝑢〉) | 
| 343 | 340, 341,
342 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢) = (𝐴 ·s 𝑢)) | 
| 344 | 319, 343 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) = ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢))) | 
| 345 | 298, 324 | opelxpd 5723 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝑡, 𝑢〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 346 | 329 | olcd 874 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵)) | 
| 347 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑢 ∈ V | 
| 348 | 306, 347 | opthne 5486 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑡, 𝑢〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑡 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵)) | 
| 349 | 346, 348 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝑡, 𝑢〉 ≠ 〈𝐴, 𝐵〉) | 
| 350 |  | opex 5468 | . . . . . . . . . . . . . . . . 17
⊢
〈𝑡, 𝑢〉 ∈ V | 
| 351 | 350 | elsn 4640 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑡, 𝑢〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑡, 𝑢〉 = 〈𝐴, 𝐵〉) | 
| 352 | 351 | necon3bbii 2987 | . . . . . . . . . . . . . . 15
⊢ (¬
〈𝑡, 𝑢〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑡, 𝑢〉 ≠ 〈𝐴, 𝐵〉) | 
| 353 | 349, 352 | sylibr 234 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ¬ 〈𝑡, 𝑢〉 ∈ {〈𝐴, 𝐵〉}) | 
| 354 | 345, 353 | eldifd 3961 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 〈𝑡, 𝑢〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 355 | 354 | fvresd 6925 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑡, 𝑢〉) = ( ·s
‘〈𝑡, 𝑢〉)) | 
| 356 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑡( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑡, 𝑢〉) | 
| 357 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑡 ·s 𝑢) = ( ·s
‘〈𝑡, 𝑢〉) | 
| 358 | 355, 356,
357 | 3eqtr4g 2801 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢) = (𝑡 ·s 𝑢)) | 
| 359 | 344, 358 | oveq12d 7450 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) | 
| 360 | 359 | eqeq2d 2747 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))) | 
| 361 | 360 | 2rexbidva 3219 | . . . . . . . 8
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) ↔ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))) | 
| 362 | 361 | abbidv 2807 | . . . . . . 7
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} = {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}) | 
| 363 |  | elun2 4182 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ ( R ‘𝐴) → 𝑣 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 364 | 363 | ad2antrl 728 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))) | 
| 365 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑣 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 366 | 364, 365 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 367 | 153 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 368 | 366, 367 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝑣, 𝐵〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 369 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝐴 → (𝑣 ∈ ( R ‘𝐴) ↔ 𝐴 ∈ ( R ‘𝐴))) | 
| 370 | 230, 369 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐴 → ¬ 𝑣 ∈ ( R ‘𝐴)) | 
| 371 | 370 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 ∈ ( R ‘𝐴) → 𝑣 ≠ 𝐴) | 
| 372 | 371 | ad2antrl 728 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ≠ 𝐴) | 
| 373 | 372 | orcd 873 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)) | 
| 374 |  | vex 3483 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑣 ∈ V | 
| 375 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∈ V ∧ 𝐵 ∈ 
No ) → (〈𝑣, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 376 | 374, 375 | mpan 690 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ 
No  → (〈𝑣,
𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 377 | 376 | ad2antlr 727 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (〈𝑣, 𝐵〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))) | 
| 378 | 373, 377 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝑣, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 379 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝑣, 𝐵〉 ∈ V | 
| 380 | 379 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝑣, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑣, 𝐵〉 = 〈𝐴, 𝐵〉) | 
| 381 | 380 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝑣, 𝐵〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑣, 𝐵〉 ≠ 〈𝐴, 𝐵〉) | 
| 382 | 378, 381 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ¬ 〈𝑣, 𝐵〉 ∈ {〈𝐴, 𝐵〉}) | 
| 383 | 368, 382 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝑣, 𝐵〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 384 | 383 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑣, 𝐵〉) = ( ·s
‘〈𝑣, 𝐵〉)) | 
| 385 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑣( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑣, 𝐵〉) | 
| 386 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝑣 ·s 𝐵) = ( ·s
‘〈𝑣, 𝐵〉) | 
| 387 | 384, 385,
386 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) = (𝑣 ·s 𝐵)) | 
| 388 | 179 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) | 
| 389 |  | elun1 4181 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ ( L ‘𝐵) → 𝑤 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 390 | 389 | ad2antll 729 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))) | 
| 391 |  | elun1 4181 | . . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑤 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 392 | 390, 391 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) | 
| 393 | 388, 392 | opelxpd 5723 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝐴, 𝑤〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 394 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = 𝐵 → (𝑤 ∈ ( L ‘𝐵) ↔ 𝐵 ∈ ( L ‘𝐵))) | 
| 395 | 186, 394 | mtbiri 327 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝐵 → ¬ 𝑤 ∈ ( L ‘𝐵)) | 
| 396 | 395 | necon2ai 2969 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 ∈ ( L ‘𝐵) → 𝑤 ≠ 𝐵) | 
| 397 | 396 | ad2antll 729 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ≠ 𝐵) | 
| 398 | 397 | olcd 874 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵)) | 
| 399 |  | opthneg 5485 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ 
No  ∧ 𝑤 ∈
V) → (〈𝐴, 𝑤〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵))) | 
| 400 | 399 | elvd 3485 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 
No  → (〈𝐴,
𝑤〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵))) | 
| 401 | 400 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (〈𝐴, 𝑤〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵))) | 
| 402 | 398, 401 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝐴, 𝑤〉 ≠ 〈𝐴, 𝐵〉) | 
| 403 |  | opex 5468 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝐴, 𝑤〉 ∈ V | 
| 404 | 403 | elsn 4640 | . . . . . . . . . . . . . . . . 17
⊢
(〈𝐴, 𝑤〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑤〉 = 〈𝐴, 𝐵〉) | 
| 405 | 404 | necon3bbii 2987 | . . . . . . . . . . . . . . . 16
⊢ (¬
〈𝐴, 𝑤〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝑤〉 ≠ 〈𝐴, 𝐵〉) | 
| 406 | 402, 405 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ¬ 〈𝐴, 𝑤〉 ∈ {〈𝐴, 𝐵〉}) | 
| 407 | 393, 406 | eldifd 3961 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝐴, 𝑤〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 408 | 407 | fvresd 6925 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑤〉) = ( ·s
‘〈𝐴, 𝑤〉)) | 
| 409 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝐴, 𝑤〉) | 
| 410 |  | df-ov 7435 | . . . . . . . . . . . . 13
⊢ (𝐴 ·s 𝑤) = ( ·s
‘〈𝐴, 𝑤〉) | 
| 411 | 408, 409,
410 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤) = (𝐴 ·s 𝑤)) | 
| 412 | 387, 411 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) = ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤))) | 
| 413 | 366, 392 | opelxpd 5723 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝑣, 𝑤〉 ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) | 
| 414 | 397 | olcd 874 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵)) | 
| 415 |  | vex 3483 | . . . . . . . . . . . . . . . . 17
⊢ 𝑤 ∈ V | 
| 416 | 374, 415 | opthne 5486 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑣, 𝑤〉 ≠ 〈𝐴, 𝐵〉 ↔ (𝑣 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵)) | 
| 417 | 414, 416 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝑣, 𝑤〉 ≠ 〈𝐴, 𝐵〉) | 
| 418 |  | opex 5468 | . . . . . . . . . . . . . . . . 17
⊢
〈𝑣, 𝑤〉 ∈ V | 
| 419 | 418 | elsn 4640 | . . . . . . . . . . . . . . . 16
⊢
(〈𝑣, 𝑤〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑣, 𝑤〉 = 〈𝐴, 𝐵〉) | 
| 420 | 419 | necon3bbii 2987 | . . . . . . . . . . . . . . 15
⊢ (¬
〈𝑣, 𝑤〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑣, 𝑤〉 ≠ 〈𝐴, 𝐵〉) | 
| 421 | 417, 420 | sylibr 234 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ¬ 〈𝑣, 𝑤〉 ∈ {〈𝐴, 𝐵〉}) | 
| 422 | 413, 421 | eldifd 3961 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 〈𝑣, 𝑤〉 ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) | 
| 423 | 422 | fvresd 6925 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑣, 𝑤〉) = ( ·s
‘〈𝑣, 𝑤〉)) | 
| 424 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑣( ·s ↾
((((( L ‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤) = (( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))‘〈𝑣, 𝑤〉) | 
| 425 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝑣 ·s 𝑤) = ( ·s
‘〈𝑣, 𝑤〉) | 
| 426 | 423, 424,
425 | 3eqtr4g 2801 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤) = (𝑣 ·s 𝑤)) | 
| 427 | 412, 426 | oveq12d 7450 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) | 
| 428 | 427 | eqeq2d 2747 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) | 
| 429 | 428 | 2rexbidva 3219 | . . . . . . . 8
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) | 
| 430 | 429 | abbidv 2807 | . . . . . . 7
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))} = {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) | 
| 431 | 362, 430 | uneq12d 4168 | . . . . . 6
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))}) = ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) | 
| 432 | 294, 431 | oveq12d 7450 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝐵) +s (𝐴( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) | 
| 433 | 109, 145,
432 | 3eqtrd 2780 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) | 
| 434 | 69, 72, 433 | 3eqtrd 2780 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ⦋(1st
‘〈𝐴, 𝐵〉) / 𝑥⦌⦋(2nd
‘〈𝐴, 𝐵〉) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞)) -s (𝑝( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠)) -s (𝑟( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢)) -s (𝑡( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑦) +s (𝑥( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤)) -s (𝑣( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) | 
| 435 | 67, 434 | eqtrid 2788 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (〈𝐴, 𝐵〉(𝑧 ∈ V, 𝑚 ∈ V ↦
⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd
‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))})))( ·s ↾ ((((( L
‘𝐴) ∪ ( R
‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) | 
| 436 | 2, 435 | eqtrd 2776 | 1
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) |