Theorem mulsval 28500

Description: The value of surreal multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)

Assertion

Ref	Expression
mulsval	⊢ ((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))})))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝐵,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤

Proof of Theorem mulsval

Dummy variables 𝑧 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-muls 28498	. . 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 28348	. 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 5431	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		mulsfn 28499	. . . . . 6 ⊢ ·s Fn (No × No)
5		fnfun 6639	. . . . . 6 ⊢ ( ·s Fn (No × No) → Fun ·s )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun ·s
7		fvex 6898	. . . . . . . . 9 ⊢ (L‘𝐴) ∈ V
8		fvex 6898	. . . . . . . . 9 ⊢ (R‘𝐴) ∈ V
9	7, 8	unex 7761	. . . . . . . 8 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) ∈ V
10		snex 5396	. . . . . . . 8 ⊢ {𝐴} ∈ V
11	9, 10	unex 7761	. . . . . . 7 ⊢ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) ∈ V
12		fvex 6898	. . . . . . . . 9 ⊢ (L‘𝐵) ∈ V
13		fvex 6898	. . . . . . . . 9 ⊢ (R‘𝐵) ∈ V
14	12, 13	unex 7761	. . . . . . . 8 ⊢ ((L‘𝐵) ∪ (R‘𝐵)) ∈ V
15		snex 5396	. . . . . . . 8 ⊢ {𝐵} ∈ V
16	14, 15	unex 7761	. . . . . . 7 ⊢ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}) ∈ V
17	11, 16	xpex 7767	. . . . . 6 ⊢ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∈ V
18	17	difexi 5291	. . . . 5 ⊢ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ∈ V
19		resfunexg 7221	. . . . 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 705	. . . 4 ⊢ ( ·s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) ∈ V
21		fveq2 6885	. . . . . 6 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑧) = (1st ‘⟨𝐴, 𝐵⟩))
22		fveq2 6885	. . . . . . 7 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑧) = (2nd ‘⟨𝐴, 𝐵⟩))
23	22	csbeq1d 3850	. . . . . 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 3855	. . . . 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 7426	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 3227	. . . . . . . . . 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 2826	. . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 3227	. . . . . . . . . 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 2826	. . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 3227	. . . . . . . . . 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 2826	. . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 3227	. . . . . . . . . 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 2826	. . . . . . . . 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 7438	. . . . . . 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 3853	. . . . . 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 3853	. . . . 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 2760	. . . . 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 7453	. . . . . . 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 5264	. . . . . 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 5264	. . . . 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 7580	. . . 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 705	. . 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 8013	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
69	68	csbeq1d 3850	. . . 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 8014	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
71	70	csbeq1d 3850	. . . . 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 3853	. . . 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 488	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐴 ∈ No)
74		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (L‘𝑥) = (L‘𝐴))
75		oveq1 7427	. . . . . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 3186	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 2826	. . . . . . . . . 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 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (R‘𝑥) = (R‘𝐴))
83		oveq1 7427	. . . . . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 3186	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 2826	. . . . . . . . . 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 7427	. . . . . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 3186	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 2826	. . . . . . . . . 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 7427	. . . . . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 3186	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 2826	. . . . . . . . . 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 7438	. . . . . . . 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 3853	. . . . . . 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 487	. . . . . 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 3882	. . . . 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 490	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐵 ∈ No)
111		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (L‘𝑦) = (L‘𝐵))
112		oveq2 7428	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 3186	. . . . . . . . . 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 2826	. . . . . . . . 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 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (R‘𝑦) = (R‘𝐵))
120		oveq2 7428	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 3186	. . . . . . . . . 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 2826	. . . . . . . . 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 7428	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 3186	. . . . . . . . . 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 2826	. . . . . . . . 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 7428	. . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 3186	. . . . . . . . . 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 2826	. . . . . . . . 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 7438	. . . . . . 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 487	. . . . . 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 3882	. . . . 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 4180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (L‘𝐴) → 𝑝 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
147	146	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝑝 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
148		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ((L‘𝐴) ∪ (R‘𝐴)) → 𝑝 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
149	147, 148	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝑝 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
150		snidg 4617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
151	150	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐵 ∈ {𝐵})
152		elun2 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
153	151, 152	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
154	153	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
155	149, 154	opelxpd 5689	. . . . . . . . . . . . . . 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 28282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 𝐴 ∈ (L‘𝐴)
157		eleq1 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝐴 → (𝑝 ∈ (L‘𝐴) ↔ 𝐴 ∈ (L‘𝐴)))
158	156, 157	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = 𝐴 → ¬ 𝑝 ∈ (L‘𝐴))
159	158	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ (L‘𝐴) → 𝑝 ≠ 𝐴)
160	159	ad2antrl 741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝑝 ≠ 𝐴)
161	160	orcd 887	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
162		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑝 ∈ V
163		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ V ∧ 𝐵 ∈ No) → (⟨𝑝, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
164	162, 163	mpan 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → (⟨𝑝, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
165	164	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → (⟨𝑝, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑝 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
166	161, 165	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → ⟨𝑝, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
167		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑝, 𝐵⟩ ∈ V
168	167	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑝, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑝, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
169	168	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝑝, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑝, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
170	166, 169	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → ¬ ⟨𝑝, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩})
171	155, 170	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝑝 ·s 𝐵) = ( ·s ‘⟨𝑝, 𝐵⟩)
175	172, 173, 174	3eqtr4g 2820	. . . . . . . . . . . 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 4617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
177	176	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐴 ∈ {𝐴})
178		elun2 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
179	177, 178	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
180	179	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
181		elun1 4180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ (L‘𝐵) → 𝑞 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
182	181	ad2antll 742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝑞 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
183		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ((L‘𝐵) ∪ (R‘𝐵)) → 𝑞 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
184	182, 183	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝑞 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
185	180, 184	opelxpd 5689	. . . . . . . . . . . . . . 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 28282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 𝐵 ∈ (L‘𝐵)
187		eleq1 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = 𝐵 → (𝑞 ∈ (L‘𝐵) ↔ 𝐵 ∈ (L‘𝐵)))
188	186, 187	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 𝐵 → ¬ 𝑞 ∈ (L‘𝐵))
189	188	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ (L‘𝐵) → 𝑞 ≠ 𝐵)
190	189	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → 𝑞 ≠ 𝐵)
191	190	olcd 888	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵))
192		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝑞 ∈ V) → (⟨𝐴, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵)))
193	192	elvd 3456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ No → (⟨𝐴, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵)))
194	193	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → (⟨𝐴, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵)))
195	191, 194	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → ⟨𝐴, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩)
196		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝐴, 𝑞⟩ ∈ V
197	196	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐴, 𝑞⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑞⟩ = ⟨𝐴, 𝐵⟩)
198	197	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝐴, 𝑞⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩)
199	195, 198	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → ¬ ⟨𝐴, 𝑞⟩ ∈ {⟨𝐴, 𝐵⟩})
200	185, 199	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝐴 ·s 𝑞) = ( ·s ‘⟨𝐴, 𝑞⟩)
204	201, 202, 203	3eqtr4g 2820	. . . . . . . . . . . 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 7438	. . . . . . . . . . 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 5689	. . . . . . . . . . . . . 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 888	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → (𝑝 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵))
208		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑞 ∈ V
209	162, 208	opthne 5450	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑝, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑝 ≠ 𝐴 ∨ 𝑞 ≠ 𝐵))
210	207, 209	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → ⟨𝑝, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩)
211		opex 5431	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑝, 𝑞⟩ ∈ V
212	211	elsn 4595	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑝, 𝑞⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑝, 𝑞⟩ = ⟨𝐴, 𝐵⟩)
213	212	necon3bbii 3002	. . . . . . . . . . . . . . 15 ⊢ (¬ ⟨𝑝, 𝑞⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑝, 𝑞⟩ ≠ ⟨𝐴, 𝐵⟩)
214	210, 213	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑝 ∈ (L‘𝐴) ∧ 𝑞 ∈ (L‘𝐵))) → ¬ ⟨𝑝, 𝑞⟩ ∈ {⟨𝐴, 𝐵⟩})
215	206, 214	eldifd 4050	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 12 ⊢ (𝑝 ·s 𝑞) = ( ·s ‘⟨𝑝, 𝑞⟩)
219	216, 217, 218	3eqtr4g 2820	. . . . . . . . . . 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 7438	. . . . . . . . . 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 2771	. . . . . . . . 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 3225	. . . . . . . 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 2826	. . . . . . 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 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ (R‘𝐴) → 𝑟 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
225	224	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝑟 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
226		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ((L‘𝐴) ∪ (R‘𝐴)) → 𝑟 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
227	225, 226	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝑟 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
228	153	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
229	227, 228	opelxpd 5689	. . . . . . . . . . . . . . 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 28283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 𝐴 ∈ (R‘𝐴)
231		eleq1 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝐴 → (𝑟 ∈ (R‘𝐴) ↔ 𝐴 ∈ (R‘𝐴)))
232	230, 231	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝐴 → ¬ 𝑟 ∈ (R‘𝐴))
233	232	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ (R‘𝐴) → 𝑟 ≠ 𝐴)
234	233	ad2antrl 741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝑟 ≠ 𝐴)
235	234	orcd 887	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
236		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑟 ∈ V
237		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑟 ∈ V ∧ 𝐵 ∈ No) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
238	236, 237	mpan 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
239	238	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
240	235, 239	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → ⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
241		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑟, 𝐵⟩ ∈ V
242	241	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑟, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑟, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
243	242	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝑟, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
244	240, 243	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → ¬ ⟨𝑟, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩})
245	229, 244	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝑟 ·s 𝐵) = ( ·s ‘⟨𝑟, 𝐵⟩)
249	246, 247, 248	3eqtr4g 2820	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
251		elun2 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ (R‘𝐵) → 𝑠 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
252	251	ad2antll 742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝑠 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
253		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ((L‘𝐵) ∪ (R‘𝐵)) → 𝑠 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
254	252, 253	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝑠 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
255	250, 254	opelxpd 5689	. . . . . . . . . . . . . . 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 28283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 𝐵 ∈ (R‘𝐵)
257		eleq1 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝐵 → (𝑠 ∈ (R‘𝐵) ↔ 𝐵 ∈ (R‘𝐵)))
258	256, 257	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝐵 → ¬ 𝑠 ∈ (R‘𝐵))
259	258	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (R‘𝐵) → 𝑠 ≠ 𝐵)
260	259	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → 𝑠 ≠ 𝐵)
261	260	olcd 888	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵))
262		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝑠 ∈ V) → (⟨𝐴, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵)))
263	262	elvd 3456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ No → (⟨𝐴, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵)))
264	263	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → (⟨𝐴, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵)))
265	261, 264	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → ⟨𝐴, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩)
266		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝐴, 𝑠⟩ ∈ V
267	266	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐴, 𝑠⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑠⟩ = ⟨𝐴, 𝐵⟩)
268	267	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝐴, 𝑠⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩)
269	265, 268	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → ¬ ⟨𝐴, 𝑠⟩ ∈ {⟨𝐴, 𝐵⟩})
270	255, 269	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝐴 ·s 𝑠) = ( ·s ‘⟨𝐴, 𝑠⟩)
274	271, 272, 273	3eqtr4g 2820	. . . . . . . . . . . 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 7438	. . . . . . . . . . 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 5689	. . . . . . . . . . . . . 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 888	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → (𝑟 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵))
278		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑠 ∈ V
279	236, 278	opthne 5450	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑟, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝑠 ≠ 𝐵))
280	277, 279	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → ⟨𝑟, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩)
281		opex 5431	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑟, 𝑠⟩ ∈ V
282	281	elsn 4595	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑟, 𝑠⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑟, 𝑠⟩ = ⟨𝐴, 𝐵⟩)
283	282	necon3bbii 3002	. . . . . . . . . . . . . . 15 ⊢ (¬ ⟨𝑟, 𝑠⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑟, 𝑠⟩ ≠ ⟨𝐴, 𝐵⟩)
284	280, 283	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑟 ∈ (R‘𝐴) ∧ 𝑠 ∈ (R‘𝐵))) → ¬ ⟨𝑟, 𝑠⟩ ∈ {⟨𝐴, 𝐵⟩})
285	276, 284	eldifd 4050	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 12 ⊢ (𝑟 ·s 𝑠) = ( ·s ‘⟨𝑟, 𝑠⟩)
289	286, 287, 288	3eqtr4g 2820	. . . . . . . . . . 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 7438	. . . . . . . . . 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 2771	. . . . . . . . 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 3225	. . . . . . . 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 2826	. . . . . . 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 4180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (L‘𝐴) → 𝑡 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
296	295	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝑡 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
297		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ ((L‘𝐴) ∪ (R‘𝐴)) → 𝑡 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
298	296, 297	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝑡 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
299	153	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
300	298, 299	opelxpd 5689	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝐴 → (𝑡 ∈ (L‘𝐴) ↔ 𝐴 ∈ (L‘𝐴)))
302	156, 301	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝐴 → ¬ 𝑡 ∈ (L‘𝐴))
303	302	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (L‘𝐴) → 𝑡 ≠ 𝐴)
304	303	ad2antrl 741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝑡 ≠ 𝐴)
305	304	orcd 887	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
306		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑡 ∈ V
307		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∈ V ∧ 𝐵 ∈ No) → (⟨𝑡, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
308	306, 307	mpan 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → (⟨𝑡, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
309	308	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → (⟨𝑡, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑡 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
310	305, 309	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → ⟨𝑡, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
311		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑡, 𝐵⟩ ∈ V
312	311	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑡, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑡, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
313	312	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝑡, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑡, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
314	310, 313	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → ¬ ⟨𝑡, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩})
315	300, 314	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝑡 ·s 𝐵) = ( ·s ‘⟨𝑡, 𝐵⟩)
319	316, 317, 318	3eqtr4g 2820	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
321		elun2 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (R‘𝐵) → 𝑢 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
322	321	ad2antll 742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝑢 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
323		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ ((L‘𝐵) ∪ (R‘𝐵)) → 𝑢 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
324	322, 323	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝑢 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
325	320, 324	opelxpd 5689	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝐵 → (𝑢 ∈ (R‘𝐵) ↔ 𝐵 ∈ (R‘𝐵)))
327	256, 326	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝐵 → ¬ 𝑢 ∈ (R‘𝐵))
328	327	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (R‘𝐵) → 𝑢 ≠ 𝐵)
329	328	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → 𝑢 ≠ 𝐵)
330	329	olcd 888	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵))
331		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝑢 ∈ V) → (⟨𝐴, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵)))
332	331	elvd 3456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ No → (⟨𝐴, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵)))
333	332	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → (⟨𝐴, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵)))
334	330, 333	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → ⟨𝐴, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩)
335		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝐴, 𝑢⟩ ∈ V
336	335	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐴, 𝑢⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑢⟩ = ⟨𝐴, 𝐵⟩)
337	336	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝐴, 𝑢⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩)
338	334, 337	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → ¬ ⟨𝐴, 𝑢⟩ ∈ {⟨𝐴, 𝐵⟩})
339	325, 338	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝐴 ·s 𝑢) = ( ·s ‘⟨𝐴, 𝑢⟩)
343	340, 341, 342	3eqtr4g 2820	. . . . . . . . . . . 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 7438	. . . . . . . . . . 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 5689	. . . . . . . . . . . . . 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 888	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → (𝑡 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵))
347		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
348	306, 347	opthne 5450	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑡, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑡 ≠ 𝐴 ∨ 𝑢 ≠ 𝐵))
349	346, 348	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → ⟨𝑡, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩)
350		opex 5431	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑡, 𝑢⟩ ∈ V
351	350	elsn 4595	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑡, 𝑢⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑡, 𝑢⟩ = ⟨𝐴, 𝐵⟩)
352	351	necon3bbii 3002	. . . . . . . . . . . . . . 15 ⊢ (¬ ⟨𝑡, 𝑢⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑡, 𝑢⟩ ≠ ⟨𝐴, 𝐵⟩)
353	349, 352	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑡 ∈ (L‘𝐴) ∧ 𝑢 ∈ (R‘𝐵))) → ¬ ⟨𝑡, 𝑢⟩ ∈ {⟨𝐴, 𝐵⟩})
354	345, 353	eldifd 4050	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 12 ⊢ (𝑡 ·s 𝑢) = ( ·s ‘⟨𝑡, 𝑢⟩)
358	355, 356, 357	3eqtr4g 2820	. . . . . . . . . . 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 7438	. . . . . . . . . 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 2771	. . . . . . . . 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 3225	. . . . . . . 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 2826	. . . . . . 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 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ (R‘𝐴) → 𝑣 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
364	363	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝑣 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
365		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ((L‘𝐴) ∪ (R‘𝐴)) → 𝑣 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
366	364, 365	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝑣 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
367	153	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
368	366, 367	opelxpd 5689	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ (R‘𝐴) ↔ 𝐴 ∈ (R‘𝐴)))
370	230, 369	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐴 → ¬ 𝑣 ∈ (R‘𝐴))
371	370	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ (R‘𝐴) → 𝑣 ≠ 𝐴)
372	371	ad2antrl 741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝑣 ≠ 𝐴)
373	372	orcd 887	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
374		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
375		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ V ∧ 𝐵 ∈ No) → (⟨𝑣, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
376	374, 375	mpan 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → (⟨𝑣, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
377	376	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → (⟨𝑣, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑣 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
378	373, 377	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → ⟨𝑣, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
379		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑣, 𝐵⟩ ∈ V
380	379	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑣, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑣, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
381	380	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝑣, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑣, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
382	378, 381	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → ¬ ⟨𝑣, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩})
383	368, 382	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝑣 ·s 𝐵) = ( ·s ‘⟨𝑣, 𝐵⟩)
387	384, 385, 386	3eqtr4g 2820	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
389		elun1 4180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (L‘𝐵) → 𝑤 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
390	389	ad2antll 742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝑤 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
391		elun1 4180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ((L‘𝐵) ∪ (R‘𝐵)) → 𝑤 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
392	390, 391	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝑤 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
393	388, 392	opelxpd 5689	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝐵 → (𝑤 ∈ (L‘𝐵) ↔ 𝐵 ∈ (L‘𝐵)))
395	186, 394	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝐵 → ¬ 𝑤 ∈ (L‘𝐵))
396	395	necon2ai 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (L‘𝐵) → 𝑤 ≠ 𝐵)
397	396	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → 𝑤 ≠ 𝐵)
398	397	olcd 888	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵))
399		opthneg 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝑤 ∈ V) → (⟨𝐴, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵)))
400	399	elvd 3456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ No → (⟨𝐴, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵)))
401	400	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → (⟨𝐴, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵)))
402	398, 401	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → ⟨𝐴, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩)
403		opex 5431	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝐴, 𝑤⟩ ∈ V
404	403	elsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐴, 𝑤⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
405	404	necon3bbii 3002	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨𝐴, 𝑤⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩)
406	402, 405	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → ¬ ⟨𝐴, 𝑤⟩ ∈ {⟨𝐴, 𝐵⟩})
407	393, 406	eldifd 4050	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝐴 ·s 𝑤) = ( ·s ‘⟨𝐴, 𝑤⟩)
411	408, 409, 410	3eqtr4g 2820	. . . . . . . . . . . 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 7438	. . . . . . . . . . 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 5689	. . . . . . . . . . . . . 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 888	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → (𝑣 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵))
415		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
416	374, 415	opthne 5450	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑣, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑣 ≠ 𝐴 ∨ 𝑤 ≠ 𝐵))
417	414, 416	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → ⟨𝑣, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩)
418		opex 5431	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑣, 𝑤⟩ ∈ V
419	418	elsn 4595	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑣, 𝑤⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑣, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
420	419	necon3bbii 3002	. . . . . . . . . . . . . . 15 ⊢ (¬ ⟨𝑣, 𝑤⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑣, 𝑤⟩ ≠ ⟨𝐴, 𝐵⟩)
421	417, 420	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (𝑣 ∈ (R‘𝐴) ∧ 𝑤 ∈ (L‘𝐵))) → ¬ ⟨𝑣, 𝑤⟩ ∈ {⟨𝐴, 𝐵⟩})
422	413, 421	eldifd 4050	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 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 7423	. . . . . . . . . . . 12 ⊢ (𝑣 ·s 𝑤) = ( ·s ‘⟨𝑣, 𝑤⟩)
426	423, 424, 425	3eqtr4g 2820	. . . . . . . . . . 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 7438	. . . . . . . . . 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 2771	. . . . . . . . 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 3225	. . . . . . . 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 2826	. . . . . . 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 7438	. . . . 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 2799	. . . 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 2799	. . 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 2807	. 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 2795	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))})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∃wrex 3086  Vcvv 3450  ⦋csb 3846   ∖ cdif 4045   ∪ cun 4149  {csn 4584  ⟨cop 4834   × cxp 5648   ↾ cres 5652  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Nocsur 28002   |s ccuts 28150  Lcleft 28216  Rcright 28217   +_s cadds 28350   −_s csubs 28411   ·_s cmuls 28497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-muls 28498

This theorem is used by:  mulsval2  28502  muls01  28503  mulsrid  28504  mulsproplem10  28516  mulscom  28530  mulsuniflem  28540