mulsval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mulsval		Structured version Visualization version GIF version

Theorem mulsval 28136

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

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 {cab 2713 ≠ wne 2939 ∃wrex 3069 Vcvv 3479 ⦋csb 3898 ∖ cdif 3947 ∪ cun 3948 {csn 4625 ⟨cop 4631 × cxp 5682 ↾ cres 5686 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1^st c1st 8013 2^nd c2nd 8014 No csur 27685 |s cscut 27828 L cleft 27885 R cright 27886 +_s cadds 27993 -_s csubs 28053 ·_s cmuls 28133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-1o 8507 df-2o 8508 df-no 27688 df-slt 27689 df-bday 27690 df-sslt 27827 df-scut 27829 df-made 27887 df-old 27888 df-left 27890 df-right 27891 df-norec2 27983 df-muls 28134

This theorem is referenced by: mulsval2 28138 muls01 28139 mulsrid 28140 mulsproplem10 28152 mulscom 28166 mulsuniflem 28176

W3C validator