Theorem addsval 28348

Description: The value of surreal addition. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)

Assertion

Ref	Expression
addsval	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)})))

Distinct variable groups:   𝐴,𝑙,𝑟,𝑦,𝑧   𝐵,𝑙,𝑟,𝑦,𝑧

Proof of Theorem addsval

Dummy variables 𝑎 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-adds 28346	. . 3 ⊢ +s = norec2 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}))))
2	1	norec2ov 28343	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 +s 𝐵) = (⟨𝐴, 𝐵⟩(𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})))( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
3		opex 5431	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		addsfn 28347	. . . . . 6 ⊢ +s Fn (No × No)
5		fnfun 6639	. . . . . 6 ⊢ ( +s Fn (No × No) → Fun +s )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun +s
7		fvex 6898	. . . . . . . . 9 ⊢ (L‘𝐴) ∈ V
8		fvex 6898	. . . . . . . . 9 ⊢ (R‘𝐴) ∈ V
9	7, 8	unex 7761	. . . . . . . 8 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) ∈ V
10		snex 5396	. . . . . . . 8 ⊢ {𝐴} ∈ V
11	9, 10	unex 7761	. . . . . . 7 ⊢ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) ∈ V
12		fvex 6898	. . . . . . . . 9 ⊢ (L‘𝐵) ∈ V
13		fvex 6898	. . . . . . . . 9 ⊢ (R‘𝐵) ∈ V
14	12, 13	unex 7761	. . . . . . . 8 ⊢ ((L‘𝐵) ∪ (R‘𝐵)) ∈ V
15		snex 5396	. . . . . . . 8 ⊢ {𝐵} ∈ V
16	14, 15	unex 7761	. . . . . . 7 ⊢ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}) ∈ V
17	11, 16	xpex 7767	. . . . . 6 ⊢ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∈ V
18	17	difexi 5291	. . . . 5 ⊢ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ∈ V
19		resfunexg 7221	. . . . 5 ⊢ ((Fun +s ∧ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ∈ V) → ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) ∈ V)
20	6, 18, 19	mp2an 705	. . . 4 ⊢ ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) ∈ V
21		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (L‘(1st ‘𝑥)) = (L‘(1st ‘⟨𝐴, 𝐵⟩)))
22		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
23	22	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑙𝑎(2nd ‘𝑥)) = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩)))
24	23	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑦 = (𝑙𝑎(2nd ‘𝑥)) ↔ 𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
25	21, 24	rexeqbidv 3335	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥)) ↔ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
26	25	abbidv 2826	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → {𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} = {𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))})
27		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (L‘(2nd ‘𝑥)) = (L‘(2nd ‘⟨𝐴, 𝐵⟩)))
28		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
29	28	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)𝑎𝑙) = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙))
30	29	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑧 = ((1st ‘𝑥)𝑎𝑙) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)))
31	27, 30	rexeqbidv 3335	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙) ↔ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)))
32	31	abbidv 2826	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)} = {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)})
33	26, 32	uneq12d 4168	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) = ({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)}))
34		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (R‘(1st ‘𝑥)) = (R‘(1st ‘⟨𝐴, 𝐵⟩)))
35	22	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑟𝑎(2nd ‘𝑥)) = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩)))
36	35	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑦 = (𝑟𝑎(2nd ‘𝑥)) ↔ 𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
37	34, 36	rexeqbidv 3335	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥)) ↔ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
38	37	abbidv 2826	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → {𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} = {𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))})
39		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (R‘(2nd ‘𝑥)) = (R‘(2nd ‘⟨𝐴, 𝐵⟩)))
40	28	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)𝑎𝑟) = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟))
41	40	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑧 = ((1st ‘𝑥)𝑎𝑟) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)))
42	39, 41	rexeqbidv 3335	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟) ↔ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)))
43	42	abbidv 2826	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)} = {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)})
44	38, 43	uneq12d 4168	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}) = ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)}))
45	33, 44	oveq12d 7438	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})) = (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)})))
46		oveq 7426	. . . . . . . . . 10 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩)) = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)))
47	46	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))))
48	47	rexbidv 3186	. . . . . . . 8 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩)) ↔ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))))
49	48	abbidv 2826	. . . . . . 7 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → {𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))} = {𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))})
50		oveq 7426	. . . . . . . . . 10 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙) = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙))
51	50	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)))
52	51	rexbidv 3186	. . . . . . . 8 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙) ↔ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)))
53	52	abbidv 2826	. . . . . . 7 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)} = {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)})
54	49, 53	uneq12d 4168	. . . . . 6 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → ({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)}) = ({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}))
55		oveq 7426	. . . . . . . . . 10 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩)) = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)))
56	55	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))))
57	56	rexbidv 3186	. . . . . . . 8 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩)) ↔ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))))
58	57	abbidv 2826	. . . . . . 7 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → {𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))} = {𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))})
59		oveq 7426	. . . . . . . . . 10 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟) = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟))
60	59	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)))
61	60	rexbidv 3186	. . . . . . . 8 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟) ↔ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)))
62	61	abbidv 2826	. . . . . . 7 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)} = {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)})
63	58, 62	uneq12d 4168	. . . . . 6 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)}) = ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)}))
64	54, 63	oveq12d 7438	. . . . 5 ⊢ (𝑎 = ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) → (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)})) = (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)})))
65		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}))) = (𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})))
66		ovex 7453	. . . . 5 ⊢ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)})) ∈ V
67	45, 64, 65, 66	ovmpo 7580	. . . 4 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ ( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) ∈ V) → (⟨𝐴, 𝐵⟩(𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})))( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))) = (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)})))
68	3, 20, 67	mp2an 705	. . 3 ⊢ (⟨𝐴, 𝐵⟩(𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})))( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))) = (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)}))
69		op1stg 8013	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
70	69	fveq2d 6889	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (L‘(1st ‘⟨𝐴, 𝐵⟩)) = (L‘𝐴))
71	70	eleq2d 2846	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩)) ↔ 𝑙 ∈ (L‘𝐴)))
72		op2ndg 8014	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
73	72	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
74	73	oveq2d 7436	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝐵))
75		elun1 4180	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (L‘𝐴) → 𝑙 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
76		elun1 4180	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ((L‘𝐴) ∪ (R‘𝐴)) → 𝑙 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
77	75, 76	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (L‘𝐴) → 𝑙 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
78	77	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → 𝑙 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
79		snidg 4620	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
80		elun2 4181	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
81	79, 80	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ No → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
82	81	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
83	82	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
84	78, 83	opelxpd 5689	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → ⟨𝑙, 𝐵⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})))
85		leftirr 28277	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐴 ∈ (L‘𝐴)
86	85	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ¬ 𝐴 ∈ (L‘𝐴))
87		eleq1 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐴 → (𝑙 ∈ (L‘𝐴) ↔ 𝐴 ∈ (L‘𝐴)))
88	87	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐴 → (¬ 𝑙 ∈ (L‘𝐴) ↔ ¬ 𝐴 ∈ (L‘𝐴)))
89	86, 88	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑙 = 𝐴 → ¬ 𝑙 ∈ (L‘𝐴)))
90	89	necon2ad 2970	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑙 ∈ (L‘𝐴) → 𝑙 ≠ 𝐴))
91	90	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → 𝑙 ≠ 𝐴)
92	91	orcd 887	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
93		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → 𝑙 ∈ (L‘𝐴))
94		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → 𝐵 ∈ No)
95		opthneg 5449	. . . . . . . . . . . . . . 15 ⊢ ((𝑙 ∈ (L‘𝐴) ∧ 𝐵 ∈ No) → (⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
96	93, 94, 95	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
97	92, 96	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → ⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
98		eldifsn 4747	. . . . . . . . . . . . 13 ⊢ (⟨𝑙, 𝐵⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ↔ (⟨𝑙, 𝐵⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∧ ⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩))
99	84, 97, 98	sylanbrc 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → ⟨𝑙, 𝐵⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
100	99	fvresd 6905	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))‘⟨𝑙, 𝐵⟩) = ( +s ‘⟨𝑙, 𝐵⟩))
101		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝑙( +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⟩}))‘⟨𝑙, 𝐵⟩)
102		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝑙 +s 𝐵) = ( +s ‘⟨𝑙, 𝐵⟩)
103	100, 101, 102	3eqtr4g 2820	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝐵) = (𝑙 +s 𝐵))
104	74, 103	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) = (𝑙 +s 𝐵))
105	104	eqeq2d 2771	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐴)) → (𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑦 = (𝑙 +s 𝐵)))
106	71, 105	sylbida 604	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))) → (𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑦 = (𝑙 +s 𝐵)))
107	70, 106	rexeqbidva 3326	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) ↔ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)))
108	107	abbidv 2826	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → {𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} = {𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)})
109	72	fveq2d 6889	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (L‘(2nd ‘⟨𝐴, 𝐵⟩)) = (L‘𝐵))
110	109	eleq2d 2846	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑙 ∈ (L‘𝐵)))
111	69	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
112	111	oveq1d 7435	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) = (𝐴( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙))
113		snidg 4620	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
114	113	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐴 ∈ {𝐴})
115		elun2 4181	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
116	114, 115	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
117	116	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
118		elun1 4180	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (L‘𝐵) → 𝑙 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
119		elun1 4180	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ((L‘𝐵) ∪ (R‘𝐵)) → 𝑙 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
120	118, 119	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (L‘𝐵) → 𝑙 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
121	120	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → 𝑙 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
122	117, 121	opelxpd 5689	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → ⟨𝐴, 𝑙⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})))
123		leftirr 28277	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐵 ∈ (L‘𝐵)
124	123	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ¬ 𝐵 ∈ (L‘𝐵))
125		eleq1 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐵 → (𝑙 ∈ (L‘𝐵) ↔ 𝐵 ∈ (L‘𝐵)))
126	125	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐵 → (¬ 𝑙 ∈ (L‘𝐵) ↔ ¬ 𝐵 ∈ (L‘𝐵)))
127	124, 126	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑙 = 𝐵 → ¬ 𝑙 ∈ (L‘𝐵)))
128	127	necon2ad 2970	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑙 ∈ (L‘𝐵) → 𝑙 ≠ 𝐵))
129	128	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → 𝑙 ≠ 𝐵)
130	129	olcd 888	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵))
131		opthneg 5449	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑙 ∈ (L‘𝐵)) → (⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵)))
132	131	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → (⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵)))
133	130, 132	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → ⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩)
134		eldifsn 4747	. . . . . . . . . . . . 13 ⊢ (⟨𝐴, 𝑙⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ↔ (⟨𝐴, 𝑙⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∧ ⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩))
135	122, 133, 134	sylanbrc 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → ⟨𝐴, 𝑙⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
136	135	fvresd 6905	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → (( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))‘⟨𝐴, 𝑙⟩) = ( +s ‘⟨𝐴, 𝑙⟩))
137		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝐴( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) = (( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))‘⟨𝐴, 𝑙⟩)
138		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝐴 +s 𝑙) = ( +s ‘⟨𝐴, 𝑙⟩)
139	136, 137, 138	3eqtr4g 2820	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → (𝐴( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) = (𝐴 +s 𝑙))
140	112, 139	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) = (𝐴 +s 𝑙))
141	140	eqeq2d 2771	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘𝐵)) → (𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) ↔ 𝑧 = (𝐴 +s 𝑙)))
142	110, 141	sylbida 604	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))) → (𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) ↔ 𝑧 = (𝐴 +s 𝑙)))
143	109, 142	rexeqbidva 3326	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙) ↔ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)))
144	143	abbidv 2826	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)} = {𝑧 ∣ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)})
145	108, 144	uneq12d 4168	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}) = ({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)}))
146	69	fveq2d 6889	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (R‘(1st ‘⟨𝐴, 𝐵⟩)) = (R‘𝐴))
147	146	eleq2d 2846	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩)) ↔ 𝑟 ∈ (R‘𝐴)))
148	72	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
149	148	oveq2d 7436	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝐵))
150		elun2 4181	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (R‘𝐴) → 𝑟 ∈ ((L‘𝐴) ∪ (R‘𝐴)))
151		elun1 4180	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ((L‘𝐴) ∪ (R‘𝐴)) → 𝑟 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
152	150, 151	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (R‘𝐴) → 𝑟 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
153	152	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → 𝑟 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
154	82	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → 𝐵 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
155	153, 154	opelxpd 5689	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → ⟨𝑟, 𝐵⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})))
156		rightirr 28278	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐴 ∈ (R‘𝐴)
157	156	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ¬ 𝐴 ∈ (R‘𝐴))
158		eleq1 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝐴 → (𝑟 ∈ (R‘𝐴) ↔ 𝐴 ∈ (R‘𝐴)))
159	158	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝐴 → (¬ 𝑟 ∈ (R‘𝐴) ↔ ¬ 𝐴 ∈ (R‘𝐴)))
160	157, 159	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑟 = 𝐴 → ¬ 𝑟 ∈ (R‘𝐴)))
161	160	necon2ad 2970	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑟 ∈ (R‘𝐴) → 𝑟 ≠ 𝐴))
162	161	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → 𝑟 ≠ 𝐴)
163	162	orcd 887	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
164		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → 𝑟 ∈ (R‘𝐴))
165		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → 𝐵 ∈ No)
166		opthneg 5449	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ (R‘𝐴) ∧ 𝐵 ∈ No) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
167	164, 165, 166	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
168	163, 167	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → ⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
169		eldifsn 4747	. . . . . . . . . . . . 13 ⊢ (⟨𝑟, 𝐵⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ↔ (⟨𝑟, 𝐵⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∧ ⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩))
170	155, 168, 169	sylanbrc 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → ⟨𝑟, 𝐵⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
171	170	fvresd 6905	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))‘⟨𝑟, 𝐵⟩) = ( +s ‘⟨𝑟, 𝐵⟩))
172		df-ov 7423	. . . . . . . . . . 11 ⊢ (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⟩)
173		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝑟 +s 𝐵) = ( +s ‘⟨𝑟, 𝐵⟩)
174	171, 172, 173	3eqtr4g 2820	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝐵) = (𝑟 +s 𝐵))
175	149, 174	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) = (𝑟 +s 𝐵))
176	175	eqeq2d 2771	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐴)) → (𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑦 = (𝑟 +s 𝐵)))
177	147, 176	sylbida 604	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))) → (𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑦 = (𝑟 +s 𝐵)))
178	146, 177	rexeqbidva 3326	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩)) ↔ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)))
179	178	abbidv 2826	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → {𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} = {𝑦 ∣ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)})
180	72	fveq2d 6889	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (R‘(2nd ‘⟨𝐴, 𝐵⟩)) = (R‘𝐵))
181	180	eleq2d 2846	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑟 ∈ (R‘𝐵)))
182	69	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
183	182	oveq1d 7435	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) = (𝐴( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟))
184	114	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → 𝐴 ∈ {𝐴})
185	184, 115	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → 𝐴 ∈ (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
186		elun2 4181	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (R‘𝐵) → 𝑟 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
187	186	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → 𝑟 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
188		elun1 4180	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ ((L‘𝐵) ∪ (R‘𝐵)) → 𝑟 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
189	187, 188	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → 𝑟 ∈ (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
190	185, 189	opelxpd 5689	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → ⟨𝐴, 𝑟⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})))
191		rightirr 28278	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐵 ∈ (R‘𝐵)
192	191	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ¬ 𝐵 ∈ (R‘𝐵))
193		eleq1 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝐵 → (𝑟 ∈ (R‘𝐵) ↔ 𝐵 ∈ (R‘𝐵)))
194	193	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝐵 → (¬ 𝑟 ∈ (R‘𝐵) ↔ ¬ 𝐵 ∈ (R‘𝐵)))
195	192, 194	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑟 = 𝐵 → ¬ 𝑟 ∈ (R‘𝐵)))
196	195	necon2ad 2970	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝑟 ∈ (R‘𝐵) → 𝑟 ≠ 𝐵))
197	196	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → 𝑟 ≠ 𝐵)
198	197	olcd 888	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵))
199		opthneg 5449	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑟 ∈ (R‘𝐵)) → (⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵)))
200	199	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → (⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵)))
201	198, 200	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → ⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩)
202		eldifsn 4747	. . . . . . . . . . . . 13 ⊢ (⟨𝐴, 𝑟⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ↔ (⟨𝐴, 𝑟⟩ ∈ ((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∧ ⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩))
203	190, 201, 202	sylanbrc 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → ⟨𝐴, 𝑟⟩ ∈ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
204	203	fvresd 6905	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → (( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))‘⟨𝐴, 𝑟⟩) = ( +s ‘⟨𝐴, 𝑟⟩))
205		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝐴( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) = (( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))‘⟨𝐴, 𝑟⟩)
206		df-ov 7423	. . . . . . . . . . 11 ⊢ (𝐴 +s 𝑟) = ( +s ‘⟨𝐴, 𝑟⟩)
207	204, 205, 206	3eqtr4g 2820	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → (𝐴( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) = (𝐴 +s 𝑟))
208	183, 207	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) = (𝐴 +s 𝑟))
209	208	eqeq2d 2771	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘𝐵)) → (𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) ↔ 𝑧 = (𝐴 +s 𝑟)))
210	181, 209	sylbida 604	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))) → (𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) ↔ 𝑧 = (𝐴 +s 𝑟)))
211	180, 210	rexeqbidva 3326	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟) ↔ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)))
212	211	abbidv 2826	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)} = {𝑧 ∣ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)})
213	179, 212	uneq12d 4168	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)}) = ({𝑦 ∣ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)}))
214	145, 213	oveq12d 7438	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑙( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘⟨𝐴, 𝐵⟩))𝑦 = (𝑟( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))(2nd ‘⟨𝐴, 𝐵⟩))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘⟨𝐴, 𝐵⟩))𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))𝑟)})) = (({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)})))
215	68, 214	eqtrid 2807	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (⟨𝐴, 𝐵⟩(𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ (L‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})))( +s ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))) = (({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)})))
216	2, 215	eqtrd 2795	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ (R‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ (R‘𝐵)𝑧 = (𝐴 +s 𝑟)})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∃wrex 3086  Vcvv 3450   ∖ cdif 4045   ∪ cun 4149  {csn 4583  ⟨cop 4589   × cxp 5648   ↾ cres 5652  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Nocsur 27997   |s ccuts 28145  Lcleft 28211  Rcright 28212   +_s cadds 28345

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec2 28335  df-adds 28346

This theorem is used by:  addsval2  28349  addsrid  28350  addscom  28352