Theorem addsval 34126

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 34119	. . 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 34114	. 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 5379	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		addsfn 34125	. . . . . 6 ⊢ +s Fn ( No × No )
5		fnfun 6533	. . . . . 6 ⊢ ( +s Fn ( No × No ) → Fun +s )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun +s
7		fvex 6787	. . . . . . . . 9 ⊢ ( L ‘𝐴) ∈ V
8		fvex 6787	. . . . . . . . 9 ⊢ ( R ‘𝐴) ∈ V
9	7, 8	unex 7596	. . . . . . . 8 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10		snex 5354	. . . . . . . 8 ⊢ {𝐴} ∈ V
11	9, 10	unex 7596	. . . . . . 7 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) ∈ V
12		fvex 6787	. . . . . . . . 9 ⊢ ( L ‘𝐵) ∈ V
13		fvex 6787	. . . . . . . . 9 ⊢ ( R ‘𝐵) ∈ V
14	12, 13	unex 7596	. . . . . . . 8 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ∈ V
15		snex 5354	. . . . . . . 8 ⊢ {𝐵} ∈ V
16	14, 15	unex 7596	. . . . . . 7 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}) ∈ V
17	11, 16	xpex 7603	. . . . . 6 ⊢ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∈ V
18	17	difexi 5252	. . . . 5 ⊢ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ∈ V
19		resfunexg 7091	. . . . 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 689	. . . 4 ⊢ ( +s ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) ∈ V
21		2fveq3 6779	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( L ‘(1st ‘𝑥)) = ( L ‘(1st ‘⟨𝐴, 𝐵⟩)))
22		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
23	22	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑙𝑎(2nd ‘𝑥)) = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩)))
24	23	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑦 = (𝑙𝑎(2nd ‘𝑥)) ↔ 𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
25	21, 24	rexeqbidv 3337	. . . . . . . 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 2807	. . . . . . 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 6779	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( L ‘(2nd ‘𝑥)) = ( L ‘(2nd ‘⟨𝐴, 𝐵⟩)))
28		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
29	28	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)𝑎𝑙) = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙))
30	29	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑧 = ((1st ‘𝑥)𝑎𝑙) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)))
31	27, 30	rexeqbidv 3337	. . . . . . . 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 2807	. . . . . . 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 4098	. . . . . 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 6779	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( R ‘(1st ‘𝑥)) = ( R ‘(1st ‘⟨𝐴, 𝐵⟩)))
35	22	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑟𝑎(2nd ‘𝑥)) = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩)))
36	35	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑦 = (𝑟𝑎(2nd ‘𝑥)) ↔ 𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
37	34, 36	rexeqbidv 3337	. . . . . . . 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 2807	. . . . . . 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 6779	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( R ‘(2nd ‘𝑥)) = ( R ‘(2nd ‘⟨𝐴, 𝐵⟩)))
40	28	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)𝑎𝑟) = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟))
41	40	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑧 = ((1st ‘𝑥)𝑎𝑟) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)))
42	39, 41	rexeqbidv 3337	. . . . . . . 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 2807	. . . . . . 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 4098	. . . . . 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 7293	. . . . 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 7281	. . . . . . . . . 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 2749	. . . . . . . . 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 3226	. . . . . . . 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 2807	. . . . . . 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 7281	. . . . . . . . . 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 2749	. . . . . . . . 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 3226	. . . . . . . 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 2807	. . . . . . 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 4098	. . . . . 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 7281	. . . . . . . . . 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 2749	. . . . . . . . 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 3226	. . . . . . . 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 2807	. . . . . . 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 7281	. . . . . . . . . 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 2749	. . . . . . . . 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 3226	. . . . . . . 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 2807	. . . . . . 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 4098	. . . . . 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 7293	. . . . 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 2738	. . . . 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 7308	. . . . 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 7433	. . . 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 689	. . 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 7843	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
70	69	fveq2d 6778	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( L ‘(1st ‘⟨𝐴, 𝐵⟩)) = ( L ‘𝐴))
71	70	eleq2d 2824	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘(1st ‘⟨𝐴, 𝐵⟩)) ↔ 𝑙 ∈ ( L ‘𝐴)))
72		op2ndg 7844	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
73	72	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
74	73	oveq2d 7291	. . . . . . . . . 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 4110	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ( L ‘𝐴) → 𝑙 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
76		elun1 4110	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑙 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
77	75, 76	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ ( L ‘𝐴) → 𝑙 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
78	77	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝑙 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
79		snidg 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
80		elun2 4111	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
81	79, 80	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ No → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
82	81	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
83	82	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
84	78, 83	opelxpd 5627	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → ⟨𝑙, 𝐵⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
85		leftirr 34073	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐴 ∈ ( L ‘𝐴)
86	85	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐴 ∈ ( L ‘𝐴))
87		eleq1 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐴 → (𝑙 ∈ ( L ‘𝐴) ↔ 𝐴 ∈ ( L ‘𝐴)))
88	87	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐴 → (¬ 𝑙 ∈ ( L ‘𝐴) ↔ ¬ 𝐴 ∈ ( L ‘𝐴)))
89	86, 88	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 = 𝐴 → ¬ 𝑙 ∈ ( L ‘𝐴)))
90	89	necon2ad 2958	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘𝐴) → 𝑙 ≠ 𝐴))
91	90	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝑙 ≠ 𝐴)
92	91	orcd 870	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
93		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝑙 ∈ ( L ‘𝐴))
94		simplr 766	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝐵 ∈ No )
95		opthneg 5396	. . . . . . . . . . . . . . 15 ⊢ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝐵 ∈ No ) → (⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
96	93, 94, 95	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → (⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
97	92, 96	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → ⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
98		eldifsn 4720	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 6794	. . . . . . . . . . 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 7278	. . . . . . . . . . 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 7278	. . . . . . . . . . 11 ⊢ (𝑙 +s 𝐵) = ( +s ‘⟨𝑙, 𝐵⟩)
103	100, 101, 102	3eqtr4g 2803	. . . . . . . . . 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 2778	. . . . . . . . 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 2749	. . . . . . . 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 592	. . . . . . 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 3355	. . . . . 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 2807	. . . . 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 6778	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( L ‘(2nd ‘⟨𝐴, 𝐵⟩)) = ( L ‘𝐵))
110	109	eleq2d 2824	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑙 ∈ ( L ‘𝐵)))
111	69	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
112	111	oveq1d 7290	. . . . . . . . . 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 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
114	113	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ {𝐴})
115		elun2 4111	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
116	114, 115	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
117	116	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
118		elun1 4110	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ( L ‘𝐵) → 𝑙 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
119		elun1 4110	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑙 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
120	118, 119	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ ( L ‘𝐵) → 𝑙 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
121	120	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → 𝑙 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
122	117, 121	opelxpd 5627	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → ⟨𝐴, 𝑙⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
123		leftirr 34073	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐵 ∈ ( L ‘𝐵)
124	123	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐵 ∈ ( L ‘𝐵))
125		eleq1 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐵 → (𝑙 ∈ ( L ‘𝐵) ↔ 𝐵 ∈ ( L ‘𝐵)))
126	125	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐵 → (¬ 𝑙 ∈ ( L ‘𝐵) ↔ ¬ 𝐵 ∈ ( L ‘𝐵)))
127	124, 126	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 = 𝐵 → ¬ 𝑙 ∈ ( L ‘𝐵)))
128	127	necon2ad 2958	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘𝐵) → 𝑙 ≠ 𝐵))
129	128	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → 𝑙 ≠ 𝐵)
130	129	olcd 871	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵))
131		opthneg 5396	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑙 ∈ ( L ‘𝐵)) → (⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵)))
132	131	adantlr 712	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → (⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵)))
133	130, 132	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → ⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩)
134		eldifsn 4720	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 6794	. . . . . . . . . . 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 7278	. . . . . . . . . . 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 7278	. . . . . . . . . . 11 ⊢ (𝐴 +s 𝑙) = ( +s ‘⟨𝐴, 𝑙⟩)
139	136, 137, 138	3eqtr4g 2803	. . . . . . . . . 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 2778	. . . . . . . . 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 2749	. . . . . . . 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 592	. . . . . . 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 3355	. . . . . 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 2807	. . . . 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 4098	. . . 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 6778	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( R ‘(1st ‘⟨𝐴, 𝐵⟩)) = ( R ‘𝐴))
147	146	eleq2d 2824	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘(1st ‘⟨𝐴, 𝐵⟩)) ↔ 𝑟 ∈ ( R ‘𝐴)))
148	72	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
149	148	oveq2d 7291	. . . . . . . . . 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 4111	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ( R ‘𝐴) → 𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
151		elun1 4110	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
152	150, 151	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ ( R ‘𝐴) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
153	152	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
154	82	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
155	153, 154	opelxpd 5627	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → ⟨𝑟, 𝐵⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
156		rightirr 34074	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐴 ∈ ( R ‘𝐴)
157	156	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐴 ∈ ( R ‘𝐴))
158		eleq1 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝐴 → (𝑟 ∈ ( R ‘𝐴) ↔ 𝐴 ∈ ( R ‘𝐴)))
159	158	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝐴 → (¬ 𝑟 ∈ ( R ‘𝐴) ↔ ¬ 𝐴 ∈ ( R ‘𝐴)))
160	157, 159	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 = 𝐴 → ¬ 𝑟 ∈ ( R ‘𝐴)))
161	160	necon2ad 2958	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘𝐴) → 𝑟 ≠ 𝐴))
162	161	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝑟 ≠ 𝐴)
163	162	orcd 870	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
164		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝑟 ∈ ( R ‘𝐴))
165		simplr 766	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝐵 ∈ No )
166		opthneg 5396	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝐵 ∈ No ) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
167	164, 165, 166	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
168	163, 167	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → ⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
169		eldifsn 4720	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 6794	. . . . . . . . . . 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 7278	. . . . . . . . . . 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 7278	. . . . . . . . . . 11 ⊢ (𝑟 +s 𝐵) = ( +s ‘⟨𝑟, 𝐵⟩)
174	171, 172, 173	3eqtr4g 2803	. . . . . . . . . 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 2778	. . . . . . . . 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 2749	. . . . . . . 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 592	. . . . . . 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 3355	. . . . . 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 2807	. . . . 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 6778	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( R ‘(2nd ‘⟨𝐴, 𝐵⟩)) = ( R ‘𝐵))
181	180	eleq2d 2824	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑟 ∈ ( R ‘𝐵)))
182	69	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
183	182	oveq1d 7290	. . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝐴 ∈ {𝐴})
185	184, 115	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
186		elun2 4111	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ( R ‘𝐵) → 𝑟 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
187	186	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝑟 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
188		elun1 4110	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑟 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
189	187, 188	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝑟 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
190	185, 189	opelxpd 5627	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → ⟨𝐴, 𝑟⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
191		rightirr 34074	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝐵 ∈ ( R ‘𝐵)
192	191	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐵 ∈ ( R ‘𝐵))
193		eleq1 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝐵 → (𝑟 ∈ ( R ‘𝐵) ↔ 𝐵 ∈ ( R ‘𝐵)))
194	193	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝐵 → (¬ 𝑟 ∈ ( R ‘𝐵) ↔ ¬ 𝐵 ∈ ( R ‘𝐵)))
195	192, 194	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 = 𝐵 → ¬ 𝑟 ∈ ( R ‘𝐵)))
196	195	necon2ad 2958	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘𝐵) → 𝑟 ≠ 𝐵))
197	196	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝑟 ≠ 𝐵)
198	197	olcd 871	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵))
199		opthneg 5396	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑟 ∈ ( R ‘𝐵)) → (⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵)))
200	199	adantlr 712	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → (⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵)))
201	198, 200	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → ⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩)
202		eldifsn 4720	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 6794	. . . . . . . . . . 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 7278	. . . . . . . . . . 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 7278	. . . . . . . . . . 11 ⊢ (𝐴 +s 𝑟) = ( +s ‘⟨𝐴, 𝑟⟩)
207	204, 205, 206	3eqtr4g 2803	. . . . . . . . . 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 2778	. . . . . . . . 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 2749	. . . . . . . 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 592	. . . . . . 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 3355	. . . . . 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 2807	. . . . 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 4098	. . . 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 7293	. . 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 2790	. 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 2778	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)})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885  {csn 4561  ⟨cop 4567   × cxp 5587   ↾ cres 5591  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830   No csur 33843   |s cscut 33977   L cleft 34029   R cright 34030   +s cadds 34116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-made 34031  df-old 34032  df-left 34034  df-right 34035  df-norec2 34106  df-adds 34119

This theorem is referenced by:  addsid1  34127  addscom  34129  addscllem1  34131