Theorem addsov 33709

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

Assertion

Ref	Expression
addsov	⊢ ((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 addsov

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

Step	Hyp	Ref	Expression
1		df-adds 33702	. . 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 33696	. 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 5328	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		addsfn 33708	. . . . . 6 ⊢ +s Fn ( No × No )
5		fnfun 6439	. . . . . 6 ⊢ ( +s Fn ( No × No ) → Fun +s )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun +s
7		fvex 6676	. . . . . . . . 9 ⊢ ( L ‘𝐴) ∈ V
8		fvex 6676	. . . . . . . . 9 ⊢ ( R ‘𝐴) ∈ V
9	7, 8	unex 7473	. . . . . . . 8 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10		snex 5304	. . . . . . . 8 ⊢ {𝐴} ∈ V
11	9, 10	unex 7473	. . . . . . 7 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) ∈ V
12		fvex 6676	. . . . . . . . 9 ⊢ ( L ‘𝐵) ∈ V
13		fvex 6676	. . . . . . . . 9 ⊢ ( R ‘𝐵) ∈ V
14	12, 13	unex 7473	. . . . . . . 8 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ∈ V
15		snex 5304	. . . . . . . 8 ⊢ {𝐵} ∈ V
16	14, 15	unex 7473	. . . . . . 7 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}) ∈ V
17	11, 16	xpex 7480	. . . . . 6 ⊢ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∈ V
18	17	difexi 5202	. . . . 5 ⊢ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) ∈ V
19		resfunexg 6975	. . . . 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 691	. . . 4 ⊢ ( +s ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) ∈ V
21		2fveq3 6668	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( L ‘(1st ‘𝑥)) = ( L ‘(1st ‘⟨𝐴, 𝐵⟩)))
22		fveq2 6663	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
23	22	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑙𝑎(2nd ‘𝑥)) = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩)))
24	23	eqeq2d 2769	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑦 = (𝑙𝑎(2nd ‘𝑥)) ↔ 𝑦 = (𝑙𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
25	21, 24	rexeqbidv 3320	. . . . . . . 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 2822	. . . . . . 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 6668	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( L ‘(2nd ‘𝑥)) = ( L ‘(2nd ‘⟨𝐴, 𝐵⟩)))
28		fveq2 6663	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
29	28	oveq1d 7171	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)𝑎𝑙) = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙))
30	29	eqeq2d 2769	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑧 = ((1st ‘𝑥)𝑎𝑙) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑙)))
31	27, 30	rexeqbidv 3320	. . . . . . . 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 2822	. . . . . . 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 4071	. . . . . 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 6668	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( R ‘(1st ‘𝑥)) = ( R ‘(1st ‘⟨𝐴, 𝐵⟩)))
35	22	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑟𝑎(2nd ‘𝑥)) = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩)))
36	35	eqeq2d 2769	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑦 = (𝑟𝑎(2nd ‘𝑥)) ↔ 𝑦 = (𝑟𝑎(2nd ‘⟨𝐴, 𝐵⟩))))
37	34, 36	rexeqbidv 3320	. . . . . . . 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 2822	. . . . . . 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 6668	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ( R ‘(2nd ‘𝑥)) = ( R ‘(2nd ‘⟨𝐴, 𝐵⟩)))
40	28	oveq1d 7171	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)𝑎𝑟) = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟))
41	40	eqeq2d 2769	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑧 = ((1st ‘𝑥)𝑎𝑟) ↔ 𝑧 = ((1st ‘⟨𝐴, 𝐵⟩)𝑎𝑟)))
42	39, 41	rexeqbidv 3320	. . . . . . . 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 2822	. . . . . . 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 4071	. . . . . 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 7174	. . . . 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 7162	. . . . . . . . . 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 2769	. . . . . . . . 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 3221	. . . . . . . 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 2822	. . . . . . 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 7162	. . . . . . . . . 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 2769	. . . . . . . . 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 3221	. . . . . . . 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 2822	. . . . . . 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 4071	. . . . . 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 7162	. . . . . . . . . 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 2769	. . . . . . . . 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 3221	. . . . . . . 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 2822	. . . . . . 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 7162	. . . . . . . . . 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 2769	. . . . . . . . 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 3221	. . . . . . . 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 2822	. . . . . . 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 4071	. . . . . 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 7174	. . . . 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 2758	. . . . 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 7189	. . . . 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 7311	. . . 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 691	. . 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 7711	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
70	69	fveq2d 6667	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( L ‘(1st ‘⟨𝐴, 𝐵⟩)) = ( L ‘𝐴))
71	70	eleq2d 2837	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘(1st ‘⟨𝐴, 𝐵⟩)) ↔ 𝑙 ∈ ( L ‘𝐴)))
72		op2ndg 7712	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
73	72	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
74	73	oveq2d 7172	. . . . . . . . . . . 12 ⊢ (((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 4083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ( L ‘𝐴) → 𝑙 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
76		elun1 4083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑙 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
77	75, 76	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 ∈ ( L ‘𝐴) → 𝑙 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
78	77	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝑙 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
79		snidg 4559	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
80		elun2 4084	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
81	79, 80	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ No → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
82	81	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
83	82	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
84	78, 83	opelxpd 5566	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → ⟨𝑙, 𝐵⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
85		leftirr 33664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ No → ¬ 𝐴 ∈ ( L ‘𝐴))
86	85	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐴 ∈ ( L ‘𝐴))
87		eleq1 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝐴 → (𝑙 ∈ ( L ‘𝐴) ↔ 𝐴 ∈ ( L ‘𝐴)))
88	87	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝐴 → (¬ 𝑙 ∈ ( L ‘𝐴) ↔ ¬ 𝐴 ∈ ( L ‘𝐴)))
89	86, 88	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 = 𝐴 → ¬ 𝑙 ∈ ( L ‘𝐴)))
90	89	necon2ad 2966	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘𝐴) → 𝑙 ≠ 𝐴))
91	90	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝑙 ≠ 𝐴)
92	91	orcd 870	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
93		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝑙 ∈ ( L ‘𝐴))
94		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → 𝐵 ∈ No )
95		opthneg 5345	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝐵 ∈ No ) → (⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
96	93, 94, 95	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → (⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑙 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
97	92, 96	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → ⟨𝑙, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
98		eldifsn 4680	. . . . . . . . . . . . . . 15 ⊢ (⟨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 586	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐴)) → ⟨𝑙, 𝐵⟩ ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
100	99	fvresd 6683	. . . . . . . . . . . . 13 ⊢ (((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 7159	. . . . . . . . . . . . 13 ⊢ (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 7159	. . . . . . . . . . . . 13 ⊢ (𝑙 +s 𝐵) = ( +s ‘⟨𝑙, 𝐵⟩)
103	100, 101, 102	3eqtr4g 2818	. . . . . . . . . . . 12 ⊢ (((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 2793	. . . . . . . . . . 11 ⊢ (((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 2769	. . . . . . . . . 10 ⊢ (((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	105	ex 416	. . . . . . . . 9 ⊢ ((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))))
107	71, 106	sylbid 243	. . . . . . . 8 ⊢ ((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))))
108	107	imp 410	. . . . . . 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)))
109	70, 108	rexeqbidva 3336	. . . . . 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)))
110	109	abbidv 2822	. . . . 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)})
111	72	fveq2d 6667	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( L ‘(2nd ‘⟨𝐴, 𝐵⟩)) = ( L ‘𝐵))
112	111	eleq2d 2837	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑙 ∈ ( L ‘𝐵)))
113	69	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
114	113	oveq1d 7171	. . . . . . . . . . . 12 ⊢ (((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))
115		snidg 4559	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
116	115	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ {𝐴})
117		elun2 4084	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
118	116, 117	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
119	118	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
120		elun1 4083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ( L ‘𝐵) → 𝑙 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
121		elun1 4083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑙 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
122	120, 121	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 ∈ ( L ‘𝐵) → 𝑙 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
123	122	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → 𝑙 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
124	119, 123	opelxpd 5566	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → ⟨𝐴, 𝑙⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
125		leftirr 33664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ No → ¬ 𝐵 ∈ ( L ‘𝐵))
126	125	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐵 ∈ ( L ‘𝐵))
127		eleq1 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝐵 → (𝑙 ∈ ( L ‘𝐵) ↔ 𝐵 ∈ ( L ‘𝐵)))
128	127	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝐵 → (¬ 𝑙 ∈ ( L ‘𝐵) ↔ ¬ 𝐵 ∈ ( L ‘𝐵)))
129	126, 128	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 = 𝐵 → ¬ 𝑙 ∈ ( L ‘𝐵)))
130	129	necon2ad 2966	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑙 ∈ ( L ‘𝐵) → 𝑙 ≠ 𝐵))
131	130	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → 𝑙 ≠ 𝐵)
132	131	olcd 871	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵))
133		opthneg 5345	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑙 ∈ ( L ‘𝐵)) → (⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵)))
134	133	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → (⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑙 ≠ 𝐵)))
135	132, 134	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → ⟨𝐴, 𝑙⟩ ≠ ⟨𝐴, 𝐵⟩)
136		eldifsn 4680	. . . . . . . . . . . . . . 15 ⊢ (⟨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⟩))
137	124, 135, 136	sylanbrc 586	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑙 ∈ ( L ‘𝐵)) → ⟨𝐴, 𝑙⟩ ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
138	137	fvresd 6683	. . . . . . . . . . . . 13 ⊢ (((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⟩))
139		df-ov 7159	. . . . . . . . . . . . 13 ⊢ (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⟩)
140		df-ov 7159	. . . . . . . . . . . . 13 ⊢ (𝐴 +s 𝑙) = ( +s ‘⟨𝐴, 𝑙⟩)
141	138, 139, 140	3eqtr4g 2818	. . . . . . . . . . . 12 ⊢ (((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))
142	114, 141	eqtrd 2793	. . . . . . . . . . 11 ⊢ (((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))
143	142	eqeq2d 2769	. . . . . . . . . 10 ⊢ (((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)))
144	143	ex 416	. . . . . . . . 9 ⊢ ((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))))
145	112, 144	sylbid 243	. . . . . . . 8 ⊢ ((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))))
146	145	imp 410	. . . . . . 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)))
147	111, 146	rexeqbidva 3336	. . . . . 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)))
148	147	abbidv 2822	. . . . 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)})
149	110, 148	uneq12d 4071	. . . 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)}))
150	69	fveq2d 6667	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( R ‘(1st ‘⟨𝐴, 𝐵⟩)) = ( R ‘𝐴))
151	150	eleq2d 2837	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘(1st ‘⟨𝐴, 𝐵⟩)) ↔ 𝑟 ∈ ( R ‘𝐴)))
152	72	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
153	152	oveq2d 7172	. . . . . . . . . . . 12 ⊢ (((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))
154		elun2 4084	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ( R ‘𝐴) → 𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
155		elun1 4083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
156	154, 155	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ( R ‘𝐴) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
157	156	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝑟 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
158	82	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝐵 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
159	157, 158	opelxpd 5566	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → ⟨𝑟, 𝐵⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
160		rightirr 33665	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ No → ¬ 𝐴 ∈ ( R ‘𝐴))
161	160	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐴 ∈ ( R ‘𝐴))
162		eleq1 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝐴 → (𝑟 ∈ ( R ‘𝐴) ↔ 𝐴 ∈ ( R ‘𝐴)))
163	162	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝐴 → (¬ 𝑟 ∈ ( R ‘𝐴) ↔ ¬ 𝐴 ∈ ( R ‘𝐴)))
164	161, 163	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 = 𝐴 → ¬ 𝑟 ∈ ( R ‘𝐴)))
165	164	necon2ad 2966	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘𝐴) → 𝑟 ≠ 𝐴))
166	165	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝑟 ≠ 𝐴)
167	166	orcd 870	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵))
168		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝑟 ∈ ( R ‘𝐴))
169		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → 𝐵 ∈ No )
170		opthneg 5345	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝐵 ∈ No ) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
171	168, 169, 170	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → (⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝑟 ≠ 𝐴 ∨ 𝐵 ≠ 𝐵)))
172	167, 171	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → ⟨𝑟, 𝐵⟩ ≠ ⟨𝐴, 𝐵⟩)
173		eldifsn 4680	. . . . . . . . . . . . . . 15 ⊢ (⟨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⟩))
174	159, 172, 173	sylanbrc 586	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐴)) → ⟨𝑟, 𝐵⟩ ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
175	174	fvresd 6683	. . . . . . . . . . . . 13 ⊢ (((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⟩))
176		df-ov 7159	. . . . . . . . . . . . 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⟩)
177		df-ov 7159	. . . . . . . . . . . . 13 ⊢ (𝑟 +s 𝐵) = ( +s ‘⟨𝑟, 𝐵⟩)
178	175, 176, 177	3eqtr4g 2818	. . . . . . . . . . . 12 ⊢ (((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))
179	153, 178	eqtrd 2793	. . . . . . . . . . 11 ⊢ (((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))
180	179	eqeq2d 2769	. . . . . . . . . 10 ⊢ (((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)))
181	180	ex 416	. . . . . . . . 9 ⊢ ((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))))
182	151, 181	sylbid 243	. . . . . . . 8 ⊢ ((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))))
183	182	imp 410	. . . . . . 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)))
184	150, 183	rexeqbidva 3336	. . . . . 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)))
185	184	abbidv 2822	. . . . 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)})
186	72	fveq2d 6667	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( R ‘(2nd ‘⟨𝐴, 𝐵⟩)) = ( R ‘𝐵))
187	186	eleq2d 2837	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘(2nd ‘⟨𝐴, 𝐵⟩)) ↔ 𝑟 ∈ ( R ‘𝐵)))
188	69	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
189	188	oveq1d 7171	. . . . . . . . . . . 12 ⊢ (((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))
190	116	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝐴 ∈ {𝐴})
191	190, 117	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝐴 ∈ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
192		elun2 4084	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ( R ‘𝐵) → 𝑟 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
193	192	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝑟 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
194		elun1 4083	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) → 𝑟 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
195	193, 194	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝑟 ∈ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
196	191, 195	opelxpd 5566	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → ⟨𝐴, 𝑟⟩ ∈ (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
197		rightirr 33665	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ No → ¬ 𝐵 ∈ ( R ‘𝐵))
198	197	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ¬ 𝐵 ∈ ( R ‘𝐵))
199		eleq1 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝐵 → (𝑟 ∈ ( R ‘𝐵) ↔ 𝐵 ∈ ( R ‘𝐵)))
200	199	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝐵 → (¬ 𝑟 ∈ ( R ‘𝐵) ↔ ¬ 𝐵 ∈ ( R ‘𝐵)))
201	198, 200	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 = 𝐵 → ¬ 𝑟 ∈ ( R ‘𝐵)))
202	201	necon2ad 2966	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑟 ∈ ( R ‘𝐵) → 𝑟 ≠ 𝐵))
203	202	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → 𝑟 ≠ 𝐵)
204	203	olcd 871	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵))
205		opthneg 5345	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑟 ∈ ( R ‘𝐵)) → (⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵)))
206	205	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → (⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝐴 ∨ 𝑟 ≠ 𝐵)))
207	204, 206	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → ⟨𝐴, 𝑟⟩ ≠ ⟨𝐴, 𝐵⟩)
208		eldifsn 4680	. . . . . . . . . . . . . . 15 ⊢ (⟨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⟩))
209	196, 207, 208	sylanbrc 586	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑟 ∈ ( R ‘𝐵)) → ⟨𝐴, 𝑟⟩ ∈ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
210	209	fvresd 6683	. . . . . . . . . . . . 13 ⊢ (((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⟩))
211		df-ov 7159	. . . . . . . . . . . . 13 ⊢ (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⟩)
212		df-ov 7159	. . . . . . . . . . . . 13 ⊢ (𝐴 +s 𝑟) = ( +s ‘⟨𝐴, 𝑟⟩)
213	210, 211, 212	3eqtr4g 2818	. . . . . . . . . . . 12 ⊢ (((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))
214	189, 213	eqtrd 2793	. . . . . . . . . . 11 ⊢ (((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))
215	214	eqeq2d 2769	. . . . . . . . . 10 ⊢ (((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)))
216	215	ex 416	. . . . . . . . 9 ⊢ ((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))))
217	187, 216	sylbid 243	. . . . . . . 8 ⊢ ((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))))
218	217	imp 410	. . . . . . 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)))
219	186, 218	rexeqbidva 3336	. . . . . 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)))
220	219	abbidv 2822	. . . . 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)})
221	185, 220	uneq12d 4071	. . . 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)}))
222	149, 221	oveq12d 7174	. . 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)})))
223	68, 222	syl5eq 2805	. 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)})))
224	2, 223	eqtrd 2793	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 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∃wrex 3071  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858  {csn 4525  ⟨cop 4531   × cxp 5526   ↾ cres 5530  Fun wfun 6334   Fn wfn 6335  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7697  2^nd c2nd 7698   No csur 33440   |s cscut 33574   L cleft 33623   R cright 33624   +s cadds 33699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-1o 8118  df-2o 8119  df-frecs 33392  df-no 33443  df-slt 33444  df-bday 33445  df-sslt 33573  df-scut 33575  df-made 33625  df-old 33626  df-left 33628  df-right 33629  df-norec2 33688  df-adds 33702

This theorem is referenced by:  addsid1  33710  addscom  33712  addscllem1  33714