Theorem addscom 34056

Description: Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)

Assertion

Ref	Expression
addscom	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Proof of Theorem addscom

Dummy variables 𝑥 𝑦 𝑟 𝑙 𝑤 𝑥_O 𝑦_O 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7262	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦) = (𝑥O +s 𝑦))
2		oveq2 7263	. . 3 ⊢ (𝑥 = 𝑥O → (𝑦 +s 𝑥) = (𝑦 +s 𝑥O))
3	1, 2	eqeq12d 2754	. 2 ⊢ (𝑥 = 𝑥O → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥O +s 𝑦) = (𝑦 +s 𝑥O)))
4		oveq2 7263	. . 3 ⊢ (𝑦 = 𝑦O → (𝑥O +s 𝑦) = (𝑥O +s 𝑦O))
5		oveq1 7262	. . 3 ⊢ (𝑦 = 𝑦O → (𝑦 +s 𝑥O) = (𝑦O +s 𝑥O))
6	4, 5	eqeq12d 2754	. 2 ⊢ (𝑦 = 𝑦O → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑥O +s 𝑦O) = (𝑦O +s 𝑥O)))
7		oveq1 7262	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦O) = (𝑥O +s 𝑦O))
8		oveq2 7263	. . 3 ⊢ (𝑥 = 𝑥O → (𝑦O +s 𝑥) = (𝑦O +s 𝑥O))
9	7, 8	eqeq12d 2754	. 2 ⊢ (𝑥 = 𝑥O → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥O +s 𝑦O) = (𝑦O +s 𝑥O)))
10		oveq1 7262	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7263	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2754	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7263	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7262	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2754	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) = (𝑦 +s 𝐴) ↔ (𝐴 +s 𝐵) = (𝐵 +s 𝐴)))
16		simpr2 1193	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O))
17		elun1 4106	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑥O +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑦 +s 𝑥O) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑥O = 𝑙 → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
21	20	rspccva 3551	. . . . . . . . . . 11 ⊢ ((∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑙 +s 𝑦) = (𝑦 +s 𝑙))
22	16, 17, 21	syl2an 595	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑙 ∈ ( L ‘𝑥)) → (𝑙 +s 𝑦) = (𝑦 +s 𝑙))
23	22	eqeq2d 2749	. . . . . . . . 9 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑙 ∈ ( L ‘𝑥)) → (𝑤 = (𝑙 +s 𝑦) ↔ 𝑤 = (𝑦 +s 𝑙)))
24	23	rexbidva 3224	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦) ↔ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)))
25	24	abbidv 2808	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} = {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)})
26		simpr3 1194	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))
27		elun1 4106	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑙))
29		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑦O +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑦O = 𝑙 → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
31	30	rspccva 3551	. . . . . . . . . . 11 ⊢ ((∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ∧ 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))) → (𝑥 +s 𝑙) = (𝑙 +s 𝑥))
32	26, 27, 31	syl2an 595	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑙 ∈ ( L ‘𝑦)) → (𝑥 +s 𝑙) = (𝑙 +s 𝑥))
33	32	eqeq2d 2749	. . . . . . . . 9 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑙 ∈ ( L ‘𝑦)) → (𝑧 = (𝑥 +s 𝑙) ↔ 𝑧 = (𝑙 +s 𝑥)))
34	33	rexbidva 3224	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙) ↔ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)))
35	34	abbidv 2808	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)} = {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)})
36	25, 35	uneq12d 4094	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) = ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)}))
37		uncom 4083	. . . . . 6 ⊢ ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)}) = ({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)})
38	36, 37	eqtrdi 2795	. . . . 5 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) = ({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}))
39		elun2 4107	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑥O +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑦 +s 𝑥O) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑥O = 𝑟 → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
43	42	rspccva 3551	. . . . . . . . . . 11 ⊢ ((∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑟 +s 𝑦) = (𝑦 +s 𝑟))
44	16, 39, 43	syl2an 595	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑟 ∈ ( R ‘𝑥)) → (𝑟 +s 𝑦) = (𝑦 +s 𝑟))
45	44	eqeq2d 2749	. . . . . . . . 9 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑟 ∈ ( R ‘𝑥)) → (𝑤 = (𝑟 +s 𝑦) ↔ 𝑤 = (𝑦 +s 𝑟)))
46	45	rexbidva 3224	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦) ↔ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)))
47	46	abbidv 2808	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} = {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})
48		elun2 4107	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑟))
50		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑦O +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑦O = 𝑟 → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
52	51	rspccva 3551	. . . . . . . . . . 11 ⊢ ((∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ∧ 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))) → (𝑥 +s 𝑟) = (𝑟 +s 𝑥))
53	26, 48, 52	syl2an 595	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑟 ∈ ( R ‘𝑦)) → (𝑥 +s 𝑟) = (𝑟 +s 𝑥))
54	53	eqeq2d 2749	. . . . . . . . 9 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) ∧ 𝑟 ∈ ( R ‘𝑦)) → (𝑧 = (𝑥 +s 𝑟) ↔ 𝑧 = (𝑟 +s 𝑥)))
55	54	rexbidva 3224	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)))
56	55	abbidv 2808	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)} = {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)})
57	47, 56	uneq12d 4094	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)}) = ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)}))
58		uncom 4083	. . . . . 6 ⊢ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)}) = ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})
59	57, 58	eqtrdi 2795	. . . . 5 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)}) = ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)}))
60	38, 59	oveq12d 7273	. . . 4 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)})) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
61		addsval 34053	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑥 +s 𝑦) = (({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)})))
62	61	adantr 480	. . . 4 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (𝑥 +s 𝑦) = (({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)})))
63		addsval 34053	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 𝑥 ∈ No ) → (𝑦 +s 𝑥) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
64	63	ancoms 458	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑦 +s 𝑥) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
65	64	adantr 480	. . . 4 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (𝑦 +s 𝑥) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
66	60, 62, 65	3eqtr4d 2788	. . 3 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))) → (𝑥 +s 𝑦) = (𝑦 +s 𝑥))
67	66	ex 412	. 2 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥O +s 𝑦O) = (𝑦O +s 𝑥O) ∧ ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ∧ ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥)) → (𝑥 +s 𝑦) = (𝑦 +s 𝑥)))
68	3, 6, 9, 12, 15, 67	no2inds 34039	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∪ cun 3881  ‘cfv 6418  (class class class)co 7255   No csur 33770   |s cscut 33904   L cleft 33956   R cright 33957   +s cadds 34043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959  df-left 33961  df-right 33962  df-norec2 34033  df-adds 34046

This theorem is referenced by:  addscomd  34057