Theorem addscom 34129

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 7282	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦) = (𝑥O +s 𝑦))
2		oveq2 7283	. . 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 7283	. . 3 ⊢ (𝑦 = 𝑦O → (𝑥O +s 𝑦) = (𝑥O +s 𝑦O))
5		oveq1 7282	. . 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 7282	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦O) = (𝑥O +s 𝑦O))
8		oveq2 7283	. . 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 7282	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7283	. . 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 7283	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7282	. . 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 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))) → ∀𝑥O ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥O +s 𝑦) = (𝑦 +s 𝑥O))
17		elun1 4110	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑥O +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7283	. . . . . . . . . . . . 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 3560	. . . . . . . . . . 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 596	. . . . . . . . . 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 3225	. . . . . . . 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 2807	. . . . . . 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 1195	. . . . . . . . . . 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 4110	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑙))
29		oveq1 7282	. . . . . . . . . . . . 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 3560	. . . . . . . . . . 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 596	. . . . . . . . . 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 3225	. . . . . . . 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 2807	. . . . . . 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 4098	. . . . . 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 4087	. . . . . 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 2794	. . . . 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 4111	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑥O +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7283	. . . . . . . . . . . . 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 3560	. . . . . . . . . . 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 596	. . . . . . . . . 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 3225	. . . . . . . 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 2807	. . . . . . 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 4111	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑟))
50		oveq1 7282	. . . . . . . . . . . . 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 3560	. . . . . . . . . . 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 596	. . . . . . . . . 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 3225	. . . . . . . 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 2807	. . . . . . 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 4098	. . . . . 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 4087	. . . . . 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 2794	. . . . 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 7293	. . . 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 34126	. . . . 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 481	. . . 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 34126	. . . . . 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 459	. . . . 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 481	. . . 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 413	. 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 34112	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ∪ cun 3885  ‘cfv 6433  (class class class)co 7275   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:  addscomd  34130