Theorem addscom 34117

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 7276	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦) = (𝑥O +s 𝑦))
2		oveq2 7277	. . 3 ⊢ (𝑥 = 𝑥O → (𝑦 +s 𝑥) = (𝑦 +s 𝑥O))
3	1, 2	eqeq12d 2756	. 2 ⊢ (𝑥 = 𝑥O → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥O +s 𝑦) = (𝑦 +s 𝑥O)))
4		oveq2 7277	. . 3 ⊢ (𝑦 = 𝑦O → (𝑥O +s 𝑦) = (𝑥O +s 𝑦O))
5		oveq1 7276	. . 3 ⊢ (𝑦 = 𝑦O → (𝑦 +s 𝑥O) = (𝑦O +s 𝑥O))
6	4, 5	eqeq12d 2756	. 2 ⊢ (𝑦 = 𝑦O → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑥O +s 𝑦O) = (𝑦O +s 𝑥O)))
7		oveq1 7276	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦O) = (𝑥O +s 𝑦O))
8		oveq2 7277	. . 3 ⊢ (𝑥 = 𝑥O → (𝑦O +s 𝑥) = (𝑦O +s 𝑥O))
9	7, 8	eqeq12d 2756	. 2 ⊢ (𝑥 = 𝑥O → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥O +s 𝑦O) = (𝑦O +s 𝑥O)))
10		oveq1 7276	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7277	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2756	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7277	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7276	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2756	. 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 4115	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18		oveq1 7276	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑥O +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7277	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑦 +s 𝑥O) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2756	. . . . . . . . . . . 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 2751	. . . . . . . . 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 3227	. . . . . . . 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 2809	. . . . . . 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 4115	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28		oveq2 7277	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑙))
29		oveq1 7276	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑦O +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2756	. . . . . . . . . . . 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 2751	. . . . . . . . 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 3227	. . . . . . . 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 2809	. . . . . . 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 4103	. . . . . 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 4092	. . . . . 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 2796	. . . . 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 4116	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40		oveq1 7276	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑥O +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7277	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑦 +s 𝑥O) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2756	. . . . . . . . . . . 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 2751	. . . . . . . . 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 3227	. . . . . . . 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 2809	. . . . . . 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 4116	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49		oveq2 7277	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑟))
50		oveq1 7276	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑦O +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2756	. . . . . . . . . . . 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 2751	. . . . . . . . 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 3227	. . . . . . . 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 2809	. . . . . . 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 4103	. . . . . 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 4092	. . . . . 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 2796	. . . . 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 7287	. . . 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 34114	. . . . 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 34114	. . . . . 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 2790	. . 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 34100	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  {cab 2717  ∀wral 3066  ∃wrex 3067   ∪ cun 3890  ‘cfv 6431  (class class class)co 7269   No csur 33831   |s cscut 33965   L cleft 34017   R cright 34018   +s cadds 34104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  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 6200  df-ord 6267  df-on 6268  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7818  df-2nd 7819  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-1o 8282  df-2o 8283  df-no 33834  df-slt 33835  df-bday 33836  df-sslt 33964  df-scut 33966  df-made 34019  df-old 34020  df-left 34022  df-right 34023  df-norec2 34094  df-adds 34107

This theorem is referenced by:  addscomd  34118