Theorem addscom 33704

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 7162	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦) = (𝑥O +s 𝑦))
2		oveq2 7163	. . 3 ⊢ (𝑥 = 𝑥O → (𝑦 +s 𝑥) = (𝑦 +s 𝑥O))
3	1, 2	eqeq12d 2774	. 2 ⊢ (𝑥 = 𝑥O → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥O +s 𝑦) = (𝑦 +s 𝑥O)))
4		oveq2 7163	. . 3 ⊢ (𝑦 = 𝑦O → (𝑥O +s 𝑦) = (𝑥O +s 𝑦O))
5		oveq1 7162	. . 3 ⊢ (𝑦 = 𝑦O → (𝑦 +s 𝑥O) = (𝑦O +s 𝑥O))
6	4, 5	eqeq12d 2774	. 2 ⊢ (𝑦 = 𝑦O → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑥O +s 𝑦O) = (𝑦O +s 𝑥O)))
7		oveq1 7162	. . 3 ⊢ (𝑥 = 𝑥O → (𝑥 +s 𝑦O) = (𝑥O +s 𝑦O))
8		oveq2 7163	. . 3 ⊢ (𝑥 = 𝑥O → (𝑦O +s 𝑥) = (𝑦O +s 𝑥O))
9	7, 8	eqeq12d 2774	. 2 ⊢ (𝑥 = 𝑥O → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥O +s 𝑦O) = (𝑦O +s 𝑥O)))
10		oveq1 7162	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7163	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2774	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7163	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7162	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2774	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) = (𝑦 +s 𝐴) ↔ (𝐴 +s 𝐵) = (𝐵 +s 𝐴)))
16		simpr2 1192	. . . . . . . . . . 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 4083	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18		oveq1 7162	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑥O +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7163	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑙 → (𝑦 +s 𝑥O) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2774	. . . . . . . . . . . 12 ⊢ (𝑥O = 𝑙 → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
21	20	rspccva 3542	. . . . . . . . . . 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 598	. . . . . . . . . 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 2769	. . . . . . . . 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 3220	. . . . . . . 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 2822	. . . . . . 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 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))) → ∀𝑦O ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦O) = (𝑦O +s 𝑥))
27		elun1 4083	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28		oveq2 7163	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑙))
29		oveq1 7162	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑙 → (𝑦O +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2774	. . . . . . . . . . . 12 ⊢ (𝑦O = 𝑙 → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
31	30	rspccva 3542	. . . . . . . . . . 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 598	. . . . . . . . . 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 2769	. . . . . . . . 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 3220	. . . . . . . 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 2822	. . . . . . 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 4071	. . . . . 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 4060	. . . . . 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 2809	. . . . 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 4084	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40		oveq1 7162	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑥O +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7163	. . . . . . . . . . . . 13 ⊢ (𝑥O = 𝑟 → (𝑦 +s 𝑥O) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2774	. . . . . . . . . . . 12 ⊢ (𝑥O = 𝑟 → ((𝑥O +s 𝑦) = (𝑦 +s 𝑥O) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
43	42	rspccva 3542	. . . . . . . . . . 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 598	. . . . . . . . . 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 2769	. . . . . . . . 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 3220	. . . . . . . 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 2822	. . . . . . 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 4084	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49		oveq2 7163	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑥 +s 𝑦O) = (𝑥 +s 𝑟))
50		oveq1 7162	. . . . . . . . . . . . 13 ⊢ (𝑦O = 𝑟 → (𝑦O +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2774	. . . . . . . . . . . 12 ⊢ (𝑦O = 𝑟 → ((𝑥 +s 𝑦O) = (𝑦O +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
52	51	rspccva 3542	. . . . . . . . . . 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 598	. . . . . . . . . 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 2769	. . . . . . . . 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 3220	. . . . . . . 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 2822	. . . . . . 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 4071	. . . . . 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 4060	. . . . . 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 2809	. . . . 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 7173	. . . 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		addsov 33701	. . . . 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 484	. . . 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		addsov 33701	. . . . . 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 462	. . . . 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 484	. . . 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 2803	. . 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 416	. 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 33686	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071   ∪ cun 3858  ‘cfv 6339  (class class class)co 7155   No csur 33432   |s cscut 33566   L cleft 33615   R cright 33616   +s cadds 33691

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 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

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 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-1o 8117  df-2o 8118  df-frecs 33384  df-no 33435  df-slt 33436  df-bday 33437  df-sslt 33565  df-scut 33567  df-made 33617  df-old 33618  df-left 33620  df-right 33621  df-norec2 33680  df-adds 33694

This theorem is referenced by:  addscomd  33705