Theorem addsass 28156

Description: Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)

Assertion

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

Proof of Theorem addsass

Dummy variables 𝑥 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑥_𝑂 𝑦_𝑂 𝑧_𝑂 𝑥_𝐿 𝑦_𝐿 𝑧_𝐿 𝑥_𝑅 𝑦_𝑅 𝑧_𝑅 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7407	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2	1	oveq1d 7415	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) +s 𝑧) = ((𝑥𝑂 +s 𝑦) +s 𝑧))
3		oveq1 7407	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s (𝑦 +s 𝑧)) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
4	2, 3	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝑥𝑂 → (((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧))))
5		oveq2 7408	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
6	5	oveq1d 7415	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧))
7		oveq1 7407	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
8	7	oveq2d 7416	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)))
9	6, 8	eqeq12d 2781	. 2 ⊢ (𝑦 = 𝑦𝑂 → (((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧))))
10		oveq2 7408	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
11		oveq2 7408	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
12	11	oveq2d 7416	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
13	10, 12	eqeq12d 2781	. 2 ⊢ (𝑧 = 𝑧𝑂 → (((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂))))
14		oveq1 7407	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
15	14	oveq1d 7415	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
16		oveq1 7407	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
17	15, 16	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝑥𝑂 → (((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂))))
18		oveq2 7408	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 +s 𝑦) = (𝑥 +s 𝑦𝑂))
19	18	oveq1d 7415	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂))
20		oveq1 7407	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
21	20	oveq2d 7416	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)))
22	19, 21	eqeq12d 2781	. 2 ⊢ (𝑦 = 𝑦𝑂 → (((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂))))
23	5	oveq1d 7415	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
24	20	oveq2d 7416	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
25	23, 24	eqeq12d 2781	. 2 ⊢ (𝑦 = 𝑦𝑂 → (((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂))))
26		oveq2 7408	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂))
27	11	oveq2d 7416	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)))
28	26, 27	eqeq12d 2781	. 2 ⊢ (𝑧 = 𝑧𝑂 → (((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂))))
29		oveq1 7407	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
30	29	oveq1d 7415	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) +s 𝑧) = ((𝐴 +s 𝑦) +s 𝑧))
31		oveq1 7407	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s (𝑦 +s 𝑧)) = (𝐴 +s (𝑦 +s 𝑧)))
32	30, 31	eqeq12d 2781	. 2 ⊢ (𝑥 = 𝐴 → (((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)) ↔ ((𝐴 +s 𝑦) +s 𝑧) = (𝐴 +s (𝑦 +s 𝑧))))
33		oveq2 7408	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
34	33	oveq1d 7415	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) +s 𝑧) = ((𝐴 +s 𝐵) +s 𝑧))
35		oveq1 7407	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
36	35	oveq2d 7416	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s (𝑦 +s 𝑧)) = (𝐴 +s (𝐵 +s 𝑧)))
37	34, 36	eqeq12d 2781	. 2 ⊢ (𝑦 = 𝐵 → (((𝐴 +s 𝑦) +s 𝑧) = (𝐴 +s (𝑦 +s 𝑧)) ↔ ((𝐴 +s 𝐵) +s 𝑧) = (𝐴 +s (𝐵 +s 𝑧))))
38		oveq2 7408	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴 +s 𝐵) +s 𝑧) = ((𝐴 +s 𝐵) +s 𝐶))
39		oveq2 7408	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
40	39	oveq2d 7416	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 +s (𝐵 +s 𝑧)) = (𝐴 +s (𝐵 +s 𝐶)))
41	38, 40	eqeq12d 2781	. 2 ⊢ (𝑧 = 𝐶 → (((𝐴 +s 𝐵) +s 𝑧) = (𝐴 +s (𝐵 +s 𝑧)) ↔ ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶))))
42		simp21 1223	. . . 4 ⊢ (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
43		simp23 1225	. . . 4 ⊢ (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)))
44		simp3 1154	. . . 4 ⊢ (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))
45	42, 43, 44	3jca 1144	. . 3 ⊢ (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))))
46		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s 𝑦) = (𝑥𝐿 +s 𝑦))
47	46	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝐿 +s 𝑦) +s 𝑧))
48		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝐿 +s (𝑦 +s 𝑧)))
49	47, 48	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑥𝐿 → (((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝐿 +s 𝑦) +s 𝑧) = (𝑥𝐿 +s (𝑦 +s 𝑧))))
50		simplr1 1232	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
51		elun1 4137	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
52	51	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
53	49, 50, 52	rspcdva 3585	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ((𝑥𝐿 +s 𝑦) +s 𝑧) = (𝑥𝐿 +s (𝑦 +s 𝑧)))
54	53	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧) ↔ 𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))))
55	54	rexbidva 3187	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))))
56	55	abbidv 2831	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))})
57		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑦𝐿))
58	57	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝐿 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝐿) +s 𝑧))
59		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝑦𝑂 +s 𝑧) = (𝑦𝐿 +s 𝑧))
60	59	oveq2d 7416	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝐿 +s 𝑧)))
61	58, 60	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑦𝐿 → (((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥 +s 𝑦𝐿) +s 𝑧) = (𝑥 +s (𝑦𝐿 +s 𝑧))))
62		simplr2 1233	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)))
63		elun1 4137	. . . . . . . . . . . . 13 ⊢ (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
64	63	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
65	61, 62, 64	rspcdva 3585	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → ((𝑥 +s 𝑦𝐿) +s 𝑧) = (𝑥 +s (𝑦𝐿 +s 𝑧)))
66	65	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → (𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧) ↔ 𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))))
67	66	rexbidva 3187	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧) ↔ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))))
68	67	abbidv 2831	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)} = {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))})
69	56, 68	uneq12d 4125	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}))
70		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝐿 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦) +s 𝑧𝐿))
71		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑧𝑂 = 𝑧𝐿 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝐿))
72	71	oveq2d 7416	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝐿 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦 +s 𝑧𝐿)))
73	70, 72	eqeq12d 2781	. . . . . . . . . . 11 ⊢ (𝑧𝑂 = 𝑧𝐿 → (((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥 +s 𝑦) +s 𝑧𝐿) = (𝑥 +s (𝑦 +s 𝑧𝐿))))
74		simplr3 1234	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))
75		elun1 4137	. . . . . . . . . . . 12 ⊢ (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
76	75	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
77	73, 74, 76	rspcdva 3585	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → ((𝑥 +s 𝑦) +s 𝑧𝐿) = (𝑥 +s (𝑦 +s 𝑧𝐿)))
78	77	eqeq2d 2776	. . . . . . . . 9 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → (𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿) ↔ 𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))))
79	78	rexbidva 3187	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿) ↔ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))))
80	79	abbidv 2831	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)} = {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))})
81	69, 80	uneq12d 4125	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)}) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))}))
82		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s 𝑦) = (𝑥𝑅 +s 𝑦))
83	82	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝑅 +s 𝑦) +s 𝑧))
84		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝑅 +s (𝑦 +s 𝑧)))
85	83, 84	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑥𝑅 → (((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝑅 +s 𝑦) +s 𝑧) = (𝑥𝑅 +s (𝑦 +s 𝑧))))
86		simplr1 1232	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
87		elun2 4138	. . . . . . . . . . . . 13 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
88	87	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
89	85, 86, 88	rspcdva 3585	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ((𝑥𝑅 +s 𝑦) +s 𝑧) = (𝑥𝑅 +s (𝑦 +s 𝑧)))
90	89	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧) ↔ 𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))))
91	90	rexbidva 3187	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))))
92	91	abbidv 2831	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} = {𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))})
93		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝑅 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑦𝑅))
94	93	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝑅 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝑅) +s 𝑧))
95		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝑅 → (𝑦𝑂 +s 𝑧) = (𝑦𝑅 +s 𝑧))
96	95	oveq2d 7416	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝑅 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝑅 +s 𝑧)))
97	94, 96	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑦𝑅 → (((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥 +s 𝑦𝑅) +s 𝑧) = (𝑥 +s (𝑦𝑅 +s 𝑧))))
98		simplr2 1233	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)))
99		elun2 4138	. . . . . . . . . . . . 13 ⊢ (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
100	99	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
101	97, 98, 100	rspcdva 3585	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → ((𝑥 +s 𝑦𝑅) +s 𝑧) = (𝑥 +s (𝑦𝑅 +s 𝑧)))
102	101	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → (𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧) ↔ 𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))))
103	102	rexbidva 3187	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧) ↔ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))))
104	103	abbidv 2831	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)} = {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))})
105	92, 104	uneq12d 4125	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) = ({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}))
106		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝑅 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦) +s 𝑧𝑅))
107		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑧𝑂 = 𝑧𝑅 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝑅))
108	107	oveq2d 7416	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝑅 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦 +s 𝑧𝑅)))
109	106, 108	eqeq12d 2781	. . . . . . . . . . 11 ⊢ (𝑧𝑂 = 𝑧𝑅 → (((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥 +s 𝑦) +s 𝑧𝑅) = (𝑥 +s (𝑦 +s 𝑧𝑅))))
110		simplr3 1234	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))
111		elun2 4138	. . . . . . . . . . . 12 ⊢ (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
112	111	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
113	109, 110, 112	rspcdva 3585	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → ((𝑥 +s 𝑦) +s 𝑧𝑅) = (𝑥 +s (𝑦 +s 𝑧𝑅)))
114	113	eqeq2d 2776	. . . . . . . . 9 ⊢ ((((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → (𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅) ↔ 𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))))
115	114	rexbidva 3187	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))))
116	115	abbidv 2831	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)} = {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))})
117	105, 116	uneq12d 4125	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)}) = (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))}))
118	81, 117	oveq12d 7418	. . . . 5 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)})) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))})))
119		simpl1 1208	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → 𝑥 ∈ No )
120		simpl2 1209	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → 𝑦 ∈ No )
121		simpl3 1210	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → 𝑧 ∈ No )
122	119, 120, 121	addsasslem1 28154	. . . . 5 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ((𝑥 +s 𝑦) +s 𝑧) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)})))
123	119, 120, 121	addsasslem2 28155	. . . . 5 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (𝑥 +s (𝑦 +s 𝑧)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))})))
124	118, 122, 123	3eqtr4d 2810	. . . 4 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)))
125	124	ex 417	. . 3 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧))))
126	45, 125	syl5 35	. 2 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧))))
127	4, 9, 13, 17, 22, 25, 28, 32, 37, 41, 126	no3inds 28109	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089   ∪ cun 3905  ‘cfv 6525  (class class class)co 7400   No csur 27762   |s ccuts 27910   L cleft 27976   R cright 27977   +_s cadds 28110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec2 28100  df-adds 28111

This theorem is referenced by:  addsassd  28157  zsoring  28560