Theorem addsass 27715

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 7418	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2	1	oveq1d 7426	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) +s 𝑧) = ((𝑥𝑂 +s 𝑦) +s 𝑧))
3		oveq1 7418	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s (𝑦 +s 𝑧)) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
4	2, 3	eqeq12d 2748	. 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 7419	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
6	5	oveq1d 7426	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧))
7		oveq1 7418	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
8	7	oveq2d 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)))
9	6, 8	eqeq12d 2748	. 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 7419	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
11		oveq2 7419	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
12	11	oveq2d 7427	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
13	10, 12	eqeq12d 2748	. 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 7418	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
15	14	oveq1d 7426	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
16		oveq1 7418	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
17	15, 16	eqeq12d 2748	. 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 7419	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 +s 𝑦) = (𝑥 +s 𝑦𝑂))
19	18	oveq1d 7426	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂))
20		oveq1 7418	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
21	20	oveq2d 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)))
22	19, 21	eqeq12d 2748	. 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 7426	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
24	20	oveq2d 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
25	23, 24	eqeq12d 2748	. 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 7419	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂))
27	11	oveq2d 7427	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)))
28	26, 27	eqeq12d 2748	. 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 7418	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
30	29	oveq1d 7426	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) +s 𝑧) = ((𝐴 +s 𝑦) +s 𝑧))
31		oveq1 7418	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s (𝑦 +s 𝑧)) = (𝐴 +s (𝑦 +s 𝑧)))
32	30, 31	eqeq12d 2748	. 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 7419	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
34	33	oveq1d 7426	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) +s 𝑧) = ((𝐴 +s 𝐵) +s 𝑧))
35		oveq1 7418	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
36	35	oveq2d 7427	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s (𝑦 +s 𝑧)) = (𝐴 +s (𝐵 +s 𝑧)))
37	34, 36	eqeq12d 2748	. 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 7419	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴 +s 𝐵) +s 𝑧) = ((𝐴 +s 𝐵) +s 𝐶))
39		oveq2 7419	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
40	39	oveq2d 7427	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 +s (𝐵 +s 𝑧)) = (𝐴 +s (𝐵 +s 𝐶)))
41	38, 40	eqeq12d 2748	. 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 1206	. . . 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 1208	. . . 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 1138	. . . 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 1128	. . 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 7418	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s 𝑦) = (𝑥𝐿 +s 𝑦))
47	46	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝐿 +s 𝑦) +s 𝑧))
48		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝐿 +s (𝑦 +s 𝑧)))
49	47, 48	eqeq12d 2748	. . . . . . . . . . . 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 1215	. . . . . . . . . . . 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 4176	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
52	51	adantl 482	. . . . . . . . . . . 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 3613	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3176	. . . . . . . . 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 2801	. . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑦𝐿))
58	57	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝐿 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝐿) +s 𝑧))
59		oveq1 7418	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝑦𝑂 +s 𝑧) = (𝑦𝐿 +s 𝑧))
60	59	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝐿 +s 𝑧)))
61	58, 60	eqeq12d 2748	. . . . . . . . . . . 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 1216	. . . . . . . . . . . 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 4176	. . . . . . . . . . . . 13 ⊢ (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
64	63	adantl 482	. . . . . . . . . . . 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 3613	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3176	. . . . . . . . 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 2801	. . . . . . . 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 4164	. . . . . . 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 7419	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝐿 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦) +s 𝑧𝐿))
71		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑧𝑂 = 𝑧𝐿 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝐿))
72	71	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝐿 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦 +s 𝑧𝐿)))
73	70, 72	eqeq12d 2748	. . . . . . . . . . 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 1217	. . . . . . . . . . 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 4176	. . . . . . . . . . . 12 ⊢ (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
76	75	adantl 482	. . . . . . . . . . 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 3613	. . . . . . . . . 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 2743	. . . . . . . . 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 3176	. . . . . . . 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 2801	. . . . . . 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 4164	. . . . . 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 7418	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s 𝑦) = (𝑥𝑅 +s 𝑦))
83	82	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝑅 +s 𝑦) +s 𝑧))
84		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝑅 +s (𝑦 +s 𝑧)))
85	83, 84	eqeq12d 2748	. . . . . . . . . . . 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 1215	. . . . . . . . . . . 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 4177	. . . . . . . . . . . . 13 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
88	87	adantl 482	. . . . . . . . . . . 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 3613	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3176	. . . . . . . . 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 2801	. . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝑅 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑦𝑅))
94	93	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝑅 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝑅) +s 𝑧))
95		oveq1 7418	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑂 = 𝑦𝑅 → (𝑦𝑂 +s 𝑧) = (𝑦𝑅 +s 𝑧))
96	95	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑦𝑅 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝑅 +s 𝑧)))
97	94, 96	eqeq12d 2748	. . . . . . . . . . . 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 1216	. . . . . . . . . . . 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 4177	. . . . . . . . . . . . 13 ⊢ (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
100	99	adantl 482	. . . . . . . . . . . 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 3613	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3176	. . . . . . . . 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 2801	. . . . . . . 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 4164	. . . . . . 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 7419	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝑅 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦) +s 𝑧𝑅))
107		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑧𝑂 = 𝑧𝑅 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝑅))
108	107	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑧𝑂 = 𝑧𝑅 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦 +s 𝑧𝑅)))
109	106, 108	eqeq12d 2748	. . . . . . . . . . 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 1217	. . . . . . . . . . 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 4177	. . . . . . . . . . . 12 ⊢ (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
112	111	adantl 482	. . . . . . . . . . 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 3613	. . . . . . . . . 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 2743	. . . . . . . . 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 3176	. . . . . . . 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 2801	. . . . . . 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 4164	. . . . . 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 7429	. . . . 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 1191	. . . . . 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 1192	. . . . . 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 1193	. . . . . 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 27713	. . . . 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 27714	. . . . 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 2782	. . . 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 413	. . 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 34	. 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 27668	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ∪ cun 3946  ‘cfv 6543  (class class class)co 7411   No csur 27367   |s cscut 27508   L cleft 27565   R cright 27566   +_s cadds 27669

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-1o 8468  df-2o 8469  df-nadd 8667  df-no 27370  df-slt 27371  df-bday 27372  df-sle 27472  df-sslt 27507  df-scut 27509  df-0s 27550  df-made 27567  df-old 27568  df-left 27570  df-right 27571  df-norec2 27659  df-adds 27670

This theorem is referenced by:  addsassd  27716