Theorem mulsass 28552

Description: Associative law for surreal multiplication. Part of theorem 7 of [Conway] p. 19. Much like the case for additive groups, this theorem together with mulscom 28525, addsdi 28541, mulsgt0 28530, and the addition theorems would make the surreals into an ordered ring except that they are a proper class. (Contributed by Scott Fenton, 10-Mar-2025.)

Assertion

Ref	Expression
mulsass	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))

Proof of Theorem mulsass

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

Step	Hyp	Ref	Expression
1		oveq1 7427	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
2	1	oveq1d 7435	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦) ·s 𝑧))
3		oveq1 7427	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦 ·s 𝑧)))
4	2, 3	eqeq12d 2776	. 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 7428	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
6	5	oveq1d 7435	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧))
7		oveq1 7427	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑧) = (𝑦𝑂 ·s 𝑧))
8	7	oveq2d 7436	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)))
9	6, 8	eqeq12d 2776	. 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 7428	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
11		oveq2 7428	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 ·s 𝑧) = (𝑦𝑂 ·s 𝑧𝑂))
12	11	oveq2d 7436	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
13	10, 12	eqeq12d 2776	. 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 7427	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
15	14	oveq1d 7435	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
16		oveq1 7427	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
17	15, 16	eqeq12d 2776	. 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 7428	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
19	18	oveq1d 7435	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂))
20		oveq1 7427	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑧𝑂) = (𝑦𝑂 ·s 𝑧𝑂))
21	20	oveq2d 7436	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
22	19, 21	eqeq12d 2776	. 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 7435	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
24	20	oveq2d 7436	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
25	23, 24	eqeq12d 2776	. 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 7428	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 ·s 𝑦𝑂) ·s 𝑧) = ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂))
27	11	oveq2d 7436	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 ·s 𝑧)) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
28	26, 27	eqeq12d 2776	. 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 7427	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
30	29	oveq1d 7435	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) ·s 𝑧) = ((𝐴 ·s 𝑦) ·s 𝑧))
31		oveq1 7427	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s (𝑦 ·s 𝑧)) = (𝐴 ·s (𝑦 ·s 𝑧)))
32	30, 31	eqeq12d 2776	. 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 7428	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
34	33	oveq1d 7435	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) ·s 𝑧) = ((𝐴 ·s 𝐵) ·s 𝑧))
35		oveq1 7427	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ·s 𝑧) = (𝐵 ·s 𝑧))
36	35	oveq2d 7436	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ·s (𝑦 ·s 𝑧)) = (𝐴 ·s (𝐵 ·s 𝑧)))
37	34, 36	eqeq12d 2776	. 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 7428	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴 ·s 𝐵) ·s 𝑧) = ((𝐴 ·s 𝐵) ·s 𝐶))
39		oveq2 7428	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 ·s 𝑧) = (𝐵 ·s 𝐶))
40	39	oveq2d 7436	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 ·s (𝐵 ·s 𝑧)) = (𝐴 ·s (𝐵 ·s 𝐶)))
41	38, 40	eqeq12d 2776	. 2 ⊢ (𝑧 = 𝐶 → (((𝐴 ·s 𝐵) ·s 𝑧) = (𝐴 ·s (𝐵 ·s 𝑧)) ↔ ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶))))
42		simpl1 1210	. . . . . . . . . 10 ⊢ (((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 ∈ No)
43		simpl2 1211	. . . . . . . . . 10 ⊢ (((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)))) → 𝑦 ∈ No)
44		simpl3 1212	. . . . . . . . . 10 ⊢ (((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)))) → 𝑧 ∈ No)
45		ssun1 4176	. . . . . . . . . 10 ⊢ (L‘𝑥) ⊆ ((L‘𝑥) ∪ (R‘𝑥))
46		ssun1 4176	. . . . . . . . . 10 ⊢ (L‘𝑦) ⊆ ((L‘𝑦) ∪ (R‘𝑦))
47		ssun1 4176	. . . . . . . . . 10 ⊢ (L‘𝑧) ⊆ ((L‘𝑧) ∪ (R‘𝑧))
48		simpr11 1276	. . . . . . . . . 10 ⊢ (((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)))) → ∀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)))
49		simpr12 1277	. . . . . . . . . 10 ⊢ (((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)))) → ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))∀𝑦𝑂 ∈ ((L‘𝑦) ∪ (R‘𝑦))((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)))
50		simpr13 1278	. . . . . . . . . 10 ⊢ (((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)))) → ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))∀𝑧𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))((𝑥𝑂 ·s 𝑦) ·s 𝑧𝑂) = (𝑥𝑂 ·s (𝑦 ·s 𝑧𝑂)))
51		simpr22 1280	. . . . . . . . . 10 ⊢ (((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)))) → ∀𝑦𝑂 ∈ ((L‘𝑦) ∪ (R‘𝑦))∀𝑧𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
52		simpr21 1279	. . . . . . . . . 10 ⊢ (((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)))) → ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))((𝑥𝑂 ·s 𝑦) ·s 𝑧) = (𝑥𝑂 ·s (𝑦 ·s 𝑧)))
53		simpr23 1281	. . . . . . . . . 10 ⊢ (((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)))) → ∀𝑦𝑂 ∈ ((L‘𝑦) ∪ (R‘𝑦))((𝑥 ·s 𝑦𝑂) ·s 𝑧) = (𝑥 ·s (𝑦𝑂 ·s 𝑧)))
54		simpr3 1215	. . . . . . . . . 10 ⊢ (((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)))) → ∀𝑧𝑂 ∈ ((L‘𝑧) ∪ (R‘𝑧))((𝑥 ·s 𝑦) ·s 𝑧𝑂) = (𝑥 ·s (𝑦 ·s 𝑧𝑂)))
55	42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))))
56	55	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} = {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))})
57		ssun2 4177	. . . . . . . . . 10 ⊢ (R‘𝑥) ⊆ ((L‘𝑥) ∪ (R‘𝑥))
58		ssun2 4177	. . . . . . . . . 10 ⊢ (R‘𝑦) ⊆ ((L‘𝑦) ∪ (R‘𝑦))
59	42, 43, 44, 57, 58, 47, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))))
60	59	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))} = {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))})
61	56, 60	uneq12d 4168	. . . . . . 7 ⊢ (((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)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}))
62		ssun2 4177	. . . . . . . . . 10 ⊢ (R‘𝑧) ⊆ ((L‘𝑧) ∪ (R‘𝑧))
63	42, 43, 44, 45, 58, 62, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))))
64	63	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} = {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))})
65	42, 43, 44, 57, 46, 62, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))))
66	65	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))} = {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))})
67	64, 66	uneq12d 4168	. . . . . . 7 ⊢ (((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)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))}))
68	61, 67	uneq12d 4168	. . . . . 6 ⊢ (((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)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))})))
69		un4 4173	. . . . . . 7 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))}))
70		uncom 4157	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))}) = ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))})
71	70	uneq2i 4164	. . . . . . 7 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}))
72	69, 71	eqtri 2783	. . . . . 6 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}))
73	68, 72	eqtrdi 2811	. . . . 5 ⊢ (((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)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))})))
74	42, 43, 44, 45, 46, 62, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))))
75	74	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} = {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))})
76	42, 43, 44, 57, 58, 62, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))))
77	76	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))} = {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))})
78	75, 77	uneq12d 4168	. . . . . . 7 ⊢ (((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)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}))
79	42, 43, 44, 45, 58, 47, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))))
80	79	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} = {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))})
81	42, 43, 44, 57, 46, 47, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28551	. . . . . . . . 9 ⊢ (((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)))) → (∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))))
82	81	abbidv 2826	. . . . . . . 8 ⊢ (((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)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))} = {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))})
83	80, 82	uneq12d 4168	. . . . . . 7 ⊢ (((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)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))}))
84	78, 83	uneq12d 4168	. . . . . 6 ⊢ (((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)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))})))
85		un4 4173	. . . . . . 7 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))}))
86		uncom 4157	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))}) = ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))})
87	86	uneq2i 4164	. . . . . . 7 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}))
88	85, 87	eqtri 2783	. . . . . 6 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}))
89	84, 88	eqtrdi 2811	. . . . 5 ⊢ (((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)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))})))
90	73, 89	oveq12d 7438	. . . 4 ⊢ (((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)))) → ((({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}))))
91	42, 43, 44	mulsasslem1 28549	. . . 4 ⊢ (((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) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝑅)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) −s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = ((((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧) +s ((𝑥 ·s 𝑦) ·s 𝑧𝐿)) −s ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) −s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))))
92	42, 43, 44	mulsasslem2 28550	. . . 4 ⊢ (((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)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)))) −s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)))) −s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝐿 ∈ (L‘𝑦)∃𝑧𝐿 ∈ (L‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)))) −s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝑧) +s (𝑦 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)∃𝑦𝑅 ∈ (R‘𝑦)∃𝑧𝑅 ∈ (R‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 ·s 𝑧)) +s (𝑥 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)))) −s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝑧) +s (𝑦 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅))))}))))
93	90, 91, 92	3eqtr4d 2805	. . 3 ⊢ (((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)))
94	93	ex 418	. 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))))
95	4, 9, 13, 17, 22, 25, 28, 32, 37, 41, 94	no3inds 28344	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 4149  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   |s ccuts 28145  Lcleft 28211  Rcright 28212   +_s cadds 28345   −_s csubs 28406   ·_s cmuls 28492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493

This theorem is used by:  mulsassd  28553  zsoring  28795