Theorem mulsass 28579

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 28552, addsdi 28568, mulsgt0 28557, 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 7447	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
2	1	oveq1d 7455	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦) ·s 𝑧))
3		oveq1 7447	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦 ·s 𝑧)))
4	2, 3	eqeq12d 2779	. 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 7448	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
6	5	oveq1d 7455	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧))
7		oveq1 7447	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑧) = (𝑦𝑂 ·s 𝑧))
8	7	oveq2d 7456	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)))
9	6, 8	eqeq12d 2779	. 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 7448	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
11		oveq2 7448	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 ·s 𝑧) = (𝑦𝑂 ·s 𝑧𝑂))
12	11	oveq2d 7456	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
13	10, 12	eqeq12d 2779	. 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 7447	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
15	14	oveq1d 7455	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
16		oveq1 7447	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
17	15, 16	eqeq12d 2779	. 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 7448	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
19	18	oveq1d 7455	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂))
20		oveq1 7447	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑧𝑂) = (𝑦𝑂 ·s 𝑧𝑂))
21	20	oveq2d 7456	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
22	19, 21	eqeq12d 2779	. 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 7455	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
24	20	oveq2d 7456	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
25	23, 24	eqeq12d 2779	. 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 7448	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 ·s 𝑦𝑂) ·s 𝑧) = ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂))
27	11	oveq2d 7456	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 ·s 𝑧)) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
28	26, 27	eqeq12d 2779	. 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 7447	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
30	29	oveq1d 7455	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) ·s 𝑧) = ((𝐴 ·s 𝑦) ·s 𝑧))
31		oveq1 7447	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s (𝑦 ·s 𝑧)) = (𝐴 ·s (𝑦 ·s 𝑧)))
32	30, 31	eqeq12d 2779	. 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 7448	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
34	33	oveq1d 7455	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) ·s 𝑧) = ((𝐴 ·s 𝐵) ·s 𝑧))
35		oveq1 7447	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ·s 𝑧) = (𝐵 ·s 𝑧))
36	35	oveq2d 7456	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ·s (𝑦 ·s 𝑧)) = (𝐴 ·s (𝐵 ·s 𝑧)))
37	34, 36	eqeq12d 2779	. 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 7448	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴 ·s 𝐵) ·s 𝑧) = ((𝐴 ·s 𝐵) ·s 𝐶))
39		oveq2 7448	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 ·s 𝑧) = (𝐵 ·s 𝐶))
40	39	oveq2d 7456	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 ·s (𝐵 ·s 𝑧)) = (𝐴 ·s (𝐵 ·s 𝐶)))
41	38, 40	eqeq12d 2779	. 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 4179	. . . . . . . . . 10 ⊢ (L‘𝑥) ⊆ ((L‘𝑥) ∪ (R‘𝑥))
46		ssun1 4179	. . . . . . . . . 10 ⊢ (L‘𝑦) ⊆ ((L‘𝑦) ∪ (R‘𝑦))
47		ssun1 4179	. . . . . . . . . 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 28578	. . . . . . . . 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 2829	. . . . . . . 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 4180	. . . . . . . . . 10 ⊢ (R‘𝑥) ⊆ ((L‘𝑥) ∪ (R‘𝑥))
58		ssun2 4180	. . . . . . . . . 10 ⊢ (R‘𝑦) ⊆ ((L‘𝑦) ∪ (R‘𝑦))
59	42, 43, 44, 57, 58, 47, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28578	. . . . . . . . 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 2829	. . . . . . . 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 4171	. . . . . . 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 4180	. . . . . . . . . 10 ⊢ (R‘𝑧) ⊆ ((L‘𝑧) ∪ (R‘𝑧))
63	42, 43, 44, 45, 58, 62, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 28578	. . . . . . . . 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 2829	. . . . . . . 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 28578	. . . . . . . . 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 2829	. . . . . . . 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 4171	. . . . . . 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 4171	. . . . . 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 4176	. . . . . . 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 4160	. . . . . . . 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 4167	. . . . . . 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 2786	. . . . . 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 2814	. . . . 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 28578	. . . . . . . . 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 2829	. . . . . . . 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 28578	. . . . . . . . 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 2829	. . . . . . . 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 4171	. . . . . . 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 28578	. . . . . . . . 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 2829	. . . . . . . 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 28578	. . . . . . . . 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 2829	. . . . . . . 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 4171	. . . . . . 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 4171	. . . . . 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 4176	. . . . . . 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 4160	. . . . . . . 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 4167	. . . . . . 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 2786	. . . . . 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 2814	. . . . 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 7458	. . . 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 28576	. . . 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 28577	. . . 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 2808	. . 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 28371	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 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  ‘cfv 6555  (class class class)co 7440  Nocsur 28024   |s ccuts 28172  Lcleft 28238  Rcright 28239   +_s cadds 28372   −_s csubs 28433   ·_s cmuls 28519

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-nadd 8700  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-muls 28520

This theorem is used by:  mulsassd  28580  zsoring  28822