Theorem mulsass 27970

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 27943, addsdi 27959, mulsgt0 27948, 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 7408	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
2	1	oveq1d 7416	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦) ·s 𝑧))
3		oveq1 7408	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦 ·s 𝑧)))
4	2, 3	eqeq12d 2740	. 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 7409	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
6	5	oveq1d 7416	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧))
7		oveq1 7408	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑧) = (𝑦𝑂 ·s 𝑧))
8	7	oveq2d 7417	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)))
9	6, 8	eqeq12d 2740	. 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 7409	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
11		oveq2 7409	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 ·s 𝑧) = (𝑦𝑂 ·s 𝑧𝑂))
12	11	oveq2d 7417	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
13	10, 12	eqeq12d 2740	. 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 7408	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
15	14	oveq1d 7416	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
16		oveq1 7408	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
17	15, 16	eqeq12d 2740	. 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 7409	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
19	18	oveq1d 7416	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂))
20		oveq1 7408	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑧𝑂) = (𝑦𝑂 ·s 𝑧𝑂))
21	20	oveq2d 7417	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
22	19, 21	eqeq12d 2740	. 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 7416	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) ·s 𝑧𝑂) = ((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂))
24	20	oveq2d 7417	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 ·s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))
25	23, 24	eqeq12d 2740	. 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 7409	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 ·s 𝑦𝑂) ·s 𝑧) = ((𝑥 ·s 𝑦𝑂) ·s 𝑧𝑂))
27	11	oveq2d 7417	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 ·s 𝑧)) = (𝑥 ·s (𝑦𝑂 ·s 𝑧𝑂)))
28	26, 27	eqeq12d 2740	. 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 7408	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
30	29	oveq1d 7416	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) ·s 𝑧) = ((𝐴 ·s 𝑦) ·s 𝑧))
31		oveq1 7408	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s (𝑦 ·s 𝑧)) = (𝐴 ·s (𝑦 ·s 𝑧)))
32	30, 31	eqeq12d 2740	. 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 7409	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
34	33	oveq1d 7416	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) ·s 𝑧) = ((𝐴 ·s 𝐵) ·s 𝑧))
35		oveq1 7408	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ·s 𝑧) = (𝐵 ·s 𝑧))
36	35	oveq2d 7417	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ·s (𝑦 ·s 𝑧)) = (𝐴 ·s (𝐵 ·s 𝑧)))
37	34, 36	eqeq12d 2740	. 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 7409	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴 ·s 𝐵) ·s 𝑧) = ((𝐴 ·s 𝐵) ·s 𝐶))
39		oveq2 7409	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 ·s 𝑧) = (𝐵 ·s 𝐶))
40	39	oveq2d 7417	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 ·s (𝐵 ·s 𝑧)) = (𝐴 ·s (𝐵 ·s 𝐶)))
41	38, 40	eqeq12d 2740	. 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 1188	. . . . . . . . . 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 1189	. . . . . . . . . 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 1190	. . . . . . . . . 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 4164	. . . . . . . . . 10 ⊢ ( L ‘𝑥) ⊆ (( L ‘𝑥) ∪ ( R ‘𝑥))
46		ssun1 4164	. . . . . . . . . 10 ⊢ ( L ‘𝑦) ⊆ (( L ‘𝑦) ∪ ( R ‘𝑦))
47		ssun1 4164	. . . . . . . . . 10 ⊢ ( L ‘𝑧) ⊆ (( L ‘𝑧) ∪ ( R ‘𝑧))
48		simpr11 1254	. . . . . . . . . 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 1255	. . . . . . . . . 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 1256	. . . . . . . . . 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 1258	. . . . . . . . . 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 1257	. . . . . . . . . 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 1259	. . . . . . . . . 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 1193	. . . . . . . . . 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 27969	. . . . . . . . 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 2793	. . . . . . . 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 4165	. . . . . . . . . 10 ⊢ ( R ‘𝑥) ⊆ (( L ‘𝑥) ∪ ( R ‘𝑥))
58		ssun2 4165	. . . . . . . . . 10 ⊢ ( R ‘𝑦) ⊆ (( L ‘𝑦) ∪ ( R ‘𝑦))
59	42, 43, 44, 57, 58, 47, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 27969	. . . . . . . . 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 2793	. . . . . . . 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 4156	. . . . . . 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 4165	. . . . . . . . . 10 ⊢ ( R ‘𝑧) ⊆ (( L ‘𝑧) ∪ ( R ‘𝑧))
63	42, 43, 44, 45, 58, 62, 48, 49, 50, 51, 52, 53, 54	mulsasslem3 27969	. . . . . . . . 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 2793	. . . . . . . 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 27969	. . . . . . . . 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 2793	. . . . . . . 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 4156	. . . . . . 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 4156	. . . . . 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 4161	. . . . . . 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 4145	. . . . . . . 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 4152	. . . . . . 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 2752	. . . . . 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 2780	. . . . 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 27969	. . . . . . . . 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 2793	. . . . . . . 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 27969	. . . . . . . . 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 2793	. . . . . . . 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 4156	. . . . . . 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 27969	. . . . . . . . 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 2793	. . . . . . . 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 27969	. . . . . . . . 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 2793	. . . . . . . 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 4156	. . . . . . 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 4156	. . . . . 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 4161	. . . . . . 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 4145	. . . . . . . 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 4152	. . . . . . 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 2752	. . . . . 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 2780	. . . . 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 7419	. . . 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 27967	. . . 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 27968	. . . 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 2774	. . 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 412	. 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 27779	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2701  ∀wral 3053  ∃wrex 3062   ∪ cun 3938  ‘cfv 6533  (class class class)co 7401   No csur 27477   |s cscut 27619   L cleft 27676   R cright 27677   +_s cadds 27780   -_s csubs 27837   ·_s cmuls 27910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-ot 4629  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-2o 8462  df-nadd 8660  df-no 27480  df-slt 27481  df-bday 27482  df-sle 27582  df-sslt 27618  df-scut 27620  df-0s 27661  df-made 27678  df-old 27679  df-left 27681  df-right 27682  df-norec 27759  df-norec2 27770  df-adds 27781  df-negs 27838  df-subs 27839  df-muls 27911

This theorem is referenced by:  mulsassd  27971