Theorem addsdi 27969

Description: Distributive law for surreal numbers. Commuted form of part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)

Assertion

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

Proof of Theorem addsdi

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 7419	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦 +s 𝑧)))
2		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
3		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧) = (𝑥𝑂 ·s 𝑧))
4	2, 3	oveq12d 7430	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)))
5	1, 4	eqeq12d 2747	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s (𝑦 +s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) ↔ (𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧))))
6		oveq1 7419	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
7	6	oveq2d 7428	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)))
8		oveq2 7420	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
9	8	oveq1d 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)))
10	7, 9	eqeq12d 2747	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ↔ (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧))))
11		oveq2 7420	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
12	11	oveq2d 7428	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
13		oveq2 7420	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s 𝑧) = (𝑥𝑂 ·s 𝑧𝑂))
14	13	oveq2d 7428	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
15	12, 14	eqeq12d 2747	. 2 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ↔ (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂))))
16		oveq1 7419	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
17		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
18		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧𝑂) = (𝑥𝑂 ·s 𝑧𝑂))
19	17, 18	oveq12d 7430	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
20	16, 19	eqeq12d 2747	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ↔ (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂))))
21		oveq1 7419	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
22	21	oveq2d 7428	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 +s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
23		oveq2 7420	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
24	23	oveq1d 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)))
25	22, 24	eqeq12d 2747	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)) ↔ (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂))))
26	21	oveq2d 7428	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
27	8	oveq1d 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
28	26, 27	eqeq12d 2747	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)) ↔ (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂))))
29	11	oveq2d 7428	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
30		oveq2 7420	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s 𝑧) = (𝑥 ·s 𝑧𝑂))
31	30	oveq2d 7428	. . 3 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)))
32	29, 31	eqeq12d 2747	. 2 ⊢ (𝑧 = 𝑧𝑂 → ((𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧)) ↔ (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂))))
33		oveq1 7419	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝑦 +s 𝑧)))
34		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
35		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑧) = (𝐴 ·s 𝑧))
36	34, 35	oveq12d 7430	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)))
37	33, 36	eqeq12d 2747	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s (𝑦 +s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) ↔ (𝐴 ·s (𝑦 +s 𝑧)) = ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧))))
38		oveq1 7419	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
39	38	oveq2d 7428	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝑧)))
40		oveq2 7420	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
41	40	oveq1d 7427	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)))
42	39, 41	eqeq12d 2747	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 ·s (𝑦 +s 𝑧)) = ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)) ↔ (𝐴 ·s (𝐵 +s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧))))
43		oveq2 7420	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
44	43	oveq2d 7428	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 ·s (𝐵 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝐶)))
45		oveq2 7420	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴 ·s 𝑧) = (𝐴 ·s 𝐶))
46	45	oveq2d 7428	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
47	44, 46	eqeq12d 2747	. 2 ⊢ (𝑧 = 𝐶 → ((𝐴 ·s (𝐵 +s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)) ↔ (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶))))
48		simpl1 1190	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → 𝑥 ∈ No )
49		simpl2 1191	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → 𝑦 ∈ No )
50		simpl3 1192	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → 𝑧 ∈ No )
51		simpr21 1259	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)))
52		simpr23 1261	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧)))
53		simpr12 1257	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)))
54		elun1 4176	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
55	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
56		elun1 4176	. . . . . . . . . . . . 13 ⊢ (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
57	56	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
58	48, 49, 50, 51, 52, 53, 55, 57	addsdilem3 27967	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦))) → (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧))) = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧)))
59	58	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦))) → (𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧))) ↔ 𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))))
60	59	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))))
61	60	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧)))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))})
62		simpr3 1195	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))
63		simpr13 1258	. . . . . . . . . . . 12 ⊢ (((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)))
64	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
65		elun1 4176	. . . . . . . . . . . . 13 ⊢ (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
66	65	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
67	48, 49, 50, 51, 62, 63, 64, 66	addsdilem4 27968	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧))) → (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿))) = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
68	67	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧))) → (𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿))) ↔ 𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))))
69	68	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))))
70	69	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿)))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))})
71	61, 70	uneq12d 4164	. . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿)))}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))}))
72		elun2 4177	. . . . . . . . . . . . 13 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
73	72	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
74		elun2 4177	. . . . . . . . . . . . 13 ⊢ (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
75	74	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
76	48, 49, 50, 51, 52, 53, 73, 75	addsdilem3 27967	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦))) → (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧))) = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧)))
77	76	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦))) → (𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧))) ↔ 𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))))
78	77	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))))
79	78	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧)))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))})
80	72	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
81		elun2 4177	. . . . . . . . . . . . 13 ⊢ (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
82	81	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
83	48, 49, 50, 51, 62, 63, 80, 82	addsdilem4 27968	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧))) → (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅))) = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
84	83	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧))) → (𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅))) ↔ 𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))))
85	84	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))))
86	85	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅)))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})
87	79, 86	uneq12d 4164	. . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅)))}) = ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))}))
88	71, 87	uneq12d 4164	. . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅)))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})))
89		un4 4169	. . . . . 6 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))}))
90	88, 89	eqtrdi 2787	. . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅)))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})))
91	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
92	74	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
93	48, 49, 50, 51, 52, 53, 91, 92	addsdilem3 27967	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦))) → (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧))) = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧)))
94	93	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦))) → (𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧))) ↔ 𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))))
95	94	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))))
96	95	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧)))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))})
97	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
98	81	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
99	48, 49, 50, 51, 62, 63, 97, 98	addsdilem4 27968	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧))) → (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅))) = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
100	99	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧))) → (𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅))) ↔ 𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))))
101	100	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))))
102	101	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅)))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))})
103	96, 102	uneq12d 4164	. . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅)))}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))}))
104	72	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
105	56	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
106	48, 49, 50, 51, 52, 53, 104, 105	addsdilem3 27967	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦))) → (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧))) = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧)))
107	106	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦))) → (𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧))) ↔ 𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))))
108	107	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))))
109	108	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧)))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))})
110	72	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
111	65	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
112	48, 49, 50, 51, 62, 63, 110, 111	addsdilem4 27968	. . . . . . . . . . 11 ⊢ ((((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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧))) → (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿))) = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
113	112	eqeq2d 2742	. . . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) ∧ (𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧))) → (𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿))) ↔ 𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))))
114	113	2rexbidva 3216	. . . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))))
115	114	abbidv 2800	. . . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿)))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))})
116	109, 115	uneq12d 4164	. . . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿)))}) = ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))
117	103, 116	uneq12d 4164	. . . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿)))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))})))
118		un4 4169	. . . . . 6 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))
119	117, 118	eqtrdi 2787	. . . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿)))})) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))})))
120	90, 119	oveq12d 7430	. . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿)))}))) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))))
121	48, 49, 50	addsdilem1 27965	. . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (𝑥 ·s (𝑦 +s 𝑧)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝑅 +s 𝑧))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = (((𝑥𝐿 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝑦 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦𝐿 +s 𝑧))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝑧)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = (((𝑥𝑅 ·s (𝑦 +s 𝑧)) +s (𝑥 ·s (𝑦 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝑦 +s 𝑧𝐿)))}))))
122	48, 49, 50	addsdilem2 27966	. . . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑦𝑅 ∈ ( R ‘𝑦)𝑎 = ((((𝑥𝐿 ·s 𝑦) +s (𝑥 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝑥 ·s 𝑧))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑦𝐿 ∈ ( L ‘𝑦)𝑎 = ((((𝑥𝑅 ·s 𝑦) +s (𝑥 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝑥 ·s 𝑧))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑧𝑅 ∈ ( R ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝐿 ·s 𝑧) +s (𝑥 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)∃𝑧𝐿 ∈ ( L ‘𝑧)𝑎 = ((𝑥 ·s 𝑦) +s (((𝑥𝑅 ·s 𝑧) +s (𝑥 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))))
123	120, 121, 122	3eqtr4d 2781	. . 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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)))) → (𝑥 ·s (𝑦 +s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)))
124	123	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 (𝑥𝑂 ·s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂))) → (𝑥 ·s (𝑦 +s 𝑧)) = ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧))))
125	5, 10, 15, 20, 25, 28, 32, 37, 42, 47, 124	no3inds 27789	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069   ∪ cun 3946  ‘cfv 6543  (class class class)co 7412   No csur 27487   |s cscut 27629   L cleft 27686   R cright 27687   +_s cadds 27790   -_s csubs 27847   ·_s cmuls 27920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-1o 8472  df-2o 8473  df-nadd 8671  df-no 27490  df-slt 27491  df-bday 27492  df-sle 27592  df-sslt 27628  df-scut 27630  df-0s 27671  df-made 27688  df-old 27689  df-left 27691  df-right 27692  df-norec 27769  df-norec2 27780  df-adds 27791  df-negs 27848  df-subs 27849  df-muls 27921

This theorem is referenced by:  addsdid  27970