Theorem addsdi 27517

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 7397	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦 +s 𝑧)))
2		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
3		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧) = (𝑥𝑂 ·s 𝑧))
4	2, 3	oveq12d 7408	. . 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 7397	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
7	6	oveq2d 7406	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)))
8		oveq2 7398	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
9	8	oveq1d 7405	. . 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 7398	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
12	11	oveq2d 7406	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
13		oveq2 7398	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s 𝑧) = (𝑥𝑂 ·s 𝑧𝑂))
14	13	oveq2d 7406	. . 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 7397	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
17		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
18		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧𝑂) = (𝑥𝑂 ·s 𝑧𝑂))
19	17, 18	oveq12d 7408	. . 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 7397	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
22	21	oveq2d 7406	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 +s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
23		oveq2 7398	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
24	23	oveq1d 7405	. . 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 7406	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
27	8	oveq1d 7405	. . 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 7406	. . 3 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
30		oveq2 7398	. . . 4 ⊢ (𝑧 = 𝑧𝑂 → (𝑥 ·s 𝑧) = (𝑥 ·s 𝑧𝑂))
31	30	oveq2d 7406	. . 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 7397	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝑦 +s 𝑧)))
34		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
35		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 𝑧) = (𝐴 ·s 𝑧))
36	34, 35	oveq12d 7408	. . 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 7397	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
39	38	oveq2d 7406	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝑧)))
40		oveq2 7398	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
41	40	oveq1d 7405	. . 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 7398	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
44	43	oveq2d 7406	. . 3 ⊢ (𝑧 = 𝐶 → (𝐴 ·s (𝐵 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝐶)))
45		oveq2 7398	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴 ·s 𝑧) = (𝐴 ·s 𝐶))
46	45	oveq2d 7406	. . 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 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)))) → 𝑥 ∈ No )
49		simpl2 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)))) → 𝑦 ∈ No )
50		simpl3 1193	. . . . . . . . . . . 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 1260	. . . . . . . . . . . 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 1262	. . . . . . . . . . . 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 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))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)))
54		elun1 4169	. . . . . . . . . . . . 13 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
55	54	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
56		elun1 4169	. . . . . . . . . . . . 13 ⊢ (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
57	56	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
58	48, 49, 50, 51, 52, 53, 55, 57	addsdilem3 27515	. . . . . . . . . . 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 1196	. . . . . . . . . . . 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 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))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)))
64	54	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
65		elun1 4169	. . . . . . . . . . . . 13 ⊢ (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
66	65	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
67	48, 49, 50, 51, 62, 63, 64, 66	addsdilem4 27516	. . . . . . . . . . 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 4157	. . . . . . 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 4170	. . . . . . . . . . . . 13 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
73	72	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
74		elun2 4170	. . . . . . . . . . . . 13 ⊢ (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
75	74	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
76	48, 49, 50, 51, 52, 53, 73, 75	addsdilem3 27515	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
81		elun2 4170	. . . . . . . . . . . . 13 ⊢ (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
82	81	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
83	48, 49, 50, 51, 62, 63, 80, 82	addsdilem4 27516	. . . . . . . . . . 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 4157	. . . . . . 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 4157	. . . . . 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 4162	. . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
92	74	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
93	48, 49, 50, 51, 52, 53, 91, 92	addsdilem3 27515	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
98	81	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
99	48, 49, 50, 51, 62, 63, 97, 98	addsdilem4 27516	. . . . . . . . . . 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 4157	. . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
105	56	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
106	48, 49, 50, 51, 52, 53, 104, 105	addsdilem3 27515	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
111	65	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
112	48, 49, 50, 51, 62, 63, 110, 111	addsdilem4 27516	. . . . . . . . . . 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 4157	. . . . . . 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 4157	. . . . . 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 4162	. . . . . 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 7408	. . . 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 27513	. . . 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 27514	. . . 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 413	. 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 27353	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2708  ∀wral 3060  ∃wrex 3069   ∪ cun 3939  ‘cfv 6529  (class class class)co 7390   No csur 27065   |s cscut 27205   L cleft 27258   R cright 27259   +_s cadds 27354   -_s csubs 27406   ·_s cmuls 27471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 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 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-tp 4624  df-op 4626  df-ot 4628  df-uni 4899  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 6286  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-riota 7346  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7954  df-2nd 7955  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-1o 8445  df-2o 8446  df-nadd 8645  df-no 27068  df-slt 27069  df-bday 27070  df-sle 27170  df-sslt 27204  df-scut 27206  df-0s 27246  df-made 27260  df-old 27261  df-left 27263  df-right 27264  df-norec 27333  df-norec2 27344  df-adds 27355  df-negs 27407  df-subs 27408  df-muls 27472

This theorem is referenced by:  addsdid  27518