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Theorem addsdi 28115
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 𝑥𝑂 𝑥𝐿 𝑥𝑅 𝑦 𝑦𝑂 𝑦𝐿 𝑦𝑅 𝑧 𝑧𝑂 𝑧𝐿 𝑧𝑅 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7417 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦 +s 𝑧)))
2 oveq1 7417 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
3 oveq1 7417 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧) = (𝑥𝑂 ·s 𝑧))
42, 3oveq12d 7428 . . 3 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)))
51, 4eqeq12d 2752 . 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 7417 . . . 4 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
76oveq2d 7426 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)))
8 oveq2 7418 . . . 4 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
98oveq1d 7425 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)))
107, 9eqeq12d 2752 . 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 7418 . . . 4 (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
1211oveq2d 7426 . . 3 (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
13 oveq2 7418 . . . 4 (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s 𝑧) = (𝑥𝑂 ·s 𝑧𝑂))
1413oveq2d 7426 . . 3 (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
1512, 14eqeq12d 2752 . 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 7417 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
17 oveq1 7417 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
18 oveq1 7417 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧𝑂) = (𝑥𝑂 ·s 𝑧𝑂))
1917, 18oveq12d 7428 . . 3 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
2016, 19eqeq12d 2752 . 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 7417 . . . 4 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
2221oveq2d 7426 . . 3 (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 +s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
23 oveq2 7418 . . . 4 (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
2423oveq1d 7425 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)))
2522, 24eqeq12d 2752 . 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))))
2621oveq2d 7426 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
278oveq1d 7425 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
2826, 27eqeq12d 2752 . 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))))
2911oveq2d 7426 . . 3 (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
30 oveq2 7418 . . . 4 (𝑧 = 𝑧𝑂 → (𝑥 ·s 𝑧) = (𝑥 ·s 𝑧𝑂))
3130oveq2d 7426 . . 3 (𝑧 = 𝑧𝑂 → ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)))
3229, 31eqeq12d 2752 . 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 7417 . . 3 (𝑥 = 𝐴 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝑦 +s 𝑧)))
34 oveq1 7417 . . . 4 (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
35 oveq1 7417 . . . 4 (𝑥 = 𝐴 → (𝑥 ·s 𝑧) = (𝐴 ·s 𝑧))
3634, 35oveq12d 7428 . . 3 (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)))
3733, 36eqeq12d 2752 . 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 7417 . . . 4 (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
3938oveq2d 7426 . . 3 (𝑦 = 𝐵 → (𝐴 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝑧)))
40 oveq2 7418 . . . 4 (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
4140oveq1d 7425 . . 3 (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)))
4239, 41eqeq12d 2752 . 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 7418 . . . 4 (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
4443oveq2d 7426 . . 3 (𝑧 = 𝐶 → (𝐴 ·s (𝐵 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝐶)))
45 oveq2 7418 . . . 4 (𝑧 = 𝐶 → (𝐴 ·s 𝑧) = (𝐴 ·s 𝐶))
4645oveq2d 7426 . . 3 (𝑧 = 𝐶 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
4744, 46eqeq12d 2752 . 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 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)))) → 𝑥 ∈ No )
49 simpl2 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)))) → 𝑦 ∈ No )
50 simpl3 1194 . . . . . . . . . . . 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 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)))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s (𝑦 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)))
52 simpr23 1263 . . . . . . . . . . . 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 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))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)))
54 elun1 4162 . . . . . . . . . . . . 13 (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
5554adantr 480 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
56 elun1 4162 . . . . . . . . . . . . 13 (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
5756adantl 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
5848, 49, 50, 51, 52, 53, 55, 57addsdilem3 28113 . . . . . . . . . . 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)))
5958eqeq2d 2747 . . . . . . . . . 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))))
60592rexbidva 3208 . . . . . . . . 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))))
6160abbidv 2802 . . . . . . . 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 1197 . . . . . . . . . . . 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 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))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))(𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)))
6454adantr 480 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
65 elun1 4162 . . . . . . . . . . . . 13 (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
6665adantl 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
6748, 49, 50, 51, 62, 63, 64, 66addsdilem4 28114 . . . . . . . . . . 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))))
6867eqeq2d 2747 . . . . . . . . . 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)))))
69682rexbidva 3208 . . . . . . . . 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)))))
7069abbidv 2802 . . . . . . . 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)))})
7161, 70uneq12d 4149 . . . . . . 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 4163 . . . . . . . . . . . . 13 (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
7372adantr 480 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
74 elun2 4163 . . . . . . . . . . . . 13 (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
7574adantl 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
7648, 49, 50, 51, 52, 53, 73, 75addsdilem3 28113 . . . . . . . . . . 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)))
7776eqeq2d 2747 . . . . . . . . . 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))))
78772rexbidva 3208 . . . . . . . . 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))))
7978abbidv 2802 . . . . . . . 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))})
8072adantr 480 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
81 elun2 4163 . . . . . . . . . . . . 13 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
8281adantl 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
8348, 49, 50, 51, 62, 63, 80, 82addsdilem4 28114 . . . . . . . . . . 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))))
8483eqeq2d 2747 . . . . . . . . . 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)))))
85842rexbidva 3208 . . . . . . . . 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)))))
8685abbidv 2802 . . . . . . . 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)))})
8779, 86uneq12d 4149 . . . . . . 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)))}))
8871, 87uneq12d 4149 . . . . . 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 4155 . . . . . 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)))}))
9088, 89eqtrdi 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)))})))
9154adantr 480 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
9274adantl 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
9348, 49, 50, 51, 52, 53, 91, 92addsdilem3 28113 . . . . . . . . . . 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)))
9493eqeq2d 2747 . . . . . . . . . 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))))
95942rexbidva 3208 . . . . . . . . 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))))
9695abbidv 2802 . . . . . . . 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))})
9754adantr 480 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
9881adantl 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
9948, 49, 50, 51, 62, 63, 97, 98addsdilem4 28114 . . . . . . . . . . 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))))
10099eqeq2d 2747 . . . . . . . . . 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)))))
1011002rexbidva 3208 . . . . . . . . 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)))))
102101abbidv 2802 . . . . . . . 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)))})
10396, 102uneq12d 4149 . . . . . . 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)))}))
10472adantr 480 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
10556adantl 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
10648, 49, 50, 51, 52, 53, 104, 105addsdilem3 28113 . . . . . . . . . . 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)))
107106eqeq2d 2747 . . . . . . . . . 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))))
1081072rexbidva 3208 . . . . . . . . 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))))
109108abbidv 2802 . . . . . . . 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))})
11072adantr 480 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
11165adantl 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
11248, 49, 50, 51, 62, 63, 110, 111addsdilem4 28114 . . . . . . . . . . 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))))
113112eqeq2d 2747 . . . . . . . . . 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)))))
1141132rexbidva 3208 . . . . . . . . 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)))))
115114abbidv 2802 . . . . . . . 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)))})
116109, 115uneq12d 4149 . . . . . . 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)))}))
117103, 116uneq12d 4149 . . . . . 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 4155 . . . . . 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)))}))
119117, 118eqtrdi 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)))})))
12090, 119oveq12d 7428 . . . 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)))}))))
12148, 49, 50addsdilem1 28111 . . . 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)))}))))
12248, 49, 50addsdilem2 28112 . . . 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)))}))))
123120, 121, 1223eqtr4d 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)))
124123ex 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))))
1255, 10, 15, 20, 25, 28, 32, 37, 42, 47, 124no3inds 27922 1 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061   ∪ cun 3929  ‘cfv 6536  (class class class)co 7410   No csur 27608   |s cscut 27751   L cleft 27810   R cright 27811   +s cadds 27923   -s csubs 27983   ·s cmuls 28066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec 27902  df-norec2 27913  df-adds 27924  df-negs 27984  df-subs 27985  df-muls 28067
This theorem is referenced by:  addsdid  28116
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