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Theorem addsdi 27539
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 7401 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦 +s 𝑧)))
2 oveq1 7401 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
3 oveq1 7401 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧) = (𝑥𝑂 ·s 𝑧))
42, 3oveq12d 7412 . . 3 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)))
51, 4eqeq12d 2748 . 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 7401 . . . 4 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
76oveq2d 7410 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)))
8 oveq2 7402 . . . 4 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
98oveq1d 7409 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)))
107, 9eqeq12d 2748 . 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 7402 . . . 4 (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
1211oveq2d 7410 . . 3 (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
13 oveq2 7402 . . . 4 (𝑧 = 𝑧𝑂 → (𝑥𝑂 ·s 𝑧) = (𝑥𝑂 ·s 𝑧𝑂))
1413oveq2d 7410 . . 3 (𝑧 = 𝑧𝑂 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
1512, 14eqeq12d 2748 . 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 7401 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
17 oveq1 7401 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
18 oveq1 7401 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑧𝑂) = (𝑥𝑂 ·s 𝑧𝑂))
1917, 18oveq12d 7412 . . 3 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
2016, 19eqeq12d 2748 . 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 7401 . . . 4 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
2221oveq2d 7410 . . 3 (𝑦 = 𝑦𝑂 → (𝑥 ·s (𝑦 +s 𝑧𝑂)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
23 oveq2 7402 . . . 4 (𝑦 = 𝑦𝑂 → (𝑥 ·s 𝑦) = (𝑥 ·s 𝑦𝑂))
2423oveq1d 7409 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧𝑂)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)))
2522, 24eqeq12d 2748 . 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 7410 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 ·s (𝑦𝑂 +s 𝑧𝑂)))
278oveq1d 7409 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝑧𝑂)))
2826, 27eqeq12d 2748 . 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 7410 . . 3 (𝑧 = 𝑧𝑂 → (𝑥 ·s (𝑦𝑂 +s 𝑧)) = (𝑥 ·s (𝑦𝑂 +s 𝑧𝑂)))
30 oveq2 7402 . . . 4 (𝑧 = 𝑧𝑂 → (𝑥 ·s 𝑧) = (𝑥 ·s 𝑧𝑂))
3130oveq2d 7410 . . 3 (𝑧 = 𝑧𝑂 → ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧)) = ((𝑥 ·s 𝑦𝑂) +s (𝑥 ·s 𝑧𝑂)))
3229, 31eqeq12d 2748 . 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 7401 . . 3 (𝑥 = 𝐴 → (𝑥 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝑦 +s 𝑧)))
34 oveq1 7401 . . . 4 (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
35 oveq1 7401 . . . 4 (𝑥 = 𝐴 → (𝑥 ·s 𝑧) = (𝐴 ·s 𝑧))
3634, 35oveq12d 7412 . . 3 (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) +s (𝑥 ·s 𝑧)) = ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)))
3733, 36eqeq12d 2748 . 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 7401 . . . 4 (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
3938oveq2d 7410 . . 3 (𝑦 = 𝐵 → (𝐴 ·s (𝑦 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝑧)))
40 oveq2 7402 . . . 4 (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
4140oveq1d 7409 . . 3 (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) +s (𝐴 ·s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)))
4239, 41eqeq12d 2748 . 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 7402 . . . 4 (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
4443oveq2d 7410 . . 3 (𝑧 = 𝐶 → (𝐴 ·s (𝐵 +s 𝑧)) = (𝐴 ·s (𝐵 +s 𝐶)))
45 oveq2 7402 . . . 4 (𝑧 = 𝐶 → (𝐴 ·s 𝑧) = (𝐴 ·s 𝐶))
4645oveq2d 7410 . . 3 (𝑧 = 𝐶 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
4744, 46eqeq12d 2748 . 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 4173 . . . . . . . . . . . . 13 (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
5554adantr 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
56 elun1 4173 . . . . . . . . . . . . 13 (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
5756adantl 482 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
5848, 49, 50, 51, 52, 53, 55, 57addsdilem3 27537 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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)))
6454adantr 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
65 elun1 4173 . . . . . . . . . . . . 13 (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
6665adantl 482 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
6748, 49, 50, 51, 62, 63, 64, 66addsdilem4 27538 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 4161 . . . . . . 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 4174 . . . . . . . . . . . . 13 (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
7372adantr 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
74 elun2 4174 . . . . . . . . . . . . 13 (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
7574adantl 482 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
7648, 49, 50, 51, 52, 53, 73, 75addsdilem3 27537 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
81 elun2 4174 . . . . . . . . . . . . 13 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
8281adantl 482 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
8348, 49, 50, 51, 62, 63, 80, 82addsdilem4 27538 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 4161 . . . . . . 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 4161 . . . . . 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 4166 . . . . . 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 2788 . . . . 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 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
9274adantl 482 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
9348, 49, 50, 51, 52, 53, 91, 92addsdilem3 27537 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
9881adantl 482 . . . . . . . . . . . 12 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
9948, 49, 50, 51, 62, 63, 97, 98addsdilem4 27538 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 4161 . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
10556adantl 482 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
10648, 49, 50, 51, 52, 53, 104, 105addsdilem3 27537 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
11165adantl 482 . . . . . . . . . . . 12 ((𝑥𝑅 ∈ ( R ‘𝑥) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
11248, 49, 50, 51, 62, 63, 110, 111addsdilem4 27538 . . . . . . . . . . 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 2743 . . . . . . . . . 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 3217 . . . . . . . . 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 2801 . . . . . . . 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 4161 . . . . . . 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 4161 . . . . . 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 4166 . . . . . 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 2788 . . . . 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 7412 . . . 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 27535 . . . 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 27536 . . . 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 2782 . . 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 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))))
1255, 10, 15, 20, 25, 28, 32, 37, 42, 47, 124no3inds 27371 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ∪ cun 3943  ‘cfv 6533  (class class class)co 7394   No csur 27072   |s cscut 27213   L cleft 27269   R cright 27270   +s cadds 27372   -s csubs 27424   ·s cmuls 27491
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 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492
This theorem is referenced by:  addsdid  27540
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