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Theorem addsass 28053
Description: Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
Assertion
Ref Expression
addsass ((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶)))

Proof of Theorem addsass
Dummy variables 𝑥 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑥𝑂 𝑦𝑂 𝑧𝑂 𝑥𝐿 𝑦𝐿 𝑧𝐿 𝑥𝑅 𝑦𝑅 𝑧𝑅 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
21oveq1d 7446 . . 3 (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) +s 𝑧) = ((𝑥𝑂 +s 𝑦) +s 𝑧))
3 oveq1 7438 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 +s (𝑦 +s 𝑧)) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
42, 3eqeq12d 2751 . 2 (𝑥 = 𝑥𝑂 → (((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧))))
5 oveq2 7439 . . . 4 (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
65oveq1d 7446 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧))
7 oveq1 7438 . . . 4 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
87oveq2d 7447 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)))
96, 8eqeq12d 2751 . 2 (𝑦 = 𝑦𝑂 → (((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧))))
10 oveq2 7439 . . 3 (𝑧 = 𝑧𝑂 → ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
11 oveq2 7439 . . . 4 (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
1211oveq2d 7447 . . 3 (𝑧 = 𝑧𝑂 → (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
1310, 12eqeq12d 2751 . 2 (𝑧 = 𝑧𝑂 → (((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂))))
14 oveq1 7438 . . . 4 (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
1514oveq1d 7446 . . 3 (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
16 oveq1 7438 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
1715, 16eqeq12d 2751 . 2 (𝑥 = 𝑥𝑂 → (((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂))))
18 oveq2 7439 . . . 4 (𝑦 = 𝑦𝑂 → (𝑥 +s 𝑦) = (𝑥 +s 𝑦𝑂))
1918oveq1d 7446 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂))
20 oveq1 7438 . . . 4 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
2120oveq2d 7447 . . 3 (𝑦 = 𝑦𝑂 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)))
2219, 21eqeq12d 2751 . 2 (𝑦 = 𝑦𝑂 → (((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂))))
235oveq1d 7446 . . 3 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂))
2420oveq2d 7447 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s (𝑦 +s 𝑧𝑂)) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)))
2523, 24eqeq12d 2751 . 2 (𝑦 = 𝑦𝑂 → (((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂))))
26 oveq2 7439 . . 3 (𝑧 = 𝑧𝑂 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂))
2711oveq2d 7447 . . 3 (𝑧 = 𝑧𝑂 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)))
2826, 27eqeq12d 2751 . 2 (𝑧 = 𝑧𝑂 → (((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂))))
29 oveq1 7438 . . . 4 (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
3029oveq1d 7446 . . 3 (𝑥 = 𝐴 → ((𝑥 +s 𝑦) +s 𝑧) = ((𝐴 +s 𝑦) +s 𝑧))
31 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥 +s (𝑦 +s 𝑧)) = (𝐴 +s (𝑦 +s 𝑧)))
3230, 31eqeq12d 2751 . 2 (𝑥 = 𝐴 → (((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)) ↔ ((𝐴 +s 𝑦) +s 𝑧) = (𝐴 +s (𝑦 +s 𝑧))))
33 oveq2 7439 . . . 4 (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
3433oveq1d 7446 . . 3 (𝑦 = 𝐵 → ((𝐴 +s 𝑦) +s 𝑧) = ((𝐴 +s 𝐵) +s 𝑧))
35 oveq1 7438 . . . 4 (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
3635oveq2d 7447 . . 3 (𝑦 = 𝐵 → (𝐴 +s (𝑦 +s 𝑧)) = (𝐴 +s (𝐵 +s 𝑧)))
3734, 36eqeq12d 2751 . 2 (𝑦 = 𝐵 → (((𝐴 +s 𝑦) +s 𝑧) = (𝐴 +s (𝑦 +s 𝑧)) ↔ ((𝐴 +s 𝐵) +s 𝑧) = (𝐴 +s (𝐵 +s 𝑧))))
38 oveq2 7439 . . 3 (𝑧 = 𝐶 → ((𝐴 +s 𝐵) +s 𝑧) = ((𝐴 +s 𝐵) +s 𝐶))
39 oveq2 7439 . . . 4 (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
4039oveq2d 7447 . . 3 (𝑧 = 𝐶 → (𝐴 +s (𝐵 +s 𝑧)) = (𝐴 +s (𝐵 +s 𝐶)))
4138, 40eqeq12d 2751 . 2 (𝑧 = 𝐶 → (((𝐴 +s 𝐵) +s 𝑧) = (𝐴 +s (𝐵 +s 𝑧)) ↔ ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶))))
42 simp21 1205 . . . 4 (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
43 simp23 1207 . . . 4 (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)))
44 simp3 1137 . . . 4 (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))
4542, 43, 443jca 1127 . . 3 (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))))
46 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s 𝑦) = (𝑥𝐿 +s 𝑦))
4746oveq1d 7446 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝐿 +s 𝑦) +s 𝑧))
48 oveq1 7438 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝐿 +s (𝑦 +s 𝑧)))
4947, 48eqeq12d 2751 . . . . . . . . . . . 12 (𝑥𝑂 = 𝑥𝐿 → (((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝐿 +s 𝑦) +s 𝑧) = (𝑥𝐿 +s (𝑦 +s 𝑧))))
50 simplr1 1214 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
51 elun1 4192 . . . . . . . . . . . . 13 (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
5251adantl 481 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
5349, 50, 52rspcdva 3623 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → ((𝑥𝐿 +s 𝑦) +s 𝑧) = (𝑥𝐿 +s (𝑦 +s 𝑧)))
5453eqeq2d 2746 . . . . . . . . . 10 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧) ↔ 𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))))
5554rexbidva 3175 . . . . . . . . 9 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))))
5655abbidv 2806 . . . . . . . 8 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))})
57 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑦𝐿 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑦𝐿))
5857oveq1d 7446 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑦𝐿 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝐿) +s 𝑧))
59 oveq1 7438 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑦𝐿 → (𝑦𝑂 +s 𝑧) = (𝑦𝐿 +s 𝑧))
6059oveq2d 7447 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑦𝐿 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝐿 +s 𝑧)))
6158, 60eqeq12d 2751 . . . . . . . . . . . 12 (𝑦𝑂 = 𝑦𝐿 → (((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥 +s 𝑦𝐿) +s 𝑧) = (𝑥 +s (𝑦𝐿 +s 𝑧))))
62 simplr2 1215 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)))
63 elun1 4192 . . . . . . . . . . . . 13 (𝑦𝐿 ∈ ( L ‘𝑦) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
6463adantl 481 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → 𝑦𝐿 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
6561, 62, 64rspcdva 3623 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → ((𝑥 +s 𝑦𝐿) +s 𝑧) = (𝑥 +s (𝑦𝐿 +s 𝑧)))
6665eqeq2d 2746 . . . . . . . . . 10 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝐿 ∈ ( L ‘𝑦)) → (𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧) ↔ 𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))))
6766rexbidva 3175 . . . . . . . . 9 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧) ↔ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))))
6867abbidv 2806 . . . . . . . 8 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)} = {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))})
6956, 68uneq12d 4179 . . . . . . 7 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}))
70 oveq2 7439 . . . . . . . . . . . 12 (𝑧𝑂 = 𝑧𝐿 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦) +s 𝑧𝐿))
71 oveq2 7439 . . . . . . . . . . . . 13 (𝑧𝑂 = 𝑧𝐿 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝐿))
7271oveq2d 7447 . . . . . . . . . . . 12 (𝑧𝑂 = 𝑧𝐿 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦 +s 𝑧𝐿)))
7370, 72eqeq12d 2751 . . . . . . . . . . 11 (𝑧𝑂 = 𝑧𝐿 → (((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥 +s 𝑦) +s 𝑧𝐿) = (𝑥 +s (𝑦 +s 𝑧𝐿))))
74 simplr3 1216 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))
75 elun1 4192 . . . . . . . . . . . 12 (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
7675adantl 481 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
7773, 74, 76rspcdva 3623 . . . . . . . . . 10 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → ((𝑥 +s 𝑦) +s 𝑧𝐿) = (𝑥 +s (𝑦 +s 𝑧𝐿)))
7877eqeq2d 2746 . . . . . . . . 9 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝐿 ∈ ( L ‘𝑧)) → (𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿) ↔ 𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))))
7978rexbidva 3175 . . . . . . . 8 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿) ↔ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))))
8079abbidv 2806 . . . . . . 7 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)} = {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))})
8169, 80uneq12d 4179 . . . . . 6 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)}) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))}))
82 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s 𝑦) = (𝑥𝑅 +s 𝑦))
8382oveq1d 7446 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 +s 𝑦) +s 𝑧) = ((𝑥𝑅 +s 𝑦) +s 𝑧))
84 oveq1 7438 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 +s (𝑦 +s 𝑧)) = (𝑥𝑅 +s (𝑦 +s 𝑧)))
8583, 84eqeq12d 2751 . . . . . . . . . . . 12 (𝑥𝑂 = 𝑥𝑅 → (((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ↔ ((𝑥𝑅 +s 𝑦) +s 𝑧) = (𝑥𝑅 +s (𝑦 +s 𝑧))))
86 simplr1 1214 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)))
87 elun2 4193 . . . . . . . . . . . . 13 (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
8887adantl 481 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
8985, 86, 88rspcdva 3623 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → ((𝑥𝑅 +s 𝑦) +s 𝑧) = (𝑥𝑅 +s (𝑦 +s 𝑧)))
9089eqeq2d 2746 . . . . . . . . . 10 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧) ↔ 𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))))
9190rexbidva 3175 . . . . . . . . 9 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))))
9291abbidv 2806 . . . . . . . 8 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} = {𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))})
93 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑦𝑅 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑦𝑅))
9493oveq1d 7446 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑦𝑅 → ((𝑥 +s 𝑦𝑂) +s 𝑧) = ((𝑥 +s 𝑦𝑅) +s 𝑧))
95 oveq1 7438 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑦𝑅 → (𝑦𝑂 +s 𝑧) = (𝑦𝑅 +s 𝑧))
9695oveq2d 7447 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑦𝑅 → (𝑥 +s (𝑦𝑂 +s 𝑧)) = (𝑥 +s (𝑦𝑅 +s 𝑧)))
9794, 96eqeq12d 2751 . . . . . . . . . . . 12 (𝑦𝑂 = 𝑦𝑅 → (((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ↔ ((𝑥 +s 𝑦𝑅) +s 𝑧) = (𝑥 +s (𝑦𝑅 +s 𝑧))))
98 simplr2 1215 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)))
99 elun2 4193 . . . . . . . . . . . . 13 (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
10099adantl 481 . . . . . . . . . . . 12 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → 𝑦𝑅 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
10197, 98, 100rspcdva 3623 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → ((𝑥 +s 𝑦𝑅) +s 𝑧) = (𝑥 +s (𝑦𝑅 +s 𝑧)))
102101eqeq2d 2746 . . . . . . . . . 10 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑦𝑅 ∈ ( R ‘𝑦)) → (𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧) ↔ 𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))))
103102rexbidva 3175 . . . . . . . . 9 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧) ↔ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))))
104103abbidv 2806 . . . . . . . 8 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)} = {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))})
10592, 104uneq12d 4179 . . . . . . 7 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) = ({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}))
106 oveq2 7439 . . . . . . . . . . . 12 (𝑧𝑂 = 𝑧𝑅 → ((𝑥 +s 𝑦) +s 𝑧𝑂) = ((𝑥 +s 𝑦) +s 𝑧𝑅))
107 oveq2 7439 . . . . . . . . . . . . 13 (𝑧𝑂 = 𝑧𝑅 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝑅))
108107oveq2d 7447 . . . . . . . . . . . 12 (𝑧𝑂 = 𝑧𝑅 → (𝑥 +s (𝑦 +s 𝑧𝑂)) = (𝑥 +s (𝑦 +s 𝑧𝑅)))
109106, 108eqeq12d 2751 . . . . . . . . . . 11 (𝑧𝑂 = 𝑧𝑅 → (((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)) ↔ ((𝑥 +s 𝑦) +s 𝑧𝑅) = (𝑥 +s (𝑦 +s 𝑧𝑅))))
110 simplr3 1216 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))
111 elun2 4193 . . . . . . . . . . . 12 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
112111adantl 481 . . . . . . . . . . 11 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
113109, 110, 112rspcdva 3623 . . . . . . . . . 10 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → ((𝑥 +s 𝑦) +s 𝑧𝑅) = (𝑥 +s (𝑦 +s 𝑧𝑅)))
114113eqeq2d 2746 . . . . . . . . 9 ((((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) ∧ 𝑧𝑅 ∈ ( R ‘𝑧)) → (𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅) ↔ 𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))))
115114rexbidva 3175 . . . . . . . 8 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))))
116115abbidv 2806 . . . . . . 7 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)} = {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))})
117105, 116uneq12d 4179 . . . . . 6 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)}) = (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))}))
11881, 117oveq12d 7449 . . . . 5 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)})) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))})))
119 simpl1 1190 . . . . . 6 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → 𝑥 No )
120 simpl2 1191 . . . . . 6 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → 𝑦 No )
121 simpl3 1192 . . . . . 6 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → 𝑧 No )
122119, 120, 121addsasslem1 28051 . . . . 5 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ((𝑥 +s 𝑦) +s 𝑧) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = ((𝑥𝐿 +s 𝑦) +s 𝑧)} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = ((𝑥 +s 𝑦𝐿) +s 𝑧)}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = ((𝑥 +s 𝑦) +s 𝑧𝐿)}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = ((𝑥𝑅 +s 𝑦) +s 𝑧)} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = ((𝑥 +s 𝑦𝑅) +s 𝑧)}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = ((𝑥 +s 𝑦) +s 𝑧𝑅)})))
123119, 120, 121addsasslem2 28052 . . . . 5 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → (𝑥 +s (𝑦 +s 𝑧)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s (𝑦 +s 𝑧))} ∪ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑏 = (𝑥 +s (𝑦𝐿 +s 𝑧))}) ∪ {𝑐 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑐 = (𝑥 +s (𝑦 +s 𝑧𝐿))}) |s (({𝑑 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑑 = (𝑥𝑅 +s (𝑦 +s 𝑧))} ∪ {𝑒 ∣ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑒 = (𝑥 +s (𝑦𝑅 +s 𝑧))}) ∪ {𝑓 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑓 = (𝑥 +s (𝑦 +s 𝑧𝑅))})))
124118, 122, 1233eqtr4d 2785 . . . 4 (((𝑥 No 𝑦 No 𝑧 No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂)))) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧)))
125124ex 412 . . 3 ((𝑥 No 𝑦 No 𝑧 No ) → ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧))))
12645, 125syl5 34 . 2 ((𝑥 No 𝑦 No 𝑧 No ) → (((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥𝑂 +s 𝑦𝑂) +s 𝑧) = (𝑥𝑂 +s (𝑦𝑂 +s 𝑧)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥𝑂 +s 𝑦) +s 𝑧𝑂) = (𝑥𝑂 +s (𝑦 +s 𝑧𝑂))) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))((𝑥𝑂 +s 𝑦) +s 𝑧) = (𝑥𝑂 +s (𝑦 +s 𝑧)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦𝑂) +s 𝑧𝑂) = (𝑥 +s (𝑦𝑂 +s 𝑧𝑂)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))((𝑥 +s 𝑦𝑂) +s 𝑧) = (𝑥 +s (𝑦𝑂 +s 𝑧))) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧))((𝑥 +s 𝑦) +s 𝑧𝑂) = (𝑥 +s (𝑦 +s 𝑧𝑂))) → ((𝑥 +s 𝑦) +s 𝑧) = (𝑥 +s (𝑦 +s 𝑧))))
1274, 9, 13, 17, 22, 25, 28, 32, 37, 41, 126no3inds 28006 1 ((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cun 3961  cfv 6563  (class class class)co 7431   No csur 27699   |s cscut 27842   L cleft 27899   R cright 27900   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008
This theorem is referenced by:  addsassd  28054
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