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Theorem omopth2 7818
Description: An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
omopth2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))

Proof of Theorem omopth2
StepHypRef Expression
1 simpl2l 1282 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐵 ∈ On)
2 eloni 5876 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
31, 2syl 17 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → Ord 𝐵)
4 simpl3l 1286 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐷 ∈ On)
5 eloni 5876 . . . . . . 7 (𝐷 ∈ On → Ord 𝐷)
64, 5syl 17 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → Ord 𝐷)
7 ordtri3or 5898 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐷) → (𝐵𝐷𝐵 = 𝐷𝐷𝐵))
83, 6, 7syl2anc 573 . . . . 5 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵𝐷𝐵 = 𝐷𝐷𝐵))
9 simpr 471 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
10 simpl1l 1278 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐴 ∈ On)
11 omcl 7770 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·𝑜 𝐷) ∈ On)
1210, 4, 11syl2anc 573 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐴 ·𝑜 𝐷) ∈ On)
13 simpl3r 1288 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐸𝐴)
14 onelon 5891 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐸𝐴) → 𝐸 ∈ On)
1510, 13, 14syl2anc 573 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐸 ∈ On)
16 oacl 7769 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ On)
1712, 15, 16syl2anc 573 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ On)
18 eloni 5876 . . . . . . . . . 10 (((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ On → Ord ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
19 ordirr 5884 . . . . . . . . . 10 (Ord ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) → ¬ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
2017, 18, 193syl 18 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
219, 20eqneltrd 2869 . . . . . . . 8 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
22 orc 854 . . . . . . . . 9 (𝐵𝐷 → (𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)))
23 omeulem2 7817 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2423adantr 466 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2522, 24syl5 34 . . . . . . . 8 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵𝐷 → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2621, 25mtod 189 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ 𝐵𝐷)
2726pm2.21d 119 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵𝐷𝐵 = 𝐷))
28 idd 24 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵 = 𝐷𝐵 = 𝐷))
2920, 9neleqtrrd 2872 . . . . . . . 8 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶))
30 orc 854 . . . . . . . . 9 (𝐷𝐵 → (𝐷𝐵 ∨ (𝐷 = 𝐵𝐸𝐶)))
31 simpl1r 1280 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐴 ≠ ∅)
32 simpl2r 1284 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐶𝐴)
33 omeulem2 7817 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐷 ∈ On ∧ 𝐸𝐴) ∧ (𝐵 ∈ On ∧ 𝐶𝐴)) → ((𝐷𝐵 ∨ (𝐷 = 𝐵𝐸𝐶)) → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶)))
3410, 31, 4, 13, 1, 32, 33syl222anc 1492 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐷𝐵 ∨ (𝐷 = 𝐵𝐸𝐶)) → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶)))
3530, 34syl5 34 . . . . . . . 8 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐷𝐵 → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶)))
3629, 35mtod 189 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ 𝐷𝐵)
3736pm2.21d 119 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐷𝐵𝐵 = 𝐷))
3827, 28, 373jaod 1540 . . . . 5 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐵𝐷𝐵 = 𝐷𝐷𝐵) → 𝐵 = 𝐷))
398, 38mpd 15 . . . 4 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐵 = 𝐷)
40 onelon 5891 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐶𝐴) → 𝐶 ∈ On)
41 eloni 5876 . . . . . . . 8 (𝐶 ∈ On → Ord 𝐶)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶𝐴) → Ord 𝐶)
4310, 32, 42syl2anc 573 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → Ord 𝐶)
44 eloni 5876 . . . . . . . 8 (𝐸 ∈ On → Ord 𝐸)
4514, 44syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐸𝐴) → Ord 𝐸)
4610, 13, 45syl2anc 573 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → Ord 𝐸)
47 ordtri3or 5898 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐸) → (𝐶𝐸𝐶 = 𝐸𝐸𝐶))
4843, 46, 47syl2anc 573 . . . . 5 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐶𝐸𝐶 = 𝐸𝐸𝐶))
49 olc 855 . . . . . . . . . 10 ((𝐵 = 𝐷𝐶𝐸) → (𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)))
5049, 24syl5 34 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
5139, 50mpand 675 . . . . . . . 8 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐶𝐸 → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
5221, 51mtod 189 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ 𝐶𝐸)
5352pm2.21d 119 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐶𝐸𝐶 = 𝐸))
54 idd 24 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐶 = 𝐸𝐶 = 𝐸))
5539eqcomd 2777 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐷 = 𝐵)
56 olc 855 . . . . . . . . . 10 ((𝐷 = 𝐵𝐸𝐶) → (𝐷𝐵 ∨ (𝐷 = 𝐵𝐸𝐶)))
5756, 34syl5 34 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐷 = 𝐵𝐸𝐶) → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶)))
5855, 57mpand 675 . . . . . . . 8 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐸𝐶 → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶)))
5929, 58mtod 189 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ¬ 𝐸𝐶)
6059pm2.21d 119 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐸𝐶𝐶 = 𝐸))
6153, 54, 603jaod 1540 . . . . 5 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → ((𝐶𝐸𝐶 = 𝐸𝐸𝐶) → 𝐶 = 𝐸))
6248, 61mpd 15 . . . 4 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐶 = 𝐸)
6339, 62jca 501 . . 3 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵 = 𝐷𝐶 = 𝐸))
6463ex 397 . 2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) → (𝐵 = 𝐷𝐶 = 𝐸)))
65 oveq2 6801 . . 3 (𝐵 = 𝐷 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐷))
66 id 22 . . 3 (𝐶 = 𝐸𝐶 = 𝐸)
6765, 66oveqan12d 6812 . 2 ((𝐵 = 𝐷𝐶 = 𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
6864, 67impbid1 215 1 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 834  w3o 1070  w3a 1071   = wceq 1631  wcel 2145  wne 2943  c0 4063  Ord word 5865  Oncon0 5866  (class class class)co 6793   +𝑜 coa 7710   ·𝑜 comu 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718
This theorem is referenced by:  omeu  7819  dfac12lem2  9168
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