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Theorem omopth2 7649
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 1112 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐵 ∈ On)
2 eloni 5721 . . . . . . 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 1114 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐷 ∈ On)
5 eloni 5721 . . . . . . 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 5743 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐷) → (𝐵𝐷𝐵 = 𝐷𝐷𝐵))
83, 6, 7syl2anc 692 . . . . 5 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵𝐷𝐵 = 𝐷𝐷𝐵))
9 simpr 477 . . . . . . . . 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 1110 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐴 ∈ On)
11 omcl 7601 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·𝑜 𝐷) ∈ On)
1210, 4, 11syl2anc 692 . . . . . . . . . . 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 1115 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐸𝐴)
14 onelon 5736 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐸𝐴) → 𝐸 ∈ On)
1510, 13, 14syl2anc 692 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐸 ∈ On)
16 oacl 7600 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ On)
1712, 15, 16syl2anc 692 . . . . . . . . . 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 5721 . . . . . . . . . 10 (((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ On → Ord ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
19 ordirr 5729 . . . . . . . . . 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 2718 . . . . . . . 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 400 . . . . . . . . 9 (𝐵𝐷 → (𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)))
23 omeulem2 7648 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2423adantr 481 . . . . . . . . 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 118 . . . . . 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 2721 . . . . . . . 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 400 . . . . . . . . 9 (𝐷𝐵 → (𝐷𝐵 ∨ (𝐷 = 𝐵𝐸𝐶)))
31 simpl1r 1111 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐴 ≠ ∅)
32 simpl2r 1113 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐶𝐴)
33 omeulem2 7648 . . . . . . . . . 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 1340 . . . . . . . . 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 118 . . . . . 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 1390 . . . . 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 5736 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐶𝐴) → 𝐶 ∈ On)
41 eloni 5721 . . . . . . . 8 (𝐶 ∈ On → Ord 𝐶)
4240, 41syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐶𝐴) → Ord 𝐶)
4310, 32, 42syl2anc 692 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → Ord 𝐶)
44 eloni 5721 . . . . . . . 8 (𝐸 ∈ On → Ord 𝐸)
4514, 44syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐸𝐴) → Ord 𝐸)
4610, 13, 45syl2anc 692 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → Ord 𝐸)
47 ordtri3or 5743 . . . . . 6 ((Ord 𝐶 ∧ Ord 𝐸) → (𝐶𝐸𝐶 = 𝐸𝐸𝐶))
4843, 46, 47syl2anc 692 . . . . 5 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐶𝐸𝐶 = 𝐸𝐸𝐶))
49 olc 399 . . . . . . . . . 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 710 . . . . . . . 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 118 . . . . . 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 2626 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → 𝐷 = 𝐵)
56 olc 399 . . . . . . . . . 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 710 . . . . . . . 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 118 . . . . . 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 1390 . . . . 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 554 . . 3 ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) ∧ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐵 = 𝐷𝐶 = 𝐸))
6463ex 450 . 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 6643 . . 3 (𝐵 = 𝐷 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐷))
66 id 22 . . 3 (𝐶 = 𝐸𝐶 = 𝐸)
6765, 66oveqan12d 6654 . 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  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1481  wcel 1988  wne 2791  c0 3907  Ord word 5710  Oncon0 5711  (class class class)co 6635   +𝑜 coa 7542   ·𝑜 comu 7543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-omul 7550
This theorem is referenced by:  omeu  7650  dfac12lem2  8951
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