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Theorem omopth2 7624
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 5702 . . . . . . 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 5702 . . . . . . 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 5724 . . . . . 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 7576 . . . . . . . . . . . 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 5717 . . . . . . . . . . . 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 7575 . . . . . . . . . . 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 5702 . . . . . . . . . 10 (((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ∈ On → Ord ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
19 ordirr 5710 . . . . . . . . . 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 2717 . . . . . . . 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 7623 . . . . . . . . . 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 2720 . . . . . . . 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 7623 . . . . . . . . . 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 1339 . . . . . . . . 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 1389 . . . . 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 5717 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐶𝐴) → 𝐶 ∈ On)
41 eloni 5702 . . . . . . . 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 5702 . . . . . . . 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 5724 . . . . . 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 2627 . . . . . . . . 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 1389 . . . . 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 6623 . . 3 (𝐵 = 𝐷 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐷))
66 id 22 . . 3 (𝐶 = 𝐸𝐶 = 𝐸)
6765, 66oveqan12d 6634 . 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 1480  wcel 1987  wne 2790  c0 3897  Ord word 5691  Oncon0 5692  (class class class)co 6615   +𝑜 coa 7517   ·𝑜 comu 7518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-omul 7525
This theorem is referenced by:  omeu  7625  dfac12lem2  8926
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