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Theorem omordi 8182
 Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
Assertion
Ref Expression
omordi (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem omordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 6203 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 416 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
3 eleq2 2904 . . . . . . . . . 10 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
4 oveq2 7153 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
54eleq2d 2901 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
63, 5imbi12d 348 . . . . . . . . 9 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
7 eleq2 2904 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 7153 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
98eleq2d 2901 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
107, 9imbi12d 348 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
11 eleq2 2904 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
12 oveq2 7153 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
1312eleq2d 2901 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
1411, 13imbi12d 348 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
15 eleq2 2904 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
16 oveq2 7153 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
1716eleq2d 2901 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
1815, 17imbi12d 348 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
19 noel 4279 . . . . . . . . . . 11 ¬ 𝐴 ∈ ∅
2019pm2.21i 119 . . . . . . . . . 10 (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
2120a1i 11 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
22 elsuci 6244 . . . . . . . . . . . . . . 15 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
23 omcl 8151 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o 𝑦) ∈ On)
24 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
2523, 24jca 515 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On))
26 oaword1 8168 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
2726sseld 3951 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
2827imim2d 57 . . . . . . . . . . . . . . . . . . 19 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶))))
2928imp 410 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3029adantrl 715 . . . . . . . . . . . . . . . . 17 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
31 oaord1 8167 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3231biimpa 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
33 oveq2 7153 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
3433eleq1d 2900 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3532, 34syl5ibrcom 250 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3635adantrr 716 . . . . . . . . . . . . . . . . 17 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3730, 36jaod 856 . . . . . . . . . . . . . . . 16 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴𝑦𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3825, 37sylan 583 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴𝑦𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3922, 38syl5 34 . . . . . . . . . . . . . 14 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
40 omsuc 8141 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
4140eleq2d 2901 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4241adantr 484 . . . . . . . . . . . . . 14 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4339, 42sylibrd 262 . . . . . . . . . . . . 13 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
4443exp43 440 . . . . . . . . . . . 12 (𝐶 ∈ On → (𝑦 ∈ On → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
4544com12 32 . . . . . . . . . . 11 (𝑦 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
4645adantld 494 . . . . . . . . . 10 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
4746impd 414 . . . . . . . . 9 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))))
48 id 22 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ∈ On ∧ Lim 𝑥))
4948ad2ant2r 746 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50 limsuc 7554 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
5150biimpa 480 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
52 oveq2 7153 . . . . . . . . . . . . . . . . . . 19 (𝑦 = suc 𝐴 → (𝐶 ·o 𝑦) = (𝐶 ·o suc 𝐴))
5352ssiun2s 4958 . . . . . . . . . . . . . . . . . 18 (suc 𝐴𝑥 → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
5451, 53syl 17 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
5554adantll 713 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
56 vex 3483 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
57 omlim 8148 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
5856, 57mpanr1 702 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
5958adantr 484 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
6055, 59sseqtrrd 3993 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
6149, 60sylan 583 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
62 omcl 8151 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
63 oaord1 8167 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6462, 63sylan 583 . . . . . . . . . . . . . . . . . . 19 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6564anabss1 665 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6665biimpa 480 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶))
67 omsuc 8141 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
6867adantr 484 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
6966, 68eleqtrrd 2919 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7069adantrl 715 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7170adantr 484 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7261, 71sseldd 3953 . . . . . . . . . . . . 13 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))
7372exp53 451 . . . . . . . . . . . 12 (𝐶 ∈ On → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐶 → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
7473com13 88 . . . . . . . . . . 11 (Lim 𝑥 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
7574imp4c 427 . . . . . . . . . 10 (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
7675a1dd 50 . . . . . . . . 9 (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (∀𝑦𝑥 (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)))))
776, 10, 14, 18, 21, 47, 76tfinds3 7569 . . . . . . . 8 (𝐵 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
7877com23 86 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
7978exp4a 435 . . . . . 6 (𝐵 ∈ On → (𝐴𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8079exp4a 435 . . . . 5 (𝐵 ∈ On → (𝐴𝐵 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))))
812, 80mpdd 43 . . . 4 (𝐵 ∈ On → (𝐴𝐵 → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8281com34 91 . . 3 (𝐵 ∈ On → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ On → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8382com24 95 . 2 (𝐵 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8483imp31 421 1 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ⊆ wss 3919  ∅c0 4275  ∪ ciun 4905  Oncon0 6178  Lim wlim 6179  suc csuc 6180  (class class class)co 7145   +o coa 8089   ·o comu 8090 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-oadd 8096  df-omul 8097 This theorem is referenced by:  omord2  8183  omcan  8185  odi  8195  omass  8196  oen0  8202  oeordi  8203  oeordsuc  8210
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