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Theorem omordi 8587
Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. Lemma 3.15 of [Schloeder] p. 9. (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 6396 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 411 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
3 eleq2 2814 . . . . . . . . . 10 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
4 oveq2 7427 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
54eleq2d 2811 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
63, 5imbi12d 343 . . . . . . . . 9 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
7 eleq2 2814 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 7427 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
98eleq2d 2811 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
107, 9imbi12d 343 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
11 eleq2 2814 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
12 oveq2 7427 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
1312eleq2d 2811 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
1411, 13imbi12d 343 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
15 eleq2 2814 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
16 oveq2 7427 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
1716eleq2d 2811 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
1815, 17imbi12d 343 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
19 noel 4330 . . . . . . . . . . 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 6438 . . . . . . . . . . . . . . 15 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
23 omcl 8557 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o 𝑦) ∈ On)
24 simpl 481 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
2523, 24jca 510 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On))
26 oaword1 8573 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
2726sseld 3975 . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3029adantrl 714 . . . . . . . . . . . . . . . . 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 8572 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3231biimpa 475 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
33 oveq2 7427 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
3433eleq1d 2810 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3532, 34syl5ibrcom 246 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3635adantrr 715 . . . . . . . . . . . . . . . . 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 857 . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . 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 8547 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
4140eleq2d 2811 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4241adantr 479 . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . 13 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
4443exp43 435 . . . . . . . . . . . 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 489 . . . . . . . . . 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 409 . . . . . . . . 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 745 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50 limsuc 7854 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
5150biimpa 475 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
52 oveq2 7427 . . . . . . . . . . . . . . . . . . 19 (𝑦 = suc 𝐴 → (𝐶 ·o 𝑦) = (𝐶 ·o suc 𝐴))
5352ssiun2s 5052 . . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·o 𝑦))
56 vex 3465 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
57 omlim 8554 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
5856, 57mpanr1 701 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
5958adantr 479 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o 𝑥) = 𝑦𝑥 (𝐶 ·o 𝑦))
6055, 59sseqtrrd 4018 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
6149, 60sylan 578 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
62 omcl 8557 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
63 oaord1 8572 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·o 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6462, 63sylan 578 . . . . . . . . . . . . . . . . . . 19 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6564anabss1 664 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
6665biimpa 475 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶))
67 omsuc 8547 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
6867adantr 479 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
6966, 68eleqtrrd 2828 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7069adantrl 714 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7170adantr 479 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
7261, 71sseldd 3977 . . . . . . . . . . . . 13 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))
7372exp53 446 . . . . . . . . . . . 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 422 . . . . . . . . . 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 7870 . . . . . . . 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 430 . . . . . 6 (𝐵 ∈ On → (𝐴𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8079exp4a 430 . . . . 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 416 1 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  wss 3944  c0 4322   ciun 4997  Oncon0 6371  Lim wlim 6372  suc csuc 6373  (class class class)co 7419   +o coa 8484   ·o comu 8485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-oadd 8491  df-omul 8492
This theorem is referenced by:  omord2  8588  omcan  8590  odi  8600  omass  8601  oen0  8607  oeordi  8608  oeordsuc  8615  onexoegt  42819  omord2i  42877  oaomoencom  42893  cantnftermord  42896  omcl2  42909  omltoe  42984
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