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Theorem omordi 7817
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 5909 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 449 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
3 eleq2 2828 . . . . . . . . . 10 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
4 oveq2 6822 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 ∅))
54eleq2d 2825 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
63, 5imbi12d 333 . . . . . . . . 9 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))))
7 eleq2 2828 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 6822 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝑦))
98eleq2d 2825 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))
107, 9imbi12d 333 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))))
11 eleq2 2828 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
12 oveq2 6822 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 suc 𝑦))
1312eleq2d 2825 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
1411, 13imbi12d 333 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))
15 eleq2 2828 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
16 oveq2 6822 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝐵))
1716eleq2d 2825 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
1815, 17imbi12d 333 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
19 noel 4062 . . . . . . . . . . 11 ¬ 𝐴 ∈ ∅
2019pm2.21i 116 . . . . . . . . . 10 (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))
2120a1i 11 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
22 elsuci 5952 . . . . . . . . . . . . . . 15 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
23 omcl 7787 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·𝑜 𝑦) ∈ On)
24 simpl 474 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
2523, 24jca 555 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On))
26 oaword1 7803 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·𝑜 𝑦) ⊆ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
2726sseld 3743 . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3029adantrl 754 . . . . . . . . . . . . . . . . 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 7802 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3231biimpa 502 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
33 oveq2 6822 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝑦))
3433eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶) ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3532, 34syl5ibrcom 237 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3635adantrr 755 . . . . . . . . . . . . . . . . 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 394 . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . 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 7777 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·𝑜 suc 𝑦) = ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
4140eleq2d 2825 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4241adantr 472 . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . 13 (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
4443exp43 641 . . . . . . . . . . . 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 484 . . . . . . . . . 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 446 . . . . . . . . 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 800 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50 limsuc 7215 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
5150biimpa 502 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴𝑥) → suc 𝐴𝑥)
52 oveq2 6822 . . . . . . . . . . . . . . . . . . 19 (𝑦 = suc 𝐴 → (𝐶 ·𝑜 𝑦) = (𝐶 ·𝑜 suc 𝐴))
5352ssiun2s 4716 . . . . . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ 𝑦𝑥 (𝐶 ·𝑜 𝑦))
56 vex 3343 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
57 omlim 7784 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·𝑜 𝑥) = 𝑦𝑥 (𝐶 ·𝑜 𝑦))
5856, 57mpanr1 721 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·𝑜 𝑥) = 𝑦𝑥 (𝐶 ·𝑜 𝑦))
5958adantr 472 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·𝑜 𝑥) = 𝑦𝑥 (𝐶 ·𝑜 𝑦))
6055, 59sseqtr4d 3783 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝑥))
6149, 60sylan 489 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝑥))
62 omcl 7787 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·𝑜 𝐴) ∈ On)
63 oaord1 7802 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶)))
6462, 63sylan 489 . . . . . . . . . . . . . . . . . . 19 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶)))
6564anabss1 890 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶)))
6665biimpa 502 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶))
67 omsuc 7777 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·𝑜 suc 𝐴) = ((𝐶 ·𝑜 𝐴) +𝑜 𝐶))
6867adantr 472 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 suc 𝐴) = ((𝐶 ·𝑜 𝐴) +𝑜 𝐶))
6966, 68eleqtrrd 2842 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝐴))
7069adantrl 754 . . . . . . . . . . . . . . 15 (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝐴))
7170adantr 472 . . . . . . . . . . . . . 14 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝐴))
7261, 71sseldd 3745 . . . . . . . . . . . . 13 ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴𝑥) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))
7372exp53 648 . . . . . . . . . . . 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 618 . . . . . . . . . 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 7230 . . . . . . . 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 634 . . . . . 6 (𝐵 ∈ On → (𝐴𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
8079exp4a 634 . . . . 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 447 1 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  c0 4058   ciun 4672  Oncon0 5884  Lim wlim 5885  suc csuc 5886  (class class class)co 6814   +𝑜 coa 7727   ·𝑜 comu 7728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-oadd 7734  df-omul 7735
This theorem is referenced by:  omord2  7818  omcan  7820  odi  7830  omass  7831  oen0  7837  oeordi  7838  oeordsuc  7845
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