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Theorem nnmordi 7865
Description: Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnmordi (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Proof of Theorem nnmordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 7222 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21expcom 398 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴 ∈ ω))
3 eleq2 2839 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
4 oveq2 6801 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝐵))
54eleq2d 2836 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
63, 5imbi12d 333 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
76imbi2d 329 . . . . . . . . 9 (𝑥 = 𝐵 → ((((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))) ↔ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
8 eleq2 2839 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
9 oveq2 6801 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 ∅))
109eleq2d 2836 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
118, 10imbi12d 333 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))))
12 eleq2 2839 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
13 oveq2 6801 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝑦))
1413eleq2d 2836 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))
1512, 14imbi12d 333 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))))
16 eleq2 2839 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
17 oveq2 6801 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 suc 𝑦))
1817eleq2d 2836 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
1916, 18imbi12d 333 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))
20 noel 4067 . . . . . . . . . . . 12 ¬ 𝐴 ∈ ∅
2120pm2.21i 117 . . . . . . . . . . 11 (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))
2221a1i 11 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
23 elsuci 5934 . . . . . . . . . . . . . . . 16 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
24 nnmcl 7846 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 𝑦) ∈ ω)
25 simpl 468 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
2624, 25jca 501 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27 nnaword1 7863 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝑦) ⊆ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
2827sseld 3751 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
2928imim2d 57 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))))
3029imp 393 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3130adantrl 695 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
32 nna0 7838 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·𝑜 𝑦) ∈ ω → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
3332ad2antrr 705 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
34 nnaordi 7852 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ ω ∧ (𝐶 ·𝑜 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3534ancoms 455 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3635imp 393 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
3733, 36eqeltrrd 2851 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
38 oveq2 6801 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝑦))
3938eleq1d 2835 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶) ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4037, 39syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4140adantrr 696 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4231, 41jaod 848 . . . . . . . . . . . . . . . . 17 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → ((𝐴𝑦𝐴 = 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4326, 42sylan 569 . . . . . . . . . . . . . . . 16 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → ((𝐴𝑦𝐴 = 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4423, 43syl5 34 . . . . . . . . . . . . . . 15 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
45 nnmsuc 7841 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 suc 𝑦) = ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
4645eleq2d 2836 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4746adantr 466 . . . . . . . . . . . . . . 15 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4844, 47sylibrd 249 . . . . . . . . . . . . . 14 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
4948exp43 423 . . . . . . . . . . . . 13 (𝐶 ∈ ω → (𝑦 ∈ ω → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5049com12 32 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5150adantld 478 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5251impd 396 . . . . . . . . . 10 (𝑦 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))))
5311, 15, 19, 22, 52finds2 7241 . . . . . . . . 9 (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))
547, 53vtoclga 3423 . . . . . . . 8 (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5554com23 86 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5655exp4a 418 . . . . . 6 (𝐵 ∈ ω → (𝐴𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5756exp4a 418 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵 → (𝐴 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))))
582, 57mpdd 43 . . . 4 (𝐵 ∈ ω → (𝐴𝐵 → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5958com34 91 . . 3 (𝐵 ∈ ω → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6059com24 95 . 2 (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6160imp31 404 1 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  c0 4063  suc csuc 5868  (class class class)co 6793  ωcom 7212   +𝑜 coa 7710   ·𝑜 comu 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718
This theorem is referenced by:  nnmord  7866  mulclpi  9917
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