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Theorem nnmordi 7473
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 6842 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21expcom 449 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴 ∈ ω))
3 eleq2 2581 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
4 oveq2 6433 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝐵))
54eleq2d 2577 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
63, 5imbi12d 332 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
76imbi2d 328 . . . . . . . . 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 2581 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
9 oveq2 6433 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 ∅))
109eleq2d 2577 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
118, 10imbi12d 332 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))))
12 eleq2 2581 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
13 oveq2 6433 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝑦))
1413eleq2d 2577 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))
1512, 14imbi12d 332 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))))
16 eleq2 2581 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
17 oveq2 6433 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 suc 𝑦))
1817eleq2d 2577 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
1916, 18imbi12d 332 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))
20 noel 3781 . . . . . . . . . . . 12 ¬ 𝐴 ∈ ∅
2120pm2.21i 114 . . . . . . . . . . 11 (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))
2221a1i 11 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
23 elsuci 5596 . . . . . . . . . . . . . . . 16 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
24 nnmcl 7454 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 𝑦) ∈ ω)
25 simpl 471 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
2624, 25jca 552 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27 nnaword1 7471 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝑦) ⊆ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
2827sseld 3471 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
2928imim2d 54 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))))
3029imp 443 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3130adantrl 747 . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·𝑜 𝑦) ∈ ω → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
3332ad2antrr 757 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
34 nnaordi 7460 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ ω ∧ (𝐶 ·𝑜 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3534ancoms 467 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3635imp 443 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
3733, 36eqeltrrd 2593 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
38 oveq2 6433 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝑦))
3938eleq1d 2576 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶) ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4037, 39syl5ibrcom 235 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4140adantrr 748 . . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 suc 𝑦) = ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
4645eleq2d 2577 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4746adantr 479 . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
4948exp43 637 . . . . . . . . . . . . 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 481 . . . . . . . . . . 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 445 . . . . . . . . . 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 6861 . . . . . . . . 9 (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))
547, 53vtoclga 3149 . . . . . . . 8 (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5554com23 83 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5655exp4a 630 . . . . . 6 (𝐵 ∈ ω → (𝐴𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5756exp4a 630 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵 → (𝐴 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))))
582, 57mpdd 41 . . . 4 (𝐵 ∈ ω → (𝐴𝐵 → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5958com34 88 . . 3 (𝐵 ∈ ω → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6059com24 92 . 2 (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6160imp31 446 1 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1938  c0 3777  suc csuc 5532  (class class class)co 6425  ωcom 6832   +𝑜 coa 7319   ·𝑜 comu 7320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6833  df-wrecs 7168  df-recs 7230  df-rdg 7268  df-oadd 7326  df-omul 7327
This theorem is referenced by:  nnmord  7474  mulclpi  9469
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