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Theorem nnmordi 6089
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 4328 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21expcom 109 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴 ∈ ω))
3 eleq2 2101 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
4 oveq2 5520 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝐵))
54eleq2d 2107 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
63, 5imbi12d 223 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
76imbi2d 219 . . . . . . . . 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 2101 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
9 oveq2 5520 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 ∅))
109eleq2d 2107 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
118, 10imbi12d 223 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))))
12 eleq2 2101 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
13 oveq2 5520 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝑦))
1413eleq2d 2107 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))
1512, 14imbi12d 223 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))))
16 eleq2 2101 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
17 oveq2 5520 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 suc 𝑦))
1817eleq2d 2107 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
1916, 18imbi12d 223 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))
20 noel 3228 . . . . . . . . . . . 12 ¬ 𝐴 ∈ ∅
2120pm2.21i 575 . . . . . . . . . . 11 (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))
2221a1i 9 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
23 elsuci 4140 . . . . . . . . . . . . . . . 16 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
24 nnmcl 6060 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 𝑦) ∈ ω)
25 simpl 102 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
2624, 25jca 290 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27 nnaword1 6086 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝑦) ⊆ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
2827sseld 2944 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
2928imim2d 48 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))))
3029imp 115 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3130adantrl 447 . . . . . . . . . . . . . . . . . 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 6053 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·𝑜 𝑦) ∈ ω → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
3332ad2antrr 457 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
34 nnaordi 6081 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ ω ∧ (𝐶 ·𝑜 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3534ancoms 255 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3635imp 115 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
3733, 36eqeltrrd 2115 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
38 oveq2 5520 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝑦))
3938eleq1d 2106 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶) ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4037, 39syl5ibrcom 146 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4140adantrr 448 . . . . . . . . . . . . . . . . . 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 637 . . . . . . . . . . . . . . . . 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 267 . . . . . . . . . . . . . . . 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 28 . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 suc 𝑦) = ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
4645eleq2d 2107 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4746adantr 261 . . . . . . . . . . . . . . 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 158 . . . . . . . . . . . . . 14 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
4948exp43 354 . . . . . . . . . . . . 13 (𝐶 ∈ ω → (𝑦 ∈ ω → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5049com12 27 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5150adantld 263 . . . . . . . . . . 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 242 . . . . . . . . . 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 4324 . . . . . . . . 9 (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))
547, 53vtoclga 2619 . . . . . . . 8 (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5554com23 72 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5655exp4a 348 . . . . . 6 (𝐵 ∈ ω → (𝐴𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5756exp4a 348 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵 → (𝐴 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))))
582, 57mpdd 36 . . . 4 (𝐵 ∈ ω → (𝐴𝐵 → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5958com34 77 . . 3 (𝐵 ∈ ω → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6059com24 81 . 2 (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6160imp31 243 1 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wo 629   = wceq 1243  wcel 1393  c0 3224  suc csuc 4102  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998   ·𝑜 comu 5999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006
This theorem is referenced by:  nnmord  6090  nnm00  6102  mulclpi  6424
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