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Theorem nnmordi 6119
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 4355 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21expcom 113 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴 ∈ ω))
3 eleq2 2117 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
4 oveq2 5547 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝐵))
54eleq2d 2123 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
63, 5imbi12d 227 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
76imbi2d 223 . . . . . . . . 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 2117 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
9 oveq2 5547 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 ∅))
109eleq2d 2123 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
118, 10imbi12d 227 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))))
12 eleq2 2117 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
13 oveq2 5547 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝑦))
1413eleq2d 2123 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))
1512, 14imbi12d 227 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))))
16 eleq2 2117 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
17 oveq2 5547 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 suc 𝑦))
1817eleq2d 2123 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
1916, 18imbi12d 227 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))
20 noel 3255 . . . . . . . . . . . 12 ¬ 𝐴 ∈ ∅
2120pm2.21i 585 . . . . . . . . . . 11 (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))
2221a1i 9 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
23 elsuci 4167 . . . . . . . . . . . . . . . 16 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
24 nnmcl 6090 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 𝑦) ∈ ω)
25 simpl 106 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
2624, 25jca 294 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27 nnaword1 6116 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝑦) ⊆ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
2827sseld 2971 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
2928imim2d 52 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))))
3029imp 119 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))) → (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3130adantrl 455 . . . . . . . . . . . . . . . . . 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 6083 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·𝑜 𝑦) ∈ ω → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
3332ad2antrr 465 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) = (𝐶 ·𝑜 𝑦))
34 nnaordi 6111 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ ω ∧ (𝐶 ·𝑜 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3534ancoms 259 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
3635imp 119 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝑦) +𝑜 ∅) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
3733, 36eqeltrrd 2131 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
38 oveq2 5547 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝑦))
3938eleq1d 2122 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶) ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4037, 39syl5ibrcom 150 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4140adantrr 456 . . . . . . . . . . . . . . . . . 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 647 . . . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . . 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 6086 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·𝑜 suc 𝑦) = ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
4645eleq2d 2123 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
4746adantr 265 . . . . . . . . . . . . . . 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 162 . . . . . . . . . . . . . 14 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
4948exp43 358 . . . . . . . . . . . . 13 (𝐶 ∈ ω → (𝑦 ∈ ω → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5049com12 30 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → ((𝐴𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))))
5150adantld 267 . . . . . . . . . . 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 246 . . . . . . . . . 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 4351 . . . . . . . . 9 (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))
547, 53vtoclga 2636 . . . . . . . 8 (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5554com23 76 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
5655exp4a 352 . . . . . 6 (𝐵 ∈ ω → (𝐴𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5756exp4a 352 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵 → (𝐴 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))))
582, 57mpdd 40 . . . 4 (𝐵 ∈ ω → (𝐴𝐵 → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
5958com34 81 . . 3 (𝐵 ∈ ω → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6059com24 85 . 2 (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
6160imp31 247 1 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639   = wceq 1259  wcel 1409  c0 3251  suc csuc 4129  ωcom 4340  (class class class)co 5539   +𝑜 coa 6028   ·𝑜 comu 6029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036
This theorem is referenced by:  nnmord  6120  nnm00  6132  mulclpi  6483
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