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Theorem nnmordi 6378
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 4487 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21expcom 115 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴 ∈ ω))
3 eleq2 2179 . . . . . . . . . . 11 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
4 oveq2 5748 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
54eleq2d 2185 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
63, 5imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
76imbi2d 229 . . . . . . . . 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 2179 . . . . . . . . . . 11 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
9 oveq2 5748 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
109eleq2d 2185 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
118, 10imbi12d 233 . . . . . . . . . 10 (𝑥 = ∅ → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
12 eleq2 2179 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
13 oveq2 5748 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
1413eleq2d 2185 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
1512, 14imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
16 eleq2 2179 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
17 oveq2 5748 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
1817eleq2d 2185 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
1916, 18imbi12d 233 . . . . . . . . . 10 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
20 noel 3335 . . . . . . . . . . . 12 ¬ 𝐴 ∈ ∅
2120pm2.21i 618 . . . . . . . . . . 11 (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
2221a1i 9 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
23 elsuci 4293 . . . . . . . . . . . . . . . 16 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
24 nnmcl 6343 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o 𝑦) ∈ ω)
25 simpl 108 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
2624, 25jca 302 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27 nnaword1 6375 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
2827sseld 3064 . . . . . . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3130adantrl 467 . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·o 𝑦) ∈ ω → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
3332ad2antrr 477 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
34 nnaordi 6370 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ ω ∧ (𝐶 ·o 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3534ancoms 266 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
3635imp 123 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
3733, 36eqeltrrd 2193 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
38 oveq2 5748 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
3938eleq1d 2184 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4037, 39syl5ibrcom 156 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4140adantrr 468 . . . . . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . . . 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 6339 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
4645eleq2d 2185 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
4746adantr 272 . . . . . . . . . . . . . . 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 168 . . . . . . . . . . . . . 14 (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
4948exp43 367 . . . . . . . . . . . . 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 274 . . . . . . . . . . 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 252 . . . . . . . . . 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 4483 . . . . . . . . 9 (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
547, 53vtoclga 2724 . . . . . . . 8 (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
5554com23 78 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
5655exp4a 361 . . . . . 6 (𝐵 ∈ ω → (𝐴𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
5756exp4a 361 . . . . 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 83 . . 3 (𝐵 ∈ ω → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
6059com24 87 . 2 (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
6160imp31 254 1 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 680   = wceq 1314  wcel 1463  c0 3331  suc csuc 4255  ωcom 4472  (class class class)co 5740   +o coa 6276   ·o comu 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-oadd 6283  df-omul 6284
This theorem is referenced by:  nnmord  6379  nnm00  6391  mulclpi  7100
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