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Theorem nnmord 7869
Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnmord ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Proof of Theorem nnmord
StepHypRef Expression
1 nnmordi 7868 . . . . . 6 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
21ex 449 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
32com23 86 . . . 4 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
43impd 446 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
543adant1 1122 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
6 ne0i 4052 . . . . . . . 8 ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐶 ·𝑜 𝐵) ≠ ∅)
7 nnm0r 7847 . . . . . . . . . 10 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
8 oveq1 6808 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
98eqeq1d 2750 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·𝑜 𝐵) = ∅ ↔ (∅ ·𝑜 𝐵) = ∅))
107, 9syl5ibrcom 237 . . . . . . . . 9 (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = ∅))
1110necon3d 2941 . . . . . . . 8 (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
126, 11syl5 34 . . . . . . 7 (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
1312adantr 472 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
14 nnord 7226 . . . . . . . 8 (𝐶 ∈ ω → Ord 𝐶)
15 ord0eln0 5928 . . . . . . . 8 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
1614, 15syl 17 . . . . . . 7 (𝐶 ∈ ω → (∅ ∈ 𝐶𝐶 ≠ ∅))
1716adantl 473 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶𝐶 ≠ ∅))
1813, 17sylibrd 249 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
19183adant1 1122 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
20 oveq2 6809 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))
2120a1i 11 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵)))
22 nnmordi 7868 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
23223adantl2 1153 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
2421, 23orim12d 919 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
2524con3d 148 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
26 simpl3 1208 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
27 simpl1 1204 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
28 nnmcl 7849 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) ∈ ω)
2926, 27, 28syl2anc 696 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ω)
30 simpl2 1206 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
31 nnmcl 7849 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) ∈ ω)
3226, 30, 31syl2anc 696 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐵) ∈ ω)
33 nnord 7226 . . . . . . . . 9 ((𝐶 ·𝑜 𝐴) ∈ ω → Ord (𝐶 ·𝑜 𝐴))
34 nnord 7226 . . . . . . . . 9 ((𝐶 ·𝑜 𝐵) ∈ ω → Ord (𝐶 ·𝑜 𝐵))
35 ordtri2 5907 . . . . . . . . 9 ((Ord (𝐶 ·𝑜 𝐴) ∧ Ord (𝐶 ·𝑜 𝐵)) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
3633, 34, 35syl2an 495 . . . . . . . 8 (((𝐶 ·𝑜 𝐴) ∈ ω ∧ (𝐶 ·𝑜 𝐵) ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
3729, 32, 36syl2anc 696 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
38 nnord 7226 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
39 nnord 7226 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
40 ordtri2 5907 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
4138, 39, 40syl2an 495 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
4227, 30, 41syl2anc 696 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
4325, 37, 423imtr4d 283 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵))
4443ex 449 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵)))
4544com23 86 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (∅ ∈ 𝐶𝐴𝐵)))
4619, 45mpdd 43 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵))
4746, 19jcad 556 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐴𝐵 ∧ ∅ ∈ 𝐶)))
485, 47impbid 202 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  c0 4046  Ord word 5871  (class class class)co 6801  ωcom 7218   ·𝑜 comu 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-oadd 7721  df-omul 7722
This theorem is referenced by:  nnmword  7870  nnneo  7888  ltmpi  9889
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