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Theorem nnmord 6090
 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 6089 . . . . . 6 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
21ex 108 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
32com23 72 . . . 4 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
43impd 242 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
543adant1 922 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
6 ne0i 3230 . . . . . . . 8 ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐶 ·𝑜 𝐵) ≠ ∅)
7 nnm0r 6058 . . . . . . . . . 10 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
8 oveq1 5519 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
98eqeq1d 2048 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·𝑜 𝐵) = ∅ ↔ (∅ ·𝑜 𝐵) = ∅))
107, 9syl5ibrcom 146 . . . . . . . . 9 (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = ∅))
1110necon3d 2249 . . . . . . . 8 (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
126, 11syl5 28 . . . . . . 7 (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
1312adantr 261 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
14 nn0eln0 4341 . . . . . . 7 (𝐶 ∈ ω → (∅ ∈ 𝐶𝐶 ≠ ∅))
1514adantl 262 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶𝐶 ≠ ∅))
1613, 15sylibrd 158 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
17163adant1 922 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
18 oveq2 5520 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))
1918a1i 9 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵)))
20 nnmordi 6089 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
21203adantl2 1061 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
2219, 21orim12d 700 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
2322con3d 561 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
24 simpl3 909 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
25 simpl1 907 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
26 nnmcl 6060 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) ∈ ω)
2724, 25, 26syl2anc 391 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ω)
28 simpl2 908 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
29 nnmcl 6060 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) ∈ ω)
3024, 28, 29syl2anc 391 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐵) ∈ ω)
31 nntri2 6073 . . . . . . . 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))))
3227, 30, 31syl2anc 391 . . . . . . 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))))
33 nntri2 6073 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
3425, 28, 33syl2anc 391 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
3523, 32, 343imtr4d 192 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵))
3635ex 108 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵)))
3736com23 72 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (∅ ∈ 𝐶𝐴𝐵)))
3817, 37mpdd 36 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵))
3938, 17jcad 291 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐴𝐵 ∧ ∅ ∈ 𝐶)))
405, 39impbid 120 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  ∅c0 3224  ωcom 4313  (class class class)co 5512   ·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-3or 886  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:  nnmword  6091  ltmpig  6437
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