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Theorem nnmord 6406
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 6405 . . . . . 6 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
21ex 114 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
32com23 78 . . . 4 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
43impd 252 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
543adant1 999 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
6 ne0i 3364 . . . . . . . 8 ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐶 ·o 𝐵) ≠ ∅)
7 nnm0r 6368 . . . . . . . . . 10 (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
8 oveq1 5774 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
98eqeq1d 2146 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·o 𝐵) = ∅ ↔ (∅ ·o 𝐵) = ∅))
107, 9syl5ibrcom 156 . . . . . . . . 9 (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·o 𝐵) = ∅))
1110necon3d 2350 . . . . . . . 8 (𝐵 ∈ ω → ((𝐶 ·o 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
126, 11syl5 32 . . . . . . 7 (𝐵 ∈ ω → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅))
1312adantr 274 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅))
14 nn0eln0 4528 . . . . . . 7 (𝐶 ∈ ω → (∅ ∈ 𝐶𝐶 ≠ ∅))
1514adantl 275 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶𝐶 ≠ ∅))
1613, 15sylibrd 168 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶))
17163adant1 999 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶))
18 oveq2 5775 . . . . . . . . . 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 6405 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
21203adantl2 1138 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
2219, 21orim12d 775 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
2322con3d 620 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
24 simpl3 986 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
25 simpl1 984 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
26 nnmcl 6370 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·o 𝐴) ∈ ω)
2724, 25, 26syl2anc 408 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ω)
28 simpl2 985 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
29 nnmcl 6370 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o 𝐵) ∈ ω)
3024, 28, 29syl2anc 408 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐵) ∈ ω)
31 nntri2 6383 . . . . . . . 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 408 . . . . . . 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 6383 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
3425, 28, 33syl2anc 408 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
3523, 32, 343imtr4d 202 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴𝐵))
3635ex 114 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴𝐵)))
3736com23 78 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (∅ ∈ 𝐶𝐴𝐵)))
3817, 37mpdd 41 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴𝐵))
3938, 17jcad 305 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐴𝐵 ∧ ∅ ∈ 𝐶)))
405, 39impbid 128 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 103  wb 104  wo 697  w3a 962   = wceq 1331  wcel 1480  wne 2306  c0 3358  ωcom 4499  (class class class)co 5767   ·o comu 6304
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311
This theorem is referenced by:  nnmword  6407  ltmpig  7140
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