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Theorem nnmord 6026
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) → ((A B 𝐶) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))

Proof of Theorem nnmord
StepHypRef Expression
1 nnmordi 6025 . . . . . 6 (((B 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
21ex 108 . . . . 5 ((B 𝜔 𝐶 𝜔) → (∅ 𝐶 → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B))))
32com23 72 . . . 4 ((B 𝜔 𝐶 𝜔) → (A B → (∅ 𝐶 → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B))))
43impd 242 . . 3 ((B 𝜔 𝐶 𝜔) → ((A B 𝐶) → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
543adant1 921 . 2 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A B 𝐶) → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
6 ne0i 3224 . . . . . . . 8 ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → (𝐶 ·𝑜 B) ≠ ∅)
7 nnm0r 5997 . . . . . . . . . 10 (B 𝜔 → (∅ ·𝑜 B) = ∅)
8 oveq1 5462 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·𝑜 B) = (∅ ·𝑜 B))
98eqeq1d 2045 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·𝑜 B) = ∅ ↔ (∅ ·𝑜 B) = ∅))
107, 9syl5ibrcom 146 . . . . . . . . 9 (B 𝜔 → (𝐶 = ∅ → (𝐶 ·𝑜 B) = ∅))
1110necon3d 2243 . . . . . . . 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 4284 . . . . . . 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 921 . . . 4 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → ∅ 𝐶))
18 oveq2 5463 . . . . . . . . . 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 6025 . . . . . . . . . 10 (((A 𝜔 𝐶 𝜔) 𝐶) → (B A → (𝐶 ·𝑜 B) (𝐶 ·𝑜 A)))
21203adantl2 1060 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (B A → (𝐶 ·𝑜 B) (𝐶 ·𝑜 A)))
2219, 21orim12d 699 . . . . . . . 8 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → ((A = B B A) → ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) (𝐶 ·𝑜 B) (𝐶 ·𝑜 A))))
2322con3d 560 . . . . . . 7 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (¬ ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) (𝐶 ·𝑜 B) (𝐶 ·𝑜 A)) → ¬ (A = B B A)))
24 simpl3 908 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → 𝐶 𝜔)
25 simpl1 906 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → A 𝜔)
26 nnmcl 5999 . . . . . . . . 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 907 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → B 𝜔)
29 nnmcl 5999 . . . . . . . . 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 6012 . . . . . . . 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 6012 . . . . . . . 8 ((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))
3425, 28, 33syl2anc 391 . . . . . . 7 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (A B ↔ ¬ (A = B B A)))
3523, 32, 343imtr4d 192 . . . . . 6 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → A B))
3635ex 108 . . . . 5 ((A 𝜔 B 𝜔 𝐶 𝜔) → (∅ 𝐶 → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → A B)))
3736com23 72 . . . 4 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → (∅ 𝐶A B)))
3817, 37mpdd 36 . . 3 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → A B))
3938, 17jcad 291 . 2 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → (A B 𝐶)))
405, 39impbid 120 1 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A B 𝐶) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  wne 2201  c0 3218  𝜔com 4256  (class class class)co 5455   ·𝑜 comu 5938
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945
This theorem is referenced by:  nnmword  6027  ltmpig  6323
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