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Theorem nnmord 6001
 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 6000 . . . . . 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 910 . 2 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A B 𝐶) → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
6 ne0i 3207 . . . . . . . 8 ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → (𝐶 ·𝑜 B) ≠ ∅)
7 nnm0r 5973 . . . . . . . . . 10 (B 𝜔 → (∅ ·𝑜 B) = ∅)
8 oveq1 5443 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·𝑜 B) = (∅ ·𝑜 B))
98eqeq1d 2030 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·𝑜 B) = ∅ ↔ (∅ ·𝑜 B) = ∅))
107, 9syl5ibrcom 146 . . . . . . . . 9 (B 𝜔 → (𝐶 = ∅ → (𝐶 ·𝑜 B) = ∅))
1110necon3d 2227 . . . . . . . 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 4268 . . . . . . 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 910 . . . 4 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 B) → ∅ 𝐶))
18 oveq2 5444 . . . . . . . . . 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 6000 . . . . . . . . . 10 (((A 𝜔 𝐶 𝜔) 𝐶) → (B A → (𝐶 ·𝑜 B) (𝐶 ·𝑜 A)))
21203adantl2 1049 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (B A → (𝐶 ·𝑜 B) (𝐶 ·𝑜 A)))
2219, 21orim12d 687 . . . . . . . 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 548 . . . . . . 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 897 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → 𝐶 𝜔)
25 simpl1 895 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → A 𝜔)
26 nnmcl 5975 . . . . . . . . 9 ((𝐶 𝜔 A 𝜔) → (𝐶 ·𝑜 A) 𝜔)
2724, 25, 26syl2anc 393 . . . . . . . 8 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (𝐶 ·𝑜 A) 𝜔)
28 simpl2 896 . . . . . . . . 9 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → B 𝜔)
29 nnmcl 5975 . . . . . . . . 9 ((𝐶 𝜔 B 𝜔) → (𝐶 ·𝑜 B) 𝜔)
3024, 28, 29syl2anc 393 . . . . . . . 8 (((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (𝐶 ·𝑜 B) 𝜔)
31 nntri2 5988 . . . . . . . 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 393 . . . . . . 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 5988 . . . . . . . 8 ((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))
3425, 28, 33syl2anc 393 . . . . . . 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 616   ∧ w3a 873   = wceq 1228   ∈ wcel 1374   ≠ wne 2186  ∅c0 3201  𝜔com 4240  (class class class)co 5436   ·𝑜 comu 5914 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238 This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921 This theorem is referenced by:  nnmword  6002  ltmpig  6199
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