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Theorem nnmordi 6000
 Description: Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnmordi (((B 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))

Proof of Theorem nnmordi
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 4255 . . . . . 6 ((A B B 𝜔) → A 𝜔)
21expcom 109 . . . . 5 (B 𝜔 → (A BA 𝜔))
3 eleq2 2083 . . . . . . . . . . 11 (x = B → (A xA B))
4 oveq2 5444 . . . . . . . . . . . 12 (x = B → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 B))
54eleq2d 2089 . . . . . . . . . . 11 (x = B → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
63, 5imbi12d 223 . . . . . . . . . 10 (x = B → ((A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x)) ↔ (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B))))
76imbi2d 219 . . . . . . . . 9 (x = B → ((((A 𝜔 𝐶 𝜔) 𝐶) → (A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x))) ↔ (((A 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))))
8 eleq2 2083 . . . . . . . . . . 11 (x = ∅ → (A xA ∅))
9 oveq2 5444 . . . . . . . . . . . 12 (x = ∅ → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 ∅))
109eleq2d 2089 . . . . . . . . . . 11 (x = ∅ → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 ∅)))
118, 10imbi12d 223 . . . . . . . . . 10 (x = ∅ → ((A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x)) ↔ (A ∅ → (𝐶 ·𝑜 A) (𝐶 ·𝑜 ∅))))
12 eleq2 2083 . . . . . . . . . . 11 (x = y → (A xA y))
13 oveq2 5444 . . . . . . . . . . . 12 (x = y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 y))
1413eleq2d 2089 . . . . . . . . . . 11 (x = y → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))
1512, 14imbi12d 223 . . . . . . . . . 10 (x = y → ((A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x)) ↔ (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y))))
16 eleq2 2083 . . . . . . . . . . 11 (x = suc y → (A xA suc y))
17 oveq2 5444 . . . . . . . . . . . 12 (x = suc y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 suc y))
1817eleq2d 2089 . . . . . . . . . . 11 (x = suc y → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y)))
1916, 18imbi12d 223 . . . . . . . . . 10 (x = suc y → ((A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x)) ↔ (A suc y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y))))
20 noel 3205 . . . . . . . . . . . 12 ¬ A
2120pm2.21i 562 . . . . . . . . . . 11 (A ∅ → (𝐶 ·𝑜 A) (𝐶 ·𝑜 ∅))
2221a1i 9 . . . . . . . . . 10 (((A 𝜔 𝐶 𝜔) 𝐶) → (A ∅ → (𝐶 ·𝑜 A) (𝐶 ·𝑜 ∅)))
23 elsuci 4089 . . . . . . . . . . . . . . . 16 (A suc y → (A y A = y))
24 nnmcl 5975 . . . . . . . . . . . . . . . . . 18 ((𝐶 𝜔 y 𝜔) → (𝐶 ·𝑜 y) 𝜔)
25 simpl 102 . . . . . . . . . . . . . . . . . 18 ((𝐶 𝜔 y 𝜔) → 𝐶 𝜔)
2624, 25jca 290 . . . . . . . . . . . . . . . . 17 ((𝐶 𝜔 y 𝜔) → ((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔))
27 nnaword1 5997 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 y) ⊆ ((𝐶 ·𝑜 y) +𝑜 𝐶))
2827sseld 2921 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 y) → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
2928imim2d 48 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) → ((A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)) → (A y → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶))))
3029imp 115 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y))) → (A y → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
3130adantrl 450 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → (A y → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
32 nna0 5968 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·𝑜 y) 𝜔 → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
3332ad2antrr 460 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) 𝐶) → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
34 nnaordi 5992 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 𝜔 (𝐶 ·𝑜 y) 𝜔) → (∅ 𝐶 → ((𝐶 ·𝑜 y) +𝑜 ∅) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
3534ancoms 255 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) → (∅ 𝐶 → ((𝐶 ·𝑜 y) +𝑜 ∅) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
3635imp 115 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) 𝐶) → ((𝐶 ·𝑜 y) +𝑜 ∅) ((𝐶 ·𝑜 y) +𝑜 𝐶))
3733, 36eqeltrrd 2097 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) 𝐶) → (𝐶 ·𝑜 y) ((𝐶 ·𝑜 y) +𝑜 𝐶))
38 oveq2 5444 . . . . . . . . . . . . . . . . . . . . 21 (A = y → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 y))
3938eleq1d 2088 . . . . . . . . . . . . . . . . . . . 20 (A = y → ((𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶) ↔ (𝐶 ·𝑜 y) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4037, 39syl5ibrcom 146 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) 𝐶) → (A = y → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4140adantrr 451 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → (A = y → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4231, 41jaod 624 . . . . . . . . . . . . . . . . 17 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → ((A y A = y) → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4326, 42sylan 267 . . . . . . . . . . . . . . . 16 (((𝐶 𝜔 y 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → ((A y A = y) → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4423, 43syl5 28 . . . . . . . . . . . . . . 15 (((𝐶 𝜔 y 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → (A suc y → (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
45 nnmsuc 5971 . . . . . . . . . . . . . . . . 17 ((𝐶 𝜔 y 𝜔) → (𝐶 ·𝑜 suc y) = ((𝐶 ·𝑜 y) +𝑜 𝐶))
4645eleq2d 2089 . . . . . . . . . . . . . . . 16 ((𝐶 𝜔 y 𝜔) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y) ↔ (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4746adantr 261 . . . . . . . . . . . . . . 15 (((𝐶 𝜔 y 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → ((𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y) ↔ (𝐶 ·𝑜 A) ((𝐶 ·𝑜 y) +𝑜 𝐶)))
4844, 47sylibrd 158 . . . . . . . . . . . . . 14 (((𝐶 𝜔 y 𝜔) (∅ 𝐶 (A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)))) → (A suc y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y)))
4948exp43 354 . . . . . . . . . . . . 13 (𝐶 𝜔 → (y 𝜔 → (∅ 𝐶 → ((A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)) → (A suc y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y))))))
5049com12 27 . . . . . . . . . . . 12 (y 𝜔 → (𝐶 𝜔 → (∅ 𝐶 → ((A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)) → (A suc y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y))))))
5150adantld 263 . . . . . . . . . . 11 (y 𝜔 → ((A 𝜔 𝐶 𝜔) → (∅ 𝐶 → ((A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)) → (A suc y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y))))))
5251impd 242 . . . . . . . . . 10 (y 𝜔 → (((A 𝜔 𝐶 𝜔) 𝐶) → ((A y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 y)) → (A suc y → (𝐶 ·𝑜 A) (𝐶 ·𝑜 suc y)))))
5311, 15, 19, 22, 52finds2 4251 . . . . . . . . 9 (x 𝜔 → (((A 𝜔 𝐶 𝜔) 𝐶) → (A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x))))
547, 53vtoclga 2596 . . . . . . . 8 (B 𝜔 → (((A 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B))))
5554com23 72 . . . . . . 7 (B 𝜔 → (A B → (((A 𝜔 𝐶 𝜔) 𝐶) → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B))))
5655exp4a 348 . . . . . 6 (B 𝜔 → (A B → ((A 𝜔 𝐶 𝜔) → (∅ 𝐶 → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))))
5756exp4a 348 . . . . 5 (B 𝜔 → (A B → (A 𝜔 → (𝐶 𝜔 → (∅ 𝐶 → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B))))))
582, 57mpdd 36 . . . 4 (B 𝜔 → (A B → (𝐶 𝜔 → (∅ 𝐶 → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))))
5958com34 77 . . 3 (B 𝜔 → (A B → (∅ 𝐶 → (𝐶 𝜔 → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))))
6059com24 81 . 2 (B 𝜔 → (𝐶 𝜔 → (∅ 𝐶 → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))))
6160imp31 243 1 (((B 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1374  ∅c0 3201  suc csuc 4051  𝜔com 4240  (class class class)co 5436   +𝑜 coa 5913   ·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-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:  nnmord  6001  nnm00  6013  mulclpi  6188
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