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Theorem nnmordi 6025
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 4271 . . . . . 6 ((A B B 𝜔) → A 𝜔)
21expcom 109 . . . . 5 (B 𝜔 → (A BA 𝜔))
3 eleq2 2098 . . . . . . . . . . 11 (x = B → (A xA B))
4 oveq2 5463 . . . . . . . . . . . 12 (x = B → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 B))
54eleq2d 2104 . . . . . . . . . . 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 2098 . . . . . . . . . . 11 (x = ∅ → (A xA ∅))
9 oveq2 5463 . . . . . . . . . . . 12 (x = ∅ → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 ∅))
109eleq2d 2104 . . . . . . . . . . 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 2098 . . . . . . . . . . 11 (x = y → (A xA y))
13 oveq2 5463 . . . . . . . . . . . 12 (x = y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 y))
1413eleq2d 2104 . . . . . . . . . . 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 2098 . . . . . . . . . . 11 (x = suc y → (A xA suc y))
17 oveq2 5463 . . . . . . . . . . . 12 (x = suc y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 suc y))
1817eleq2d 2104 . . . . . . . . . . 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 3222 . . . . . . . . . . . 12 ¬ A
2120pm2.21i 574 . . . . . . . . . . 11 (A ∅ → (𝐶 ·𝑜 A) (𝐶 ·𝑜 ∅))
2221a1i 9 . . . . . . . . . 10 (((A 𝜔 𝐶 𝜔) 𝐶) → (A ∅ → (𝐶 ·𝑜 A) (𝐶 ·𝑜 ∅)))
23 elsuci 4106 . . . . . . . . . . . . . . . 16 (A suc y → (A y A = y))
24 nnmcl 5999 . . . . . . . . . . . . . . . . . 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 6022 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 y) ⊆ ((𝐶 ·𝑜 y) +𝑜 𝐶))
2827sseld 2938 . . . . . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . . . 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 5992 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ·𝑜 y) 𝜔 → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
3332ad2antrr 457 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) 𝐶) → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
34 nnaordi 6017 . . . . . . . . . . . . . . . . . . . . . . 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 2112 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶 ·𝑜 y) 𝜔 𝐶 𝜔) 𝐶) → (𝐶 ·𝑜 y) ((𝐶 ·𝑜 y) +𝑜 𝐶))
38 oveq2 5463 . . . . . . . . . . . . . . . . . . . . 21 (A = y → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 y))
3938eleq1d 2103 . . . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . 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 636 . . . . . . . . . . . . . . . . 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 5995 . . . . . . . . . . . . . . . . 17 ((𝐶 𝜔 y 𝜔) → (𝐶 ·𝑜 suc y) = ((𝐶 ·𝑜 y) +𝑜 𝐶))
4645eleq2d 2104 . . . . . . . . . . . . . . . 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 4267 . . . . . . . . 9 (x 𝜔 → (((A 𝜔 𝐶 𝜔) 𝐶) → (A x → (𝐶 ·𝑜 A) (𝐶 ·𝑜 x))))
547, 53vtoclga 2613 . . . . . . . 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 628   = wceq 1242   wcel 1390  c0 3218  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937   ·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-bndl 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-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:  nnmord  6026  nnm00  6038  mulclpi  6312
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