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Theorem nnmulcl 7696
Description: Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
Assertion
Ref Expression
nnmulcl ((A B ℕ) → (A · B) ℕ)

Proof of Theorem nnmulcl
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5463 . . . . 5 (x = 1 → (A · x) = (A · 1))
21eleq1d 2103 . . . 4 (x = 1 → ((A · x) ℕ ↔ (A · 1) ℕ))
32imbi2d 219 . . 3 (x = 1 → ((A ℕ → (A · x) ℕ) ↔ (A ℕ → (A · 1) ℕ)))
4 oveq2 5463 . . . . 5 (x = y → (A · x) = (A · y))
54eleq1d 2103 . . . 4 (x = y → ((A · x) ℕ ↔ (A · y) ℕ))
65imbi2d 219 . . 3 (x = y → ((A ℕ → (A · x) ℕ) ↔ (A ℕ → (A · y) ℕ)))
7 oveq2 5463 . . . . 5 (x = (y + 1) → (A · x) = (A · (y + 1)))
87eleq1d 2103 . . . 4 (x = (y + 1) → ((A · x) ℕ ↔ (A · (y + 1)) ℕ))
98imbi2d 219 . . 3 (x = (y + 1) → ((A ℕ → (A · x) ℕ) ↔ (A ℕ → (A · (y + 1)) ℕ)))
10 oveq2 5463 . . . . 5 (x = B → (A · x) = (A · B))
1110eleq1d 2103 . . . 4 (x = B → ((A · x) ℕ ↔ (A · B) ℕ))
1211imbi2d 219 . . 3 (x = B → ((A ℕ → (A · x) ℕ) ↔ (A ℕ → (A · B) ℕ)))
13 nncn 7683 . . . 4 (A ℕ → A ℂ)
14 mulid1 6802 . . . . . 6 (A ℂ → (A · 1) = A)
1514eleq1d 2103 . . . . 5 (A ℂ → ((A · 1) ℕ ↔ A ℕ))
1615biimprd 147 . . . 4 (A ℂ → (A ℕ → (A · 1) ℕ))
1713, 16mpcom 32 . . 3 (A ℕ → (A · 1) ℕ)
18 nnaddcl 7695 . . . . . . . 8 (((A · y) A ℕ) → ((A · y) + A) ℕ)
1918ancoms 255 . . . . . . 7 ((A (A · y) ℕ) → ((A · y) + A) ℕ)
20 nncn 7683 . . . . . . . . 9 (y ℕ → y ℂ)
21 ax-1cn 6756 . . . . . . . . . . 11 1
22 adddi 6791 . . . . . . . . . . 11 ((A y 1 ℂ) → (A · (y + 1)) = ((A · y) + (A · 1)))
2321, 22mp3an3 1220 . . . . . . . . . 10 ((A y ℂ) → (A · (y + 1)) = ((A · y) + (A · 1)))
2414oveq2d 5471 . . . . . . . . . . 11 (A ℂ → ((A · y) + (A · 1)) = ((A · y) + A))
2524adantr 261 . . . . . . . . . 10 ((A y ℂ) → ((A · y) + (A · 1)) = ((A · y) + A))
2623, 25eqtrd 2069 . . . . . . . . 9 ((A y ℂ) → (A · (y + 1)) = ((A · y) + A))
2713, 20, 26syl2an 273 . . . . . . . 8 ((A y ℕ) → (A · (y + 1)) = ((A · y) + A))
2827eleq1d 2103 . . . . . . 7 ((A y ℕ) → ((A · (y + 1)) ℕ ↔ ((A · y) + A) ℕ))
2919, 28syl5ibr 145 . . . . . 6 ((A y ℕ) → ((A (A · y) ℕ) → (A · (y + 1)) ℕ))
3029exp4b 349 . . . . 5 (A ℕ → (y ℕ → (A ℕ → ((A · y) ℕ → (A · (y + 1)) ℕ))))
3130pm2.43b 46 . . . 4 (y ℕ → (A ℕ → ((A · y) ℕ → (A · (y + 1)) ℕ)))
3231a2d 23 . . 3 (y ℕ → ((A ℕ → (A · y) ℕ) → (A ℕ → (A · (y + 1)) ℕ)))
333, 6, 9, 12, 17, 32nnind 7691 . 2 (B ℕ → (A ℕ → (A · B) ℕ))
3433impcom 116 1 ((A B ℕ) → (A · B) ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  (class class class)co 5455  cc 6689  1c1 6692   + caddc 6694   · cmul 6696  cn 7675
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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6754  ax-resscn 6755  ax-1cn 6756  ax-1re 6757  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-mulcom 6764  ax-addass 6765  ax-mulass 6766  ax-distr 6767  ax-1rid 6770  ax-cnre 6774
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7676
This theorem is referenced by:  nnmulcli  7697  nndivtr  7716  nnmulcld  7722  nn0mulcl  7974  qaddcl  8326  qmulcl  8328  nnexpcl  8902  nnsqcl  8956
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