MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulpipq2 Structured version   Visualization version   GIF version

Theorem mulpipq2 10970
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpipq2 ((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N)) โ†’ (๐ด ยทpQ ๐ต) = โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐ต)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต))โŸฉ)

Proof of Theorem mulpipq2
Dummy variables ๐‘ฅ ๐‘ฆ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . 4 (๐‘ฅ = ๐ด โ†’ (1st โ€˜๐‘ฅ) = (1st โ€˜๐ด))
21oveq1d 7441 . . 3 (๐‘ฅ = ๐ด โ†’ ((1st โ€˜๐‘ฅ) ยทN (1st โ€˜๐‘ฆ)) = ((1st โ€˜๐ด) ยทN (1st โ€˜๐‘ฆ)))
3 fveq2 6902 . . . 4 (๐‘ฅ = ๐ด โ†’ (2nd โ€˜๐‘ฅ) = (2nd โ€˜๐ด))
43oveq1d 7441 . . 3 (๐‘ฅ = ๐ด โ†’ ((2nd โ€˜๐‘ฅ) ยทN (2nd โ€˜๐‘ฆ)) = ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐‘ฆ)))
52, 4opeq12d 4886 . 2 (๐‘ฅ = ๐ด โ†’ โŸจ((1st โ€˜๐‘ฅ) ยทN (1st โ€˜๐‘ฆ)), ((2nd โ€˜๐‘ฅ) ยทN (2nd โ€˜๐‘ฆ))โŸฉ = โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐‘ฆ)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐‘ฆ))โŸฉ)
6 fveq2 6902 . . . 4 (๐‘ฆ = ๐ต โ†’ (1st โ€˜๐‘ฆ) = (1st โ€˜๐ต))
76oveq2d 7442 . . 3 (๐‘ฆ = ๐ต โ†’ ((1st โ€˜๐ด) ยทN (1st โ€˜๐‘ฆ)) = ((1st โ€˜๐ด) ยทN (1st โ€˜๐ต)))
8 fveq2 6902 . . . 4 (๐‘ฆ = ๐ต โ†’ (2nd โ€˜๐‘ฆ) = (2nd โ€˜๐ต))
98oveq2d 7442 . . 3 (๐‘ฆ = ๐ต โ†’ ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐‘ฆ)) = ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต)))
107, 9opeq12d 4886 . 2 (๐‘ฆ = ๐ต โ†’ โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐‘ฆ)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐‘ฆ))โŸฉ = โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐ต)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต))โŸฉ)
11 df-mpq 10940 . 2 ยทpQ = (๐‘ฅ โˆˆ (N ร— N), ๐‘ฆ โˆˆ (N ร— N) โ†ฆ โŸจ((1st โ€˜๐‘ฅ) ยทN (1st โ€˜๐‘ฆ)), ((2nd โ€˜๐‘ฅ) ยทN (2nd โ€˜๐‘ฆ))โŸฉ)
12 opex 5470 . 2 โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐ต)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต))โŸฉ โˆˆ V
135, 10, 11, 12ovmpo 7587 1 ((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N)) โ†’ (๐ด ยทpQ ๐ต) = โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐ต)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต))โŸฉ)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 394   = wceq 1533   โˆˆ wcel 2098  โŸจcop 4638   ร— cxp 5680  โ€˜cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  Ncnpi 10875   ยทN cmi 10877   ยทpQ cmpq 10880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-mpq 10940
This theorem is referenced by:  mulpipq  10971  mulcompq  10983  mulerpqlem  10986  mulassnq  10990  distrnq  10992  ltmnq  11003
  Copyright terms: Public domain W3C validator