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Theorem mulpipq2 6230
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
Assertion
Ref Expression
mulpipq2 ((A (N × N) B (N × N)) → (A ·pQ B) = ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩)

Proof of Theorem mulpipq2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 5715 . . . 4 (A (N × N) → (1stA) N)
2 xp1st 5715 . . . 4 (B (N × N) → (1stB) N)
3 mulclpi 6188 . . . 4 (((1stA) N (1stB) N) → ((1stA) ·N (1stB)) N)
41, 2, 3syl2an 273 . . 3 ((A (N × N) B (N × N)) → ((1stA) ·N (1stB)) N)
5 xp2nd 5716 . . . 4 (A (N × N) → (2ndA) N)
6 xp2nd 5716 . . . 4 (B (N × N) → (2ndB) N)
7 mulclpi 6188 . . . 4 (((2ndA) N (2ndB) N) → ((2ndA) ·N (2ndB)) N)
85, 6, 7syl2an 273 . . 3 ((A (N × N) B (N × N)) → ((2ndA) ·N (2ndB)) N)
9 opexg 3938 . . 3 ((((1stA) ·N (1stB)) N ((2ndA) ·N (2ndB)) N) → ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩ V)
104, 8, 9syl2anc 393 . 2 ((A (N × N) B (N × N)) → ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩ V)
11 fveq2 5103 . . . . 5 (x = A → (1stx) = (1stA))
1211oveq1d 5451 . . . 4 (x = A → ((1stx) ·N (1sty)) = ((1stA) ·N (1sty)))
13 fveq2 5103 . . . . 5 (x = A → (2ndx) = (2ndA))
1413oveq1d 5451 . . . 4 (x = A → ((2ndx) ·N (2ndy)) = ((2ndA) ·N (2ndy)))
1512, 14opeq12d 3531 . . 3 (x = A → ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩ = ⟨((1stA) ·N (1sty)), ((2ndA) ·N (2ndy))⟩)
16 fveq2 5103 . . . . 5 (y = B → (1sty) = (1stB))
1716oveq2d 5452 . . . 4 (y = B → ((1stA) ·N (1sty)) = ((1stA) ·N (1stB)))
18 fveq2 5103 . . . . 5 (y = B → (2ndy) = (2ndB))
1918oveq2d 5452 . . . 4 (y = B → ((2ndA) ·N (2ndy)) = ((2ndA) ·N (2ndB)))
2017, 19opeq12d 3531 . . 3 (y = B → ⟨((1stA) ·N (1sty)), ((2ndA) ·N (2ndy))⟩ = ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩)
21 df-mpq 6204 . . 3 ·pQ = (x (N × N), y (N × N) ↦ ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩)
2215, 20, 21ovmpt2g 5558 . 2 ((A (N × N) B (N × N) ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩ V) → (A ·pQ B) = ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩)
2310, 22mpd3an3 1218 1 ((A (N × N) B (N × N)) → (A ·pQ B) = ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  Vcvv 2535  cop 3353   × cxp 4270  cfv 4829  (class class class)co 5436  1st c1st 5688  2nd c2nd 5689  Ncnpi 6130   ·N cmi 6132   ·pQ cmpq 6135
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-dc 734  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  df-ni 6164  df-mi 6166  df-mpq 6204
This theorem is referenced by:  mulpipq  6231
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