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Theorem mulpipq2 6355
 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 5734 . . . 4 (A (N × N) → (1stA) N)
2 xp1st 5734 . . . 4 (B (N × N) → (1stB) N)
3 mulclpi 6312 . . . 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 5735 . . . 4 (A (N × N) → (2ndA) N)
6 xp2nd 5735 . . . 4 (B (N × N) → (2ndB) N)
7 mulclpi 6312 . . . 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 3955 . . 3 ((((1stA) ·N (1stB)) N ((2ndA) ·N (2ndB)) N) → ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩ V)
104, 8, 9syl2anc 391 . 2 ((A (N × N) B (N × N)) → ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩ V)
11 fveq2 5121 . . . . 5 (x = A → (1stx) = (1stA))
1211oveq1d 5470 . . . 4 (x = A → ((1stx) ·N (1sty)) = ((1stA) ·N (1sty)))
13 fveq2 5121 . . . . 5 (x = A → (2ndx) = (2ndA))
1413oveq1d 5470 . . . 4 (x = A → ((2ndx) ·N (2ndy)) = ((2ndA) ·N (2ndy)))
1512, 14opeq12d 3548 . . 3 (x = A → ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩ = ⟨((1stA) ·N (1sty)), ((2ndA) ·N (2ndy))⟩)
16 fveq2 5121 . . . . 5 (y = B → (1sty) = (1stB))
1716oveq2d 5471 . . . 4 (y = B → ((1stA) ·N (1sty)) = ((1stA) ·N (1stB)))
18 fveq2 5121 . . . . 5 (y = B → (2ndy) = (2ndB))
1918oveq2d 5471 . . . 4 (y = B → ((2ndA) ·N (2ndy)) = ((2ndA) ·N (2ndB)))
2017, 19opeq12d 3548 . . 3 (y = B → ⟨((1stA) ·N (1sty)), ((2ndA) ·N (2ndy))⟩ = ⟨((1stA) ·N (1stB)), ((2ndA) ·N (2ndB))⟩)
21 df-mpq 6329 . . 3 ·pQ = (x (N × N), y (N × N) ↦ ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩)
2215, 20, 21ovmpt2g 5577 . 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 1232 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 1242   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   × cxp 4286  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Ncnpi 6256   ·N cmi 6258   ·pQ cmpq 6261 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-bnd 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-dc 742  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  df-ni 6288  df-mi 6290  df-mpq 6329 This theorem is referenced by:  mulpipq  6356
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