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Theorem mulpipq2 7143
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 6029 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
2 xp1st 6029 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
3 mulclpi 7100 . . . 4 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
41, 2, 3syl2an 285 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
5 xp2nd 6030 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
6 xp2nd 6030 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
7 mulclpi 7100 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
85, 6, 7syl2an 285 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
9 opexg 4118 . . 3 ((((1st𝐴) ·N (1st𝐵)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N) → ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V)
104, 8, 9syl2anc 406 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V)
11 fveq2 5387 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
1211oveq1d 5755 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝑦)))
13 fveq2 5387 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
1413oveq1d 5755 . . . 4 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
1512, 14opeq12d 3681 . . 3 (𝑥 = 𝐴 → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩)
16 fveq2 5387 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1716oveq2d 5756 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝐵)))
18 fveq2 5387 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
1918oveq2d 5756 . . . 4 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
2017, 19opeq12d 3681 . . 3 (𝑦 = 𝐵 → ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
21 df-mpq 7117 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
2215, 20, 21ovmpog 5871 . 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 1299 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 103   = wceq 1314  wcel 1463  Vcvv 2658  cop 3498   × cxp 4505  cfv 5091  (class class class)co 5740  1st c1st 6002  2nd c2nd 6003  Ncnpi 7044   ·N cmi 7046   ·pQ cmpq 7049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-oadd 6283  df-omul 6284  df-ni 7076  df-mi 7078  df-mpq 7117
This theorem is referenced by:  mulpipq  7144
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