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Theorem mulpipq2 6984
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 5950 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
2 xp1st 5950 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
3 mulclpi 6941 . . . 4 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
41, 2, 3syl2an 284 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
5 xp2nd 5951 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
6 xp2nd 5951 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
7 mulclpi 6941 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
85, 6, 7syl2an 284 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
9 opexg 4064 . . 3 ((((1st𝐴) ·N (1st𝐵)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N) → ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V)
104, 8, 9syl2anc 404 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V)
11 fveq2 5318 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
1211oveq1d 5681 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝑦)))
13 fveq2 5318 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
1413oveq1d 5681 . . . 4 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
1512, 14opeq12d 3636 . . 3 (𝑥 = 𝐴 → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩)
16 fveq2 5318 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1716oveq2d 5682 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝐵)))
18 fveq2 5318 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
1918oveq2d 5682 . . . 4 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
2017, 19opeq12d 3636 . . 3 (𝑦 = 𝐵 → ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
21 df-mpq 6958 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
2215, 20, 21ovmpt2g 5793 . 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 1275 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 1290  wcel 1439  Vcvv 2620  cop 3453   × cxp 4449  cfv 5028  (class class class)co 5666  1st c1st 5923  2nd c2nd 5924  Ncnpi 6885   ·N cmi 6887   ·pQ cmpq 6890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-oadd 6199  df-omul 6200  df-ni 6917  df-mi 6919  df-mpq 6958
This theorem is referenced by:  mulpipq  6985
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