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Theorem mulpipq2 6623
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 5823 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
2 xp1st 5823 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
3 mulclpi 6580 . . . 4 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
41, 2, 3syl2an 283 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
5 xp2nd 5824 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
6 xp2nd 5824 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
7 mulclpi 6580 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
85, 6, 7syl2an 283 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
9 opexg 3991 . . 3 ((((1st𝐴) ·N (1st𝐵)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N) → ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V)
104, 8, 9syl2anc 403 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V)
11 fveq2 5209 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
1211oveq1d 5558 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝑦)))
13 fveq2 5209 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
1413oveq1d 5558 . . . 4 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
1512, 14opeq12d 3586 . . 3 (𝑥 = 𝐴 → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩)
16 fveq2 5209 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1716oveq2d 5559 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝐵)))
18 fveq2 5209 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
1918oveq2d 5559 . . . 4 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
2017, 19opeq12d 3586 . . 3 (𝑦 = 𝐵 → ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
21 df-mpq 6597 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
2215, 20, 21ovmpt2g 5666 . 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 1270 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 102   = wceq 1285  wcel 1434  Vcvv 2602  cop 3409   × cxp 4369  cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  Ncnpi 6524   ·N cmi 6526   ·pQ cmpq 6529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070  df-ni 6556  df-mi 6558  df-mpq 6597
This theorem is referenced by:  mulpipq  6624
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