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Theorem recmulnqg 6853
Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
Assertion
Ref Expression
recmulnqg ((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))

Proof of Theorem recmulnqg
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5598 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
21eqeq1d 2091 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
32anbi2d 452 . . 3 (𝑥 = 𝐴 → ((𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
4 eleq1 2145 . . . 4 (𝑦 = 𝐵 → (𝑦Q𝐵Q))
5 oveq2 5599 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
65eqeq1d 2091 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
74, 6anbi12d 457 . . 3 (𝑦 = 𝐵 → ((𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q) ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
8 recexnq 6852 . . . 4 (𝑥Q → ∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9 1nq 6828 . . . . 5 1QQ
10 mulcomnqg 6845 . . . . 5 ((𝑧Q𝑤Q) → (𝑧 ·Q 𝑤) = (𝑤 ·Q 𝑧))
11 mulassnqg 6846 . . . . 5 ((𝑧Q𝑤Q𝑣Q) → ((𝑧 ·Q 𝑤) ·Q 𝑣) = (𝑧 ·Q (𝑤 ·Q 𝑣)))
12 mulidnq 6851 . . . . 5 (𝑧Q → (𝑧 ·Q 1Q) = 𝑧)
139, 10, 11, 12caovimo 5773 . . . 4 (𝑥Q → ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
14 eu5 1990 . . . 4 (∃!𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ∧ ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
158, 13, 14sylanbrc 408 . . 3 (𝑥Q → ∃!𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
16 df-rq 6814 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
17 3anass 924 . . . . 5 ((𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
1817opabbii 3871 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
1916, 18eqtri 2103 . . 3 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
203, 7, 15, 19fvopab3g 5322 . 2 ((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
21 ibar 295 . . 3 (𝐵Q → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
2221adantl 271 . 2 ((𝐴Q𝐵Q) → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
2320, 22bitr4d 189 1 ((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434  ∃!weu 1943  ∃*wmo 1944  {copab 3864  cfv 4969  (class class class)co 5591  Qcnq 6742  1Qc1q 6743   ·Q cmq 6745  *Qcrq 6746
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 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-mi 6768  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-mqqs 6812  df-1nqqs 6813  df-rq 6814
This theorem is referenced by:  recclnq  6854  recidnq  6855  recrecnq  6856  recexprlem1ssl  7095  recexprlem1ssu  7096
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