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Theorem recmulnqg 7167
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 5749 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
21eqeq1d 2126 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
32anbi2d 459 . . 3 (𝑥 = 𝐴 → ((𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
4 eleq1 2180 . . . 4 (𝑦 = 𝐵 → (𝑦Q𝐵Q))
5 oveq2 5750 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
65eqeq1d 2126 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
74, 6anbi12d 464 . . 3 (𝑦 = 𝐵 → ((𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q) ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
8 recexnq 7166 . . . 4 (𝑥Q → ∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9 1nq 7142 . . . . 5 1QQ
10 mulcomnqg 7159 . . . . 5 ((𝑧Q𝑤Q) → (𝑧 ·Q 𝑤) = (𝑤 ·Q 𝑧))
11 mulassnqg 7160 . . . . 5 ((𝑧Q𝑤Q𝑣Q) → ((𝑧 ·Q 𝑤) ·Q 𝑣) = (𝑧 ·Q (𝑤 ·Q 𝑣)))
12 mulidnq 7165 . . . . 5 (𝑧Q → (𝑧 ·Q 1Q) = 𝑧)
139, 10, 11, 12caovimo 5932 . . . 4 (𝑥Q → ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
14 eu5 2024 . . . 4 (∃!𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ∧ ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
158, 13, 14sylanbrc 413 . . 3 (𝑥Q → ∃!𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
16 df-rq 7128 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
17 3anass 951 . . . . 5 ((𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
1817opabbii 3965 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
1916, 18eqtri 2138 . . 3 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
203, 7, 15, 19fvopab3g 5462 . 2 ((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
21 ibar 299 . . 3 (𝐵Q → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
2221adantl 275 . 2 ((𝐴Q𝐵Q) → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
2320, 22bitr4d 190 1 ((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wex 1453  wcel 1465  ∃!weu 1977  ∃*wmo 1978  {copab 3958  cfv 5093  (class class class)co 5742  Qcnq 7056  1Qc1q 7057   ·Q cmq 7059  *Qcrq 7060
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-mi 7082  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-mqqs 7126  df-1nqqs 7127  df-rq 7128
This theorem is referenced by:  recclnq  7168  recidnq  7169  recrecnq  7170  recexprlem1ssl  7409  recexprlem1ssu  7410
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