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Theorem recmulnqg 7211
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 5781 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
21eqeq1d 2148 . . . 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 2202 . . . 4 (𝑦 = 𝐵 → (𝑦Q𝐵Q))
5 oveq2 5782 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
65eqeq1d 2148 . . . 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 7210 . . . 4 (𝑥Q → ∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9 1nq 7186 . . . . 5 1QQ
10 mulcomnqg 7203 . . . . 5 ((𝑧Q𝑤Q) → (𝑧 ·Q 𝑤) = (𝑤 ·Q 𝑧))
11 mulassnqg 7204 . . . . 5 ((𝑧Q𝑤Q𝑣Q) → ((𝑧 ·Q 𝑤) ·Q 𝑣) = (𝑧 ·Q (𝑤 ·Q 𝑣)))
12 mulidnq 7209 . . . . 5 (𝑧Q → (𝑧 ·Q 1Q) = 𝑧)
139, 10, 11, 12caovimo 5964 . . . 4 (𝑥Q → ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
14 eu5 2046 . . . 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 7172 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
17 3anass 966 . . . . 5 ((𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
1817opabbii 3995 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
1916, 18eqtri 2160 . . 3 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
203, 7, 15, 19fvopab3g 5494 . 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 962   = wceq 1331  wex 1468  wcel 1480  ∃!weu 1999  ∃*wmo 2000  {copab 3988  cfv 5123  (class class class)co 5774  Qcnq 7100  1Qc1q 7101   ·Q cmq 7103  *Qcrq 7104
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-mi 7126  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-mqqs 7170  df-1nqqs 7171  df-rq 7172
This theorem is referenced by:  recclnq  7212  recidnq  7213  recrecnq  7214  recexprlem1ssl  7453  recexprlem1ssu  7454
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