ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recmulnqg GIF version

Theorem recmulnqg 7047
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 5697 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
21eqeq1d 2103 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
32anbi2d 453 . . 3 (𝑥 = 𝐴 → ((𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
4 eleq1 2157 . . . 4 (𝑦 = 𝐵 → (𝑦Q𝐵Q))
5 oveq2 5698 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
65eqeq1d 2103 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
74, 6anbi12d 458 . . 3 (𝑦 = 𝐵 → ((𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q) ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
8 recexnq 7046 . . . 4 (𝑥Q → ∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9 1nq 7022 . . . . 5 1QQ
10 mulcomnqg 7039 . . . . 5 ((𝑧Q𝑤Q) → (𝑧 ·Q 𝑤) = (𝑤 ·Q 𝑧))
11 mulassnqg 7040 . . . . 5 ((𝑧Q𝑤Q𝑣Q) → ((𝑧 ·Q 𝑤) ·Q 𝑣) = (𝑧 ·Q (𝑤 ·Q 𝑣)))
12 mulidnq 7045 . . . . 5 (𝑧Q → (𝑧 ·Q 1Q) = 𝑧)
139, 10, 11, 12caovimo 5876 . . . 4 (𝑥Q → ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
14 eu5 2002 . . . 4 (∃!𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (∃𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ∧ ∃*𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
158, 13, 14sylanbrc 409 . . 3 (𝑥Q → ∃!𝑦(𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))
16 df-rq 7008 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
17 3anass 931 . . . . 5 ((𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
1817opabbii 3927 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
1916, 18eqtri 2115 . . 3 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
203, 7, 15, 19fvopab3g 5412 . 2 ((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
21 ibar 296 . . 3 (𝐵Q → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
2221adantl 272 . 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 927   = wceq 1296  wex 1433  wcel 1445  ∃!weu 1955  ∃*wmo 1956  {copab 3920  cfv 5049  (class class class)co 5690  Qcnq 6936  1Qc1q 6937   ·Q cmq 6939  *Qcrq 6940
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-mi 6962  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-mqqs 7006  df-1nqqs 7007  df-rq 7008
This theorem is referenced by:  recclnq  7048  recidnq  7049  recrecnq  7050  recexprlem1ssl  7289  recexprlem1ssu  7290
  Copyright terms: Public domain W3C validator