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

Theorem recexnq 7210
Description: Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
Assertion
Ref Expression
recexnq (𝐴Q → ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Distinct variable group:   𝑦,𝐴

Proof of Theorem recexnq
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nqqs 7168 . 2 Q = ((N × N) / ~Q )
2 oveq1 5781 . . . . 5 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
32eqeq1d 2148 . . . 4 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
43anbi2d 459 . . 3 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
54exbidv 1797 . 2 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6 opelxpi 4571 . . . . . 6 ((𝑧N𝑥N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
76ancoms 266 . . . . 5 ((𝑥N𝑧N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8 enqex 7180 . . . . . 6 ~Q ∈ V
98ecelqsi 6483 . . . . 5 (⟨𝑧, 𝑥⟩ ∈ (N × N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
107, 9syl 14 . . . 4 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
1110, 1eleqtrrdi 2233 . . 3 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~QQ)
12 mulcompig 7151 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
1312opeq2d 3712 . . . . . 6 ((𝑥N𝑧N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
1413eceq1d 6465 . . . . 5 ((𝑥N𝑧N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15 mulclpi 7148 . . . . . 6 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) ∈ N)
16 1qec 7208 . . . . . 6 ((𝑥 ·N 𝑧) ∈ N → 1Q = [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q )
1715, 16syl 14 . . . . 5 ((𝑥N𝑧N) → 1Q = [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q )
18 mulpipqqs 7193 . . . . . . 7 (((𝑥N𝑧N) ∧ (𝑧N𝑥N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
1918an42s 578 . . . . . 6 (((𝑥N𝑧N) ∧ (𝑥N𝑧N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2019anidms 394 . . . . 5 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2114, 17, 203eqtr4rd 2183 . . . 4 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
2211, 21jca 304 . . 3 ((𝑥N𝑧N) → ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23 eleq1 2202 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦Q ↔ [⟨𝑧, 𝑥⟩] ~QQ))
24 oveq2 5782 . . . . . 6 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
2524eqeq1d 2148 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
2623, 25anbi12d 464 . . . 4 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)))
2726spcegv 2774 . . 3 ([⟨𝑧, 𝑥⟩] ~QQ → (([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q) → ∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q)))
2811, 22, 27sylc 62 . 2 ((𝑥N𝑧N) → ∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q))
291, 5, 28ecoptocl 6516 1 (𝐴Q → ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wex 1468  wcel 1480  cop 3530   × cxp 4537  (class class class)co 5774  [cec 6427   / cqs 6428  Ncnpi 7092   ·N cmi 7094   ~Q ceq 7099  Qcnq 7100  1Qc1q 7101   ·Q cmq 7103
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
This theorem is referenced by:  recmulnqg  7211  recclnq  7212
  Copyright terms: Public domain W3C validator