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Theorem recexnq 6712
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 6670 . 2 Q = ((N × N) / ~Q )
2 oveq1 5571 . . . . 5 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
32eqeq1d 2091 . . . 4 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
43anbi2d 452 . . 3 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
54exbidv 1748 . 2 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6 opelxpi 4422 . . . . . 6 ((𝑧N𝑥N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
76ancoms 264 . . . . 5 ((𝑥N𝑧N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8 enqex 6682 . . . . . 6 ~Q ∈ V
98ecelqsi 6248 . . . . 5 (⟨𝑧, 𝑥⟩ ∈ (N × N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
107, 9syl 14 . . . 4 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
1110, 1syl6eleqr 2176 . . 3 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~QQ)
12 mulcompig 6653 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
1312opeq2d 3597 . . . . . 6 ((𝑥N𝑧N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
1413eceq1d 6230 . . . . 5 ((𝑥N𝑧N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15 mulclpi 6650 . . . . . 6 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) ∈ N)
16 1qec 6710 . . . . . 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 6695 . . . . . . 7 (((𝑥N𝑧N) ∧ (𝑧N𝑥N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
1918an42s 554 . . . . . 6 (((𝑥N𝑧N) ∧ (𝑥N𝑧N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2019anidms 389 . . . . 5 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2114, 17, 203eqtr4rd 2126 . . . 4 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
2211, 21jca 300 . . 3 ((𝑥N𝑧N) → ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23 eleq1 2145 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦Q ↔ [⟨𝑧, 𝑥⟩] ~QQ))
24 oveq2 5572 . . . . . 6 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
2524eqeq1d 2091 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
2623, 25anbi12d 457 . . . 4 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)))
2726spcegv 2695 . . 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 61 . 2 ((𝑥N𝑧N) → ∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q))
291, 5, 28ecoptocl 6281 1 (𝐴Q → ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  cop 3419   × cxp 4389  (class class class)co 5564  [cec 6192   / cqs 6193  Ncnpi 6594   ·N cmi 6596   ~Q ceq 6601  Qcnq 6602  1Qc1q 6603   ·Q cmq 6605
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 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
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 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-mi 6628  df-mpq 6667  df-enq 6669  df-nqqs 6670  df-mqqs 6672  df-1nqqs 6673
This theorem is referenced by:  recmulnqg  6713  recclnq  6714
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