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Theorem recexnq 7364
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 7322 . 2 Q = ((N × N) / ~Q )
2 oveq1 5872 . . . . 5 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
32eqeq1d 2184 . . . 4 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
43anbi2d 464 . . 3 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
54exbidv 1823 . 2 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6 opelxpi 4652 . . . . . 6 ((𝑧N𝑥N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
76ancoms 268 . . . . 5 ((𝑥N𝑧N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8 enqex 7334 . . . . . 6 ~Q ∈ V
98ecelqsi 6579 . . . . 5 (⟨𝑧, 𝑥⟩ ∈ (N × N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
107, 9syl 14 . . . 4 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
1110, 1eleqtrrdi 2269 . . 3 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~QQ)
12 mulcompig 7305 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
1312opeq2d 3781 . . . . . 6 ((𝑥N𝑧N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
1413eceq1d 6561 . . . . 5 ((𝑥N𝑧N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15 mulclpi 7302 . . . . . 6 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) ∈ N)
16 1qec 7362 . . . . . 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 7347 . . . . . . 7 (((𝑥N𝑧N) ∧ (𝑧N𝑥N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
1918an42s 589 . . . . . 6 (((𝑥N𝑧N) ∧ (𝑥N𝑧N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2019anidms 397 . . . . 5 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2114, 17, 203eqtr4rd 2219 . . . 4 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
2211, 21jca 306 . . 3 ((𝑥N𝑧N) → ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23 eleq1 2238 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦Q ↔ [⟨𝑧, 𝑥⟩] ~QQ))
24 oveq2 5873 . . . . . 6 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
2524eqeq1d 2184 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
2623, 25anbi12d 473 . . . 4 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)))
2726spcegv 2823 . . 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 6612 1 (𝐴Q → ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1490  wcel 2146  cop 3592   × cxp 4618  (class class class)co 5865  [cec 6523   / cqs 6524  Ncnpi 7246   ·N cmi 7248   ~Q ceq 7253  Qcnq 7254  1Qc1q 7255   ·Q cmq 7257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-mi 7280  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-mqqs 7324  df-1nqqs 7325
This theorem is referenced by:  recmulnqg  7365  recclnq  7366
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