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Theorem recexnq 6545
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 6503 . 2 Q = ((N × N) / ~Q )
2 oveq1 5546 . . . . 5 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
32eqeq1d 2064 . . . 4 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
43anbi2d 445 . . 3 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
54exbidv 1722 . 2 ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6 opelxpi 4403 . . . . . 6 ((𝑧N𝑥N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
76ancoms 259 . . . . 5 ((𝑥N𝑧N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8 enqex 6515 . . . . . 6 ~Q ∈ V
98ecelqsi 6190 . . . . 5 (⟨𝑧, 𝑥⟩ ∈ (N × N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
107, 9syl 14 . . . 4 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
1110, 1syl6eleqr 2147 . . 3 ((𝑥N𝑧N) → [⟨𝑧, 𝑥⟩] ~QQ)
12 mulcompig 6486 . . . . . . 7 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
1312opeq2d 3583 . . . . . 6 ((𝑥N𝑧N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
1413eceq1d 6172 . . . . 5 ((𝑥N𝑧N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15 mulclpi 6483 . . . . . 6 ((𝑥N𝑧N) → (𝑥 ·N 𝑧) ∈ N)
16 1qec 6543 . . . . . 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 6528 . . . . . . 7 (((𝑥N𝑧N) ∧ (𝑧N𝑥N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
1918an42s 531 . . . . . 6 (((𝑥N𝑧N) ∧ (𝑥N𝑧N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2019anidms 383 . . . . 5 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
2114, 17, 203eqtr4rd 2099 . . . 4 ((𝑥N𝑧N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
2211, 21jca 294 . . 3 ((𝑥N𝑧N) → ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23 eleq1 2116 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦Q ↔ [⟨𝑧, 𝑥⟩] ~QQ))
24 oveq2 5547 . . . . . 6 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
2524eqeq1d 2064 . . . . 5 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
2623, 25anbi12d 450 . . . 4 (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ((𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ([⟨𝑧, 𝑥⟩] ~QQ ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)))
2726spcegv 2658 . . 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 60 . 2 ((𝑥N𝑧N) → ∃𝑦(𝑦Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q))
291, 5, 28ecoptocl 6223 1 (𝐴Q → ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  cop 3405   × cxp 4370  (class class class)co 5539  [cec 6134   / cqs 6135  Ncnpi 6427   ·N cmi 6429   ~Q ceq 6434  Qcnq 6435  1Qc1q 6436   ·Q cmq 6438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-mqqs 6505  df-1nqqs 6506
This theorem is referenced by:  recmulnqg  6546  recclnq  6547
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