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Theorem recexnq 6374
Description: Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
Assertion
Ref Expression
recexnq (A Qy(y Q (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 6332 . 2 Q = ((N × N) / ~Q )
2 oveq1 5462 . . . . 5 ([⟨x, z⟩] ~Q = A → ([⟨x, z⟩] ~Q ·Q y) = (A ·Q y))
32eqeq1d 2045 . . . 4 ([⟨x, z⟩] ~Q = A → (([⟨x, z⟩] ~Q ·Q y) = 1Q ↔ (A ·Q y) = 1Q))
43anbi2d 437 . . 3 ([⟨x, z⟩] ~Q = A → ((y Q ([⟨x, z⟩] ~Q ·Q y) = 1Q) ↔ (y Q (A ·Q y) = 1Q)))
54exbidv 1703 . 2 ([⟨x, z⟩] ~Q = A → (y(y Q ([⟨x, z⟩] ~Q ·Q y) = 1Q) ↔ y(y Q (A ·Q y) = 1Q)))
6 opelxpi 4319 . . . . . 6 ((z N x N) → ⟨z, x (N × N))
76ancoms 255 . . . . 5 ((x N z N) → ⟨z, x (N × N))
8 enqex 6344 . . . . . 6 ~Q V
98ecelqsi 6096 . . . . 5 (⟨z, x (N × N) → [⟨z, x⟩] ~Q ((N × N) / ~Q ))
107, 9syl 14 . . . 4 ((x N z N) → [⟨z, x⟩] ~Q ((N × N) / ~Q ))
1110, 1syl6eleqr 2128 . . 3 ((x N z N) → [⟨z, x⟩] ~Q Q)
12 mulcompig 6315 . . . . . . 7 ((x N z N) → (x ·N z) = (z ·N x))
1312opeq2d 3547 . . . . . 6 ((x N z N) → ⟨(x ·N z), (x ·N z)⟩ = ⟨(x ·N z), (z ·N x)⟩)
1413eceq1d 6078 . . . . 5 ((x N z N) → [⟨(x ·N z), (x ·N z)⟩] ~Q = [⟨(x ·N z), (z ·N x)⟩] ~Q )
15 mulclpi 6312 . . . . . 6 ((x N z N) → (x ·N z) N)
16 1qec 6372 . . . . . 6 ((x ·N z) N → 1Q = [⟨(x ·N z), (x ·N z)⟩] ~Q )
1715, 16syl 14 . . . . 5 ((x N z N) → 1Q = [⟨(x ·N z), (x ·N z)⟩] ~Q )
18 mulpipqqs 6357 . . . . . . 7 (((x N z N) (z N x N)) → ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = [⟨(x ·N z), (z ·N x)⟩] ~Q )
1918an42s 523 . . . . . 6 (((x N z N) (x N z N)) → ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = [⟨(x ·N z), (z ·N x)⟩] ~Q )
2019anidms 377 . . . . 5 ((x N z N) → ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = [⟨(x ·N z), (z ·N x)⟩] ~Q )
2114, 17, 203eqtr4rd 2080 . . . 4 ((x N z N) → ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = 1Q)
2211, 21jca 290 . . 3 ((x N z N) → ([⟨z, x⟩] ~Q Q ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = 1Q))
23 eleq1 2097 . . . . 5 (y = [⟨z, x⟩] ~Q → (y Q ↔ [⟨z, x⟩] ~Q Q))
24 oveq2 5463 . . . . . 6 (y = [⟨z, x⟩] ~Q → ([⟨x, z⟩] ~Q ·Q y) = ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ))
2524eqeq1d 2045 . . . . 5 (y = [⟨z, x⟩] ~Q → (([⟨x, z⟩] ~Q ·Q y) = 1Q ↔ ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = 1Q))
2623, 25anbi12d 442 . . . 4 (y = [⟨z, x⟩] ~Q → ((y Q ([⟨x, z⟩] ~Q ·Q y) = 1Q) ↔ ([⟨z, x⟩] ~Q Q ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = 1Q)))
2726spcegv 2635 . . 3 ([⟨z, x⟩] ~Q Q → (([⟨z, x⟩] ~Q Q ([⟨x, z⟩] ~Q ·Q [⟨z, x⟩] ~Q ) = 1Q) → y(y Q ([⟨x, z⟩] ~Q ·Q y) = 1Q)))
2811, 22, 27sylc 56 . 2 ((x N z N) → y(y Q ([⟨x, z⟩] ~Q ·Q y) = 1Q))
291, 5, 28ecoptocl 6129 1 (A Qy(y Q (A ·Q y) = 1Q))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  cop 3370   × cxp 4286  (class class class)co 5455  [cec 6040   / cqs 6041  Ncnpi 6256   ·N cmi 6258   ~Q ceq 6263  Qcnq 6264  1Qc1q 6265   ·Q cmq 6267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-mqqs 6334  df-1nqqs 6335
This theorem is referenced by:  recmulnqg  6375  recclnq  6376
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