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Theorem recexpr 9725
Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
recexpr (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexpr
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4576 . . . . . 6 (𝑧 = 𝑤 → (𝑧 <Q 𝑦𝑤 <Q 𝑦))
21anbi1d 736 . . . . 5 (𝑧 = 𝑤 → ((𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ (𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
32exbidv 1835 . . . 4 (𝑧 = 𝑤 → (∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
43cbvabv 2729 . . 3 {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} = {𝑤 ∣ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
54reclem2pr 9722 . 2 (𝐴P → {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P)
64reclem4pr 9724 . 2 (𝐴P → (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P)
7 oveq2 6531 . . . 4 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → (𝐴 ·P 𝑥) = (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}))
87eqeq1d 2607 . . 3 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P))
98rspcev 3277 . 2 (({𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P ∧ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P) → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
105, 6, 9syl2anc 690 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1975  {cab 2591  wrex 2892   class class class wbr 4573  cfv 5786  (class class class)co 6523  *Qcrq 9531   <Q cltq 9532  Pcnp 9533  1Pc1p 9534   ·P cmp 9536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-omul 7425  df-er 7602  df-ni 9546  df-pli 9547  df-mi 9548  df-lti 9549  df-plpq 9582  df-mpq 9583  df-ltpq 9584  df-enq 9585  df-nq 9586  df-erq 9587  df-plq 9588  df-mq 9589  df-1nq 9590  df-rq 9591  df-ltnq 9592  df-np 9655  df-1p 9656  df-mp 9658
This theorem is referenced by:  recexsrlem  9776
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