MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recexpr Structured version   Visualization version   GIF version

Theorem recexpr 9752
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 4586 . . . . . 6 (𝑧 = 𝑤 → (𝑧 <Q 𝑦𝑤 <Q 𝑦))
21anbi1d 737 . . . . 5 (𝑧 = 𝑤 → ((𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ (𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
32exbidv 1837 . . . 4 (𝑧 = 𝑤 → (∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
43cbvabv 2734 . . 3 {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} = {𝑤 ∣ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
54reclem2pr 9749 . 2 (𝐴P → {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P)
64reclem4pr 9751 . 2 (𝐴P → (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P)
7 oveq2 6557 . . . 4 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → (𝐴 ·P 𝑥) = (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}))
87eqeq1d 2612 . . 3 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P))
98rspcev 3282 . 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 691 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  *Qcrq 9558   <Q cltq 9559  Pcnp 9560  1Pc1p 9561   ·P cmp 9563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-1p 9683  df-mp 9685
This theorem is referenced by:  recexsrlem  9803
  Copyright terms: Public domain W3C validator