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Theorem genpprecl 9808
Description: Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpprecl ((𝐴P𝐵P) → ((𝐶𝐴𝐷𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpprecl
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2620 . . 3 (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2 rspceov 6677 . . 3 ((𝐶𝐴𝐷𝐵 ∧ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)) → ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺))
31, 2mp3an3 1411 . 2 ((𝐶𝐴𝐷𝐵) → ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺))
4 genp.1 . . 3 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
5 genp.2 . . 3 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
64, 5genpelv 9807 . 2 ((𝐴P𝐵P) → ((𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺)))
73, 6syl5ibr 236 1 ((𝐴P𝐵P) → ((𝐶𝐴𝐷𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  {cab 2606  wrex 2910  (class class class)co 6635  cmpt2 6637  Qcnq 9659  Pcnp 9666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-ni 9679  df-nq 9719  df-np 9788
This theorem is referenced by:  genpn0  9810  genpnmax  9814  addclprlem2  9824  mulclprlem  9826  distrlem1pr  9832  distrlem4pr  9833  ltaddpr  9841  ltexprlem7  9849
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