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Theorem genpprecl 11039
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 2735 . . 3 (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2 rspceov 7480 . . 3 ((𝐶𝐴𝐷𝐵 ∧ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)) → ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺))
31, 2mp3an3 1449 . 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 11038 . 2 ((𝐴P𝐵P) → ((𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺)))
73, 6imbitrrid 246 1 ((𝐴P𝐵P) → ((𝐶𝐴𝐷𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  (class class class)co 7431  cmpo 7433  Qcnq 10890  Pcnp 10897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-ni 10910  df-nq 10950  df-np 11019
This theorem is referenced by:  genpn0  11041  genpnmax  11045  addclprlem2  11055  mulclprlem  11057  distrlem1pr  11063  distrlem4pr  11064  ltaddpr  11072  ltexprlem7  11080
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