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Theorem genpprecl 10416
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 2801 . . 3 (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2 rspceov 7186 . . 3 ((𝐶𝐴𝐷𝐵 ∧ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)) → ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺))
31, 2mp3an3 1447 . 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 10415 . 2 ((𝐴P𝐵P) → ((𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 (𝐶𝐺𝐷) = (𝑔𝐺)))
73, 6syl5ibr 249 1 ((𝐴P𝐵P) → ((𝐶𝐴𝐷𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  (class class class)co 7139  cmpo 7141  Qcnq 10267  Pcnp 10274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-ni 10287  df-nq 10327  df-np 10396
This theorem is referenced by:  genpn0  10418  genpnmax  10422  addclprlem2  10432  mulclprlem  10434  distrlem1pr  10440  distrlem4pr  10441  ltaddpr  10449  ltexprlem7  10457
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