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Theorem genpelv 10614
Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-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
genpelv ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑔,   𝑔,𝐹   𝐶,𝑔,
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,)

Proof of Theorem genpelv
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . 4 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2 genp.2 . . . 4 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpv 10613 . . 3 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
43eleq2d 2823 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}))
5 id 22 . . . . . 6 (𝐶 = (𝑔𝐺) → 𝐶 = (𝑔𝐺))
6 ovex 7246 . . . . . 6 (𝑔𝐺) ∈ V
75, 6eqeltrdi 2846 . . . . 5 (𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
87rexlimivw 3201 . . . 4 (∃𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
98rexlimivw 3201 . . 3 (∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
10 eqeq1 2741 . . . 4 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
11102rexbidv 3219 . . 3 (𝑓 = 𝐶 → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
129, 11elab3 3595 . 2 (𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)} ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺))
134, 12bitrdi 290 1 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  wrex 3062  Vcvv 3408  (class class class)co 7213  cmpo 7215  Qcnq 10466  Pcnp 10473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-ni 10486  df-nq 10526  df-np 10595
This theorem is referenced by:  genpprecl  10615  genpss  10618  genpnnp  10619  genpcd  10620  genpnmax  10621  genpass  10623  distrlem1pr  10639  distrlem5pr  10641  1idpr  10643  ltexprlem6  10655  reclem3pr  10663  reclem4pr  10664
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