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Theorem genpelv 11041
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 11040 . . 3 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
43eleq2d 2826 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}))
5 id 22 . . . . . 6 (𝐶 = (𝑔𝐺) → 𝐶 = (𝑔𝐺))
6 ovex 7465 . . . . . 6 (𝑔𝐺) ∈ V
75, 6eqeltrdi 2848 . . . . 5 (𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
87rexlimivw 3150 . . . 4 (∃𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
98rexlimivw 3150 . . 3 (∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
10 eqeq1 2740 . . . 4 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
11102rexbidv 3221 . . 3 (𝑓 = 𝐶 → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
129, 11elab3 3685 . 2 (𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)} ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺))
134, 12bitrdi 287 1 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wrex 3069  Vcvv 3479  (class class class)co 7432  cmpo 7434  Qcnq 10893  Pcnp 10900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-ni 10913  df-nq 10953  df-np 11022
This theorem is referenced by:  genpprecl  11042  genpss  11045  genpnnp  11046  genpcd  11047  genpnmax  11048  genpass  11050  distrlem1pr  11066  distrlem5pr  11068  1idpr  11070  ltexprlem6  11082  reclem3pr  11090  reclem4pr  11091
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