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Theorem genpelv 9399
 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
genp.2
Assertion
Ref Expression
genpelv
Distinct variable groups:   ,,,,,   ,,,,,   ,,,,,,,   ,   ,,

Proof of Theorem genpelv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . 4
2 genp.2 . . . 4
31, 2genpv 9398 . . 3
43eleq2d 2527 . 2
5 id 22 . . . . . 6
6 ovex 6324 . . . . . 6
75, 6syl6eqel 2553 . . . . 5
87rexlimivw 2946 . . . 4
98rexlimivw 2946 . . 3
10 eqeq1 2461 . . . 4
11102rexbidv 2975 . . 3
129, 11elab3 3253 . 2
134, 12syl6bb 261 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109  (class class class)co 6296  e.cmpt2 6298   cnq 9251   cnp 9258 This theorem is referenced by:  genpprecl  9400  genpss  9403  genpnnp  9404  genpcd  9405  genpnmax  9406  genpass  9408  distrlem1pr  9424  distrlem5pr  9426  1idpr  9428  ltexprlem6  9440  reclem3pr  9448  reclem4pr  9449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-inf2 8079 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6701  df-ni 9271  df-nq 9311  df-np 9380
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