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Theorem genpprecl 8871
 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
genp.2
Assertion
Ref Expression
genpprecl
Distinct variable groups:   ,,,   ,,,   ,,,,,
Allowed substitution hints:   (,)   (,)   (,,,,)   (,,,,)   (,,,,)

Proof of Theorem genpprecl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3
2 rspceov 6109 . . 3
31, 2mp3an3 1268 . 2
4 genp.1 . . 3
5 genp.2 . . 3
64, 5genpelv 8870 . 2
73, 6syl5ibr 213 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cab 2422  wrex 2699  (class class class)co 6074   cmpt2 6076  cnq 8720  cnp 8727 This theorem is referenced by:  genpn0  8873  genpnmax  8877  addclprlem2  8887  mulclprlem  8889  distrlem1pr  8895  distrlem4pr  8896  ltaddpr  8904  ltexprlem7  8912 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-ni 8742  df-nq 8782  df-np 8851
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