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Theorem stoweidlem2 27727
 Description: lemma for stoweid 27788: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem2.1
stoweidlem2.2
stoweidlem2.3
stoweidlem2.4
stoweidlem2.5
stoweidlem2.6
Assertion
Ref Expression
stoweidlem2
Distinct variable groups:   ,,,   ,,   ,,   ,,,   ,,   ,,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem stoweidlem2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 stoweidlem2.1 . . 3
2 simpr 448 . . . . . 6
3 stoweidlem2.5 . . . . . . 7
43adantr 452 . . . . . 6
5 eqidd 2437 . . . . . . . 8
65cbvmptv 4300 . . . . . . 7
76fvmpt2 5812 . . . . . 6
82, 4, 7syl2anc 643 . . . . 5
98eqcomd 2441 . . . 4
109oveq1d 6096 . . 3
111, 10mpteq2da 4294 . 2
12 id 20 . . . . . . . . 9
1312mpteq2dv 4296 . . . . . . . 8
1413eleq1d 2502 . . . . . . 7
1514imbi2d 308 . . . . . 6
16 stoweidlem2.3 . . . . . . 7
1716expcom 425 . . . . . 6
1815, 17vtoclga 3017 . . . . 5
193, 18mpcom 34 . . . 4
206, 19syl5eqel 2520 . . 3
21 fveq1 5727 . . . . . . . 8
2221oveq1d 6096 . . . . . . 7
2322mpteq2dv 4296 . . . . . 6
2423eleq1d 2502 . . . . 5
2524imbi2d 308 . . . 4
26 stoweidlem2.6 . . . . . . 7
2726adantr 452 . . . . . 6
28 fveq1 5727 . . . . . . . . . . 11
2928oveq2d 6097 . . . . . . . . . 10
3029mpteq2dv 4296 . . . . . . . . 9
3130eleq1d 2502 . . . . . . . 8
3231imbi2d 308 . . . . . . 7
33 stoweidlem2.2 . . . . . . . . 9
34333comr 1161 . . . . . . . 8
35343expib 1156 . . . . . . 7
3632, 35vtoclga 3017 . . . . . 6
3727, 36mpcom 34 . . . . 5
3837expcom 425 . . . 4
3925, 38vtoclga 3017 . . 3
4020, 39mpcom 34 . 2
4111, 40eqeltrd 2510 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936  wnf 1553   wceq 1652   wcel 1725   cmpt 4266  wf 5450  cfv 5454  (class class class)co 6081  cr 8989   cmul 8995 This theorem is referenced by:  stoweidlem17  27742 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 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084
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