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Theorem stoweidlem15 27731
 Description: This lemma is used to prove the existence of a function as in Lemma 1 from [BrosowskiDeutsh] p. 90: is in the subalgebra, such that 0 ≤ p ≤ 1, p(t_0) = 0, and p > 0 on T - U. Here is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem15.1
stoweidlem15.3
stoweidlem15.4
Assertion
Ref Expression
stoweidlem15
Distinct variable groups:   ,   ,   ,   ,   ,   ,,   ,   ,,   ,,   ,
Allowed substitution hints:   (,)   ()   (,,)   (,,)   (,,)   (,)

Proof of Theorem stoweidlem15
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . 4
2 stoweidlem15.3 . . . . . 6
32fnvinran 27652 . . . . 5
4 elrabi 3082 . . . . . 6
5 stoweidlem15.1 . . . . . 6
64, 5eleq2s 2527 . . . . 5
73, 6syl 16 . . . 4
8 eleq1 2495 . . . . . . . 8
98anbi2d 685 . . . . . . 7
10 feq1 5568 . . . . . . 7
119, 10imbi12d 312 . . . . . 6
12 stoweidlem15.4 . . . . . 6
1311, 12vtoclg 3003 . . . . 5
147, 13syl 16 . . . 4
151, 7, 14mp2and 661 . . 3
1615fnvinran 27652 . 2
173, 5syl6eleq 2525 . . . . . . 7
18 fveq1 5719 . . . . . . . . . 10
1918eqeq1d 2443 . . . . . . . . 9
20 fveq1 5719 . . . . . . . . . . . 12
2120breq2d 4216 . . . . . . . . . . 11
2220breq1d 4214 . . . . . . . . . . 11
2321, 22anbi12d 692 . . . . . . . . . 10
2423ralbidv 2717 . . . . . . . . 9
2519, 24anbi12d 692 . . . . . . . 8
2625elrab 3084 . . . . . . 7
2717, 26sylib 189 . . . . . 6
2827simprd 450 . . . . 5
2928simprd 450 . . . 4
30 fveq2 5720 . . . . . . . 8
3130breq2d 4216 . . . . . . 7
3230breq1d 4214 . . . . . . 7
3331, 32anbi12d 692 . . . . . 6
3433cbvralv 2924 . . . . 5
35 fveq2 5720 . . . . . . . 8
3635breq2d 4216 . . . . . . 7
3735breq1d 4214 . . . . . . 7
3836, 37anbi12d 692 . . . . . 6
3938rspccva 3043 . . . . 5
4034, 39sylanbr 460 . . . 4
4129, 40sylan 458 . . 3
4241simpld 446 . 2
4341simprd 450 . 2
4416, 42, 433jca 1134 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  crab 2701   class class class wbr 4204  wf 5442  cfv 5446  (class class class)co 6073  cr 8981  cc0 8982  c1 8983   cle 9113  cfz 11035 This theorem is referenced by:  stoweidlem30  27746  stoweidlem38  27754  stoweidlem44  27760 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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