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Theorem cstucnd 18306
 Description: A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypotheses
Ref Expression
cstucnd.1 UnifOn
cstucnd.2 UnifOn
cstucnd.3
Assertion
Ref Expression
cstucnd Cnu

Proof of Theorem cstucnd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cstucnd.3 . . 3
2 fconst6g 5624 . . 3
31, 2syl 16 . 2
4 cstucnd.1 . . . . . 6 UnifOn
54adantr 452 . . . . 5 UnifOn
6 ustne0 18235 . . . . 5 UnifOn
75, 6syl 16 . . . 4
8 cstucnd.2 . . . . . . . . . 10 UnifOn
98ad3antrrr 711 . . . . . . . . 9 UnifOn
10 simpllr 736 . . . . . . . . 9
111ad3antrrr 711 . . . . . . . . 9
12 ustref 18240 . . . . . . . . 9 UnifOn
139, 10, 11, 12syl3anc 1184 . . . . . . . 8
14 simprl 733 . . . . . . . . 9
15 fvconst2g 5937 . . . . . . . . 9
1611, 14, 15syl2anc 643 . . . . . . . 8
17 simprr 734 . . . . . . . . 9
18 fvconst2g 5937 . . . . . . . . 9
1911, 17, 18syl2anc 643 . . . . . . . 8
2013, 16, 193brtr4d 4234 . . . . . . 7
2120a1d 23 . . . . . 6
2221ralrimivva 2790 . . . . 5
2322reximdva0 3631 . . . 4
247, 23mpdan 650 . . 3
2524ralrimiva 2781 . 2
26 isucn 18300 . . 3 UnifOn UnifOn Cnu
274, 8, 26syl2anc 643 . 2 Cnu
283, 25, 27mpbir2and 889 1 Cnu
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698  c0 3620  csn 3806   class class class wbr 4204   cxp 4868  wf 5442  cfv 5446  (class class class)co 6073  UnifOncust 18221   Cnucucn 18297 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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 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-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ust 18222  df-ucn 18298
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