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Theorem cstucnd 21079
 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 5757 . . 3
31, 2syl 17 . 2
4 cstucnd.1 . . . . . 6 UnifOn
54adantr 463 . . . . 5 UnifOn
6 ustne0 21008 . . . . 5 UnifOn
75, 6syl 17 . . . 4
8 cstucnd.2 . . . . . . . . . 10 UnifOn
98ad3antrrr 728 . . . . . . . . 9 UnifOn
10 simpllr 761 . . . . . . . . 9
111ad3antrrr 728 . . . . . . . . 9
12 ustref 21013 . . . . . . . . 9 UnifOn
139, 10, 11, 12syl3anc 1230 . . . . . . . 8
14 simprl 756 . . . . . . . . 9
15 fvconst2g 6105 . . . . . . . . 9
1611, 14, 15syl2anc 659 . . . . . . . 8
17 simprr 758 . . . . . . . . 9
18 fvconst2g 6105 . . . . . . . . 9
1911, 17, 18syl2anc 659 . . . . . . . 8
2013, 16, 193brtr4d 4425 . . . . . . 7
2120a1d 25 . . . . . 6
2221ralrimivva 2825 . . . . 5
2322reximdva0 3750 . . . 4
247, 23mpdan 666 . . 3
2524ralrimiva 2818 . 2
26 isucn 21073 . . 3 UnifOn UnifOn Cnu
274, 8, 26syl2anc 659 . 2 Cnu
283, 25, 27mpbir2and 923 1 Cnu
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842   wne 2598  wral 2754  wrex 2755  c0 3738  csn 3972   class class class wbr 4395   cxp 4821  wf 5565  cfv 5569  (class class class)co 6278  UnifOncust 20994   Cnucucn 21070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-ust 20995  df-ucn 21071 This theorem is referenced by: (None)
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