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Theorem cstucnd 21993
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 (𝜑𝑈 ∈ (UnifOn‘𝑋))
cstucnd.2 (𝜑𝑉 ∈ (UnifOn‘𝑌))
cstucnd.3 (𝜑𝐴𝑌)
Assertion
Ref Expression
cstucnd (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))

Proof of Theorem cstucnd
Dummy variables 𝑠 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cstucnd.3 . . 3 (𝜑𝐴𝑌)
2 fconst6g 6053 . . 3 (𝐴𝑌 → (𝑋 × {𝐴}):𝑋𝑌)
31, 2syl 17 . 2 (𝜑 → (𝑋 × {𝐴}):𝑋𝑌)
4 cstucnd.1 . . . . . 6 (𝜑𝑈 ∈ (UnifOn‘𝑋))
54adantr 481 . . . . 5 ((𝜑𝑠𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6 ustne0 21922 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
75, 6syl 17 . . . 4 ((𝜑𝑠𝑉) → 𝑈 ≠ ∅)
8 cstucnd.2 . . . . . . . . . 10 (𝜑𝑉 ∈ (UnifOn‘𝑌))
98ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10 simpllr 798 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑠𝑉)
111ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑌)
12 ustref 21927 . . . . . . . . 9 ((𝑉 ∈ (UnifOn‘𝑌) ∧ 𝑠𝑉𝐴𝑌) → 𝐴𝑠𝐴)
139, 10, 11, 12syl3anc 1323 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑠𝐴)
14 simprl 793 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
15 fvconst2g 6422 . . . . . . . . 9 ((𝐴𝑌𝑥𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
1611, 14, 15syl2anc 692 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17 simprr 795 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
18 fvconst2g 6422 . . . . . . . . 9 ((𝐴𝑌𝑦𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
1911, 17, 18syl2anc 692 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
2013, 16, 193brtr4d 4650 . . . . . . 7 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦))
2120a1d 25 . . . . . 6 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2221ralrimivva 2970 . . . . 5 (((𝜑𝑠𝑉) ∧ 𝑟𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2322reximdva0 3914 . . . 4 (((𝜑𝑠𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
247, 23mpdan 701 . . 3 ((𝜑𝑠𝑉) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2524ralrimiva 2965 . 2 (𝜑 → ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26 isucn 21987 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
274, 8, 26syl2anc 692 . 2 (𝜑 → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
283, 25, 27mpbir2and 956 1 (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  c0 3896  {csn 4153   class class class wbr 4618   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  UnifOncust 21908   Cnucucn 21984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-ust 21909  df-ucn 21985
This theorem is referenced by: (None)
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