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Theorem cstucnd 22308
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 6234 . . 3 (𝐴𝑌 → (𝑋 × {𝐴}):𝑋𝑌)
31, 2syl 17 . 2 (𝜑 → (𝑋 × {𝐴}):𝑋𝑌)
4 cstucnd.1 . . . . . 6 (𝜑𝑈 ∈ (UnifOn‘𝑋))
54adantr 466 . . . . 5 ((𝜑𝑠𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6 ustne0 22237 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
75, 6syl 17 . . . 4 ((𝜑𝑠𝑉) → 𝑈 ≠ ∅)
8 cstucnd.2 . . . . . . . . . 10 (𝜑𝑉 ∈ (UnifOn‘𝑌))
98ad3antrrr 709 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10 simpllr 760 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑠𝑉)
111ad3antrrr 709 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑌)
12 ustref 22242 . . . . . . . . 9 ((𝑉 ∈ (UnifOn‘𝑌) ∧ 𝑠𝑉𝐴𝑌) → 𝐴𝑠𝐴)
139, 10, 11, 12syl3anc 1476 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑠𝐴)
14 simprl 754 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
15 fvconst2g 6611 . . . . . . . . 9 ((𝐴𝑌𝑥𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
1611, 14, 15syl2anc 573 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17 simprr 756 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
18 fvconst2g 6611 . . . . . . . . 9 ((𝐴𝑌𝑦𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
1911, 17, 18syl2anc 573 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
2013, 16, 193brtr4d 4818 . . . . . . 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 3120 . . . . 5 (((𝜑𝑠𝑉) ∧ 𝑟𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2322reximdva0 4080 . . . 4 (((𝜑𝑠𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
247, 23mpdan 667 . . 3 ((𝜑𝑠𝑉) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2524ralrimiva 3115 . 2 (𝜑 → ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26 isucn 22302 . . 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 573 . 2 (𝜑 → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
283, 25, 27mpbir2and 692 1 (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  c0 4063  {csn 4316   class class class wbr 4786   × cxp 5247  wf 6027  cfv 6031  (class class class)co 6793  UnifOncust 22223   Cnucucn 22299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-ust 22224  df-ucn 22300
This theorem is referenced by: (None)
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