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Theorem cstucnd 24309
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 6798 . . 3 (𝐴𝑌 → (𝑋 × {𝐴}):𝑋𝑌)
31, 2syl 17 . 2 (𝜑 → (𝑋 × {𝐴}):𝑋𝑌)
4 cstucnd.1 . . . . . 6 (𝜑𝑈 ∈ (UnifOn‘𝑋))
54adantr 480 . . . . 5 ((𝜑𝑠𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6 ustne0 24238 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
75, 6syl 17 . . . 4 ((𝜑𝑠𝑉) → 𝑈 ≠ ∅)
8 cstucnd.2 . . . . . . . . . 10 (𝜑𝑉 ∈ (UnifOn‘𝑌))
98ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10 simpllr 776 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑠𝑉)
111ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑌)
12 ustref 24243 . . . . . . . . 9 ((𝑉 ∈ (UnifOn‘𝑌) ∧ 𝑠𝑉𝐴𝑌) → 𝐴𝑠𝐴)
139, 10, 11, 12syl3anc 1370 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑠𝐴)
14 simprl 771 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
15 fvconst2g 7222 . . . . . . . . 9 ((𝐴𝑌𝑥𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
1611, 14, 15syl2anc 584 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17 simprr 773 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
18 fvconst2g 7222 . . . . . . . . 9 ((𝐴𝑌𝑦𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
1911, 17, 18syl2anc 584 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
2013, 16, 193brtr4d 5180 . . . . . . 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 3200 . . . . 5 (((𝜑𝑠𝑉) ∧ 𝑟𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2322reximdva0 4361 . . . 4 (((𝜑𝑠𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
247, 23mpdan 687 . . 3 ((𝜑𝑠𝑉) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2524ralrimiva 3144 . 2 (𝜑 → ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26 isucn 24303 . . 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 584 . 2 (𝜑 → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
283, 25, 27mpbir2and 713 1 (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  c0 4339  {csn 4631   class class class wbr 5148   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  UnifOncust 24224   Cnucucn 24300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ust 24225  df-ucn 24301
This theorem is referenced by: (None)
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