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Theorem cstucnd 22281
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 6247 . . 3 (𝐴𝑌 → (𝑋 × {𝐴}):𝑋𝑌)
31, 2syl 17 . 2 (𝜑 → (𝑋 × {𝐴}):𝑋𝑌)
4 cstucnd.1 . . . . . 6 (𝜑𝑈 ∈ (UnifOn‘𝑋))
54adantr 472 . . . . 5 ((𝜑𝑠𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6 ustne0 22210 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
75, 6syl 17 . . . 4 ((𝜑𝑠𝑉) → 𝑈 ≠ ∅)
8 cstucnd.2 . . . . . . . . . 10 (𝜑𝑉 ∈ (UnifOn‘𝑌))
98ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10 simpllr 817 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑠𝑉)
111ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑌)
12 ustref 22215 . . . . . . . . 9 ((𝑉 ∈ (UnifOn‘𝑌) ∧ 𝑠𝑉𝐴𝑌) → 𝐴𝑠𝐴)
139, 10, 11, 12syl3anc 1473 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑠𝐴)
14 simprl 811 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
15 fvconst2g 6623 . . . . . . . . 9 ((𝐴𝑌𝑥𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
1611, 14, 15syl2anc 696 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17 simprr 813 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
18 fvconst2g 6623 . . . . . . . . 9 ((𝐴𝑌𝑦𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
1911, 17, 18syl2anc 696 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
2013, 16, 193brtr4d 4828 . . . . . . 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 3101 . . . . 5 (((𝜑𝑠𝑉) ∧ 𝑟𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2322reximdva0 4068 . . . 4 (((𝜑𝑠𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
247, 23mpdan 705 . . 3 ((𝜑𝑠𝑉) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2524ralrimiva 3096 . 2 (𝜑 → ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26 isucn 22275 . . 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 696 . 2 (𝜑 → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
283, 25, 27mpbir2and 995 1 (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wne 2924  wral 3042  wrex 3043  c0 4050  {csn 4313   class class class wbr 4796   × cxp 5256  wf 6037  cfv 6041  (class class class)co 6805  UnifOncust 22196   Cnucucn 22272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-ust 22197  df-ucn 22273
This theorem is referenced by: (None)
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