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Theorem cstucnd 24280
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 6791 . . 3 (𝐴𝑌 → (𝑋 × {𝐴}):𝑋𝑌)
31, 2syl 17 . 2 (𝜑 → (𝑋 × {𝐴}):𝑋𝑌)
4 cstucnd.1 . . . . . 6 (𝜑𝑈 ∈ (UnifOn‘𝑋))
54adantr 479 . . . . 5 ((𝜑𝑠𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6 ustne0 24209 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
75, 6syl 17 . . . 4 ((𝜑𝑠𝑉) → 𝑈 ≠ ∅)
8 cstucnd.2 . . . . . . . . . 10 (𝜑𝑉 ∈ (UnifOn‘𝑌))
98ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10 simpllr 774 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑠𝑉)
111ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑌)
12 ustref 24214 . . . . . . . . 9 ((𝑉 ∈ (UnifOn‘𝑌) ∧ 𝑠𝑉𝐴𝑌) → 𝐴𝑠𝐴)
139, 10, 11, 12syl3anc 1368 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑠𝐴)
14 simprl 769 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
15 fvconst2g 7219 . . . . . . . . 9 ((𝐴𝑌𝑥𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
1611, 14, 15syl2anc 582 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17 simprr 771 . . . . . . . . 9 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
18 fvconst2g 7219 . . . . . . . . 9 ((𝐴𝑌𝑦𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
1911, 17, 18syl2anc 582 . . . . . . . 8 ((((𝜑𝑠𝑉) ∧ 𝑟𝑈) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
2013, 16, 193brtr4d 5185 . . . . . . 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 3191 . . . . 5 (((𝜑𝑠𝑉) ∧ 𝑟𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2322reximdva0 4354 . . . 4 (((𝜑𝑠𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
247, 23mpdan 685 . . 3 ((𝜑𝑠𝑉) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
2524ralrimiva 3136 . 2 (𝜑 → ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26 isucn 24274 . . 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 582 . 2 (𝜑 → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
283, 25, 27mpbir2and 711 1 (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  c0 4325  {csn 4633   class class class wbr 5153   × cxp 5680  wf 6550  cfv 6554  (class class class)co 7424  UnifOncust 24195   Cnucucn 24271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-ust 24196  df-ucn 24272
This theorem is referenced by: (None)
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