Theorem cstucnd 23673

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	⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
cstucnd.2	⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
cstucnd.3	⊢ (𝜑 → 𝐴 ∈ 𝑌)

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.

Step	Hyp	Ref	Expression
1		cstucnd.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑌)
2		fconst6g 6736	. . 3 ⊢ (𝐴 ∈ 𝑌 → (𝑋 × {𝐴}):𝑋⟶𝑌)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝑋 × {𝐴}):𝑋⟶𝑌)
4		cstucnd.1	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6		ustne0 23602	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
7	5, 6	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉) → 𝑈 ≠ ∅)
8		cstucnd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
9	8	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑠 ∈ 𝑉)
11	1	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ 𝑌)
12		ustref 23607	. . . . . . . . 9 ⊢ ((𝑉 ∈ (UnifOn‘𝑌) ∧ 𝑠 ∈ 𝑉 ∧ 𝐴 ∈ 𝑌) → 𝐴𝑠𝐴)
13	9, 10, 11, 12	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴𝑠𝐴)
14		simprl 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
15		fvconst2g 7156	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
16	11, 14, 15	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17		simprr 771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
18		fvconst2g 7156	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
19	11, 17, 18	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
20	13, 16, 19	3brtr4d 5142	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦))
21	20	a1d 25	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
22	21	ralrimivva 3193	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
23	22	reximdva0 4316	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
24	7, 23	mpdan 685	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
25	24	ralrimiva 3139	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26		isucn 23667	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
27	4, 8, 26	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉) ↔ ((𝑋 × {𝐴}):𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))))
28	3, 25, 27	mpbir2and 711	1 ⊢ (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  ∅c0 4287  {csn 4591   class class class wbr 5110   × cxp 5636  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  UnifOncust 23588   Cn_ucucn 23664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-ust 23589  df-ucn 23665

This theorem is referenced by: (None)