Theorem cstucnd 24314

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 6810	. . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
6		ustne0 24243	. . . . 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 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑉 ∈ (UnifOn‘𝑌))
10		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑠 ∈ 𝑉)
11	1	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ 𝑌)
12		ustref 24248	. . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
15		fvconst2g 7239	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
16	11, 14, 15	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
17		simprr 772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
18		fvconst2g 7239	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
19	11, 17, 18	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑋 × {𝐴})‘𝑦) = 𝐴)
20	13, 16, 19	3brtr4d 5198	. . . . . . 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 3208	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑟 ∈ 𝑈) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
23	22	reximdva0 4378	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑉) ∧ 𝑈 ≠ ∅) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
24	7, 23	mpdan 686	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
25	24	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → ((𝑋 × {𝐴})‘𝑥)𝑠((𝑋 × {𝐴})‘𝑦)))
26		isucn 24308	. . 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 583	. 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 712	1 ⊢ (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∅c0 4352  {csn 4648   class class class wbr 5166   × cxp 5698  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  UnifOncust 24229   Cn_ucucn 24305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-ust 24230  df-ucn 24306

This theorem is referenced by: (None)