Theorem cnmpt2c 23557

Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmpt21.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt2c.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt2c.p	⊢ (𝜑 → 𝑃 ∈ 𝑍)

Assertion

Ref	Expression
cnmpt2c	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑃,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦

Allowed substitution hints:   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2c

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2730	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
2	1	mpompt 7503	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃)
3		cnmpt21.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		cnmpt21.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
5		txtopon 23478	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6	3, 4, 5	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7		cnmpt2c.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
8		cnmpt2c.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑍)
9	6, 7, 8	cnmptc 23549	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
10	2, 9	eqeltrrid 2833	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   ↦ cmpt 5188   × cxp 5636  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  TopOnctopon 22797   Cn ccn 23111   ×_t ctx 23447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cn 23114  df-cnp 23115  df-tx 23449

This theorem is referenced by:  cnrehmeo  24851  cnrehmeoOLD  24852  pcopt  24922  pcopt2  24923  vmcn  30628  dipcn  30649  cvxsconn  35230