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Theorem cnmpt2c 22282
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 (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt2c.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt2c.p (𝜑𝑃𝑍)
Assertion
Ref Expression
cnmpt2c (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((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.
StepHypRef Expression
1 eqidd 2825 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
21mpompt 7259 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥𝑋, 𝑦𝑌𝑃)
3 cnmpt21.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt21.k . . . 4 (𝜑𝐾 ∈ (TopOn‘𝑌))
5 txtopon 22203 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
63, 4, 5syl2anc 587 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7 cnmpt2c.l . . 3 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt2c.p . . 3 (𝜑𝑃𝑍)
96, 7, 8cnmptc 22274 . 2 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
102, 9eqeltrrid 2921 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cop 4556  cmpt 5132   × cxp 5540  cfv 6343  (class class class)co 7149  cmpo 7151  TopOnctopon 21522   Cn ccn 21836   ×t ctx 22172
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-topgen 16717  df-top 21506  df-topon 21523  df-bases 21558  df-cn 21839  df-cnp 21840  df-tx 22174
This theorem is referenced by:  cnrehmeo  23565  pcopt  23634  pcopt2  23635  vmcn  28489  dipcn  28510  cvxsconn  32551
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