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Theorem cnmpt2c 21801
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 2801 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
21mpt2mpt 6987 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥𝑋, 𝑦𝑌𝑃)
3 cnmpt21.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt21.k . . . 4 (𝜑𝐾 ∈ (TopOn‘𝑌))
5 txtopon 21722 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
63, 4, 5syl2anc 580 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7 cnmpt2c.l . . 3 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt2c.p . . 3 (𝜑𝑃𝑍)
96, 7, 8cnmptc 21793 . 2 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
102, 9syl5eqelr 2884 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cop 4375  cmpt 4923   × cxp 5311  cfv 6102  (class class class)co 6879  cmpt2 6881  TopOnctopon 21042   Cn ccn 21356   ×t ctx 21691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-map 8098  df-topgen 16418  df-top 21026  df-topon 21043  df-bases 21078  df-cn 21359  df-cnp 21360  df-tx 21693
This theorem is referenced by:  cnrehmeo  23079  pcopt  23148  pcopt2  23149  vmcn  28078  dipcn  28099  cvxsconn  31741
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