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Theorem cnmpt2c 23693
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 2735 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
21mpompt 7546 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥𝑋, 𝑦𝑌𝑃)
3 cnmpt21.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt21.k . . . 4 (𝜑𝐾 ∈ (TopOn‘𝑌))
5 txtopon 23614 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
63, 4, 5syl2anc 584 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7 cnmpt2c.l . . 3 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt2c.p . . 3 (𝜑𝑃𝑍)
96, 7, 8cnmptc 23685 . 2 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
102, 9eqeltrrid 2843 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cop 4636  cmpt 5230   × cxp 5686  cfv 6562  (class class class)co 7430  cmpo 7432  TopOnctopon 22931   Cn ccn 23247   ×t ctx 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cn 23250  df-cnp 23251  df-tx 23585
This theorem is referenced by:  cnrehmeo  24997  cnrehmeoOLD  24998  pcopt  25068  pcopt2  25069  vmcn  30727  dipcn  30748  cvxsconn  35227
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