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Theorem cnmpt2c 23525
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt21.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmpt2c.l (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
cnmpt2c.p (πœ‘ β†’ 𝑃 ∈ 𝑍)
Assertion
Ref Expression
cnmpt2c (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑃) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
Distinct variable groups:   π‘₯,𝑦,𝐿   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝑃,𝑦   π‘₯,π‘Œ,𝑦   π‘₯,𝑍,𝑦
Allowed substitution hints:   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)

Proof of Theorem cnmpt2c
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2727 . . 3 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ 𝑃 = 𝑃)
21mpompt 7517 . 2 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ 𝑃) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑃)
3 cnmpt21.j . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 cnmpt21.k . . . 4 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
5 txtopon 23446 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
63, 4, 5syl2anc 583 . . 3 (πœ‘ β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
7 cnmpt2c.l . . 3 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
8 cnmpt2c.p . . 3 (πœ‘ β†’ 𝑃 ∈ 𝑍)
96, 7, 8cnmptc 23517 . 2 (πœ‘ β†’ (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ 𝑃) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
102, 9eqeltrrid 2832 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑃) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4629   ↦ cmpt 5224   Γ— cxp 5667  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  TopOnctopon 22763   Cn ccn 23079   Γ—t ctx 23415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-topgen 17396  df-top 22747  df-topon 22764  df-bases 22800  df-cn 23082  df-cnp 23083  df-tx 23417
This theorem is referenced by:  cnrehmeo  24829  cnrehmeoOLD  24830  pcopt  24900  pcopt2  24901  vmcn  30457  dipcn  30478  cvxsconn  34762
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