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Theorem cnmpt2c 23592
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 2728 . . 3 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ 𝑃 = 𝑃)
21mpompt 7538 . 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 23513 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
63, 4, 5syl2anc 582 . . 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 23584 . 2 (πœ‘ β†’ (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ 𝑃) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
102, 9eqeltrrid 2833 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 4636   ↦ cmpt 5233   Γ— cxp 5678  β€˜cfv 6551  (class class class)co 7424   ∈ cmpo 7426  TopOnctopon 22830   Cn ccn 23146   Γ—t ctx 23482
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-map 8851  df-topgen 17430  df-top 22814  df-topon 22831  df-bases 22867  df-cn 23149  df-cnp 23150  df-tx 23484
This theorem is referenced by:  cnrehmeo  24896  cnrehmeoOLD  24897  pcopt  24967  pcopt2  24968  vmcn  30527  dipcn  30548  cvxsconn  34858
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