MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt2c Structured version   Visualization version   GIF version

Theorem cnmpt2c 23044
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 2734 . . 3 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ 𝑃 = 𝑃)
21mpompt 7474 . 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 22965 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Γ—t 𝐾) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
63, 4, 5syl2anc 585 . . 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 23036 . 2 (πœ‘ β†’ (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ 𝑃) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
102, 9eqeltrrid 2839 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 1542   ∈ wcel 2107  βŸ¨cop 4596   ↦ cmpt 5192   Γ— cxp 5635  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  TopOnctopon 22282   Cn ccn 22598   Γ—t ctx 22934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-cnp 22602  df-tx 22936
This theorem is referenced by:  cnrehmeo  24339  pcopt  24408  pcopt2  24409  vmcn  29690  dipcn  29711  cvxsconn  33901
  Copyright terms: Public domain W3C validator