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Theorem cnmpt2c 21696
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 2762 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
21mpt2mpt 6919 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥𝑋, 𝑦𝑌𝑃)
3 cnmpt21.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt21.k . . . 4 (𝜑𝐾 ∈ (TopOn‘𝑌))
5 txtopon 21617 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
63, 4, 5syl2anc 696 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7 cnmpt2c.l . . 3 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt2c.p . . 3 (𝜑𝑃𝑍)
96, 7, 8cnmptc 21688 . 2 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
102, 9syl5eqelr 2845 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  cop 4328  cmpt 4882   × cxp 5265  cfv 6050  (class class class)co 6815  cmpt2 6817  TopOnctopon 20938   Cn ccn 21251   ×t ctx 21586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-map 8028  df-topgen 16327  df-top 20922  df-topon 20939  df-bases 20973  df-cn 21254  df-cnp 21255  df-tx 21588
This theorem is referenced by:  cnrehmeo  22974  pcopt  23043  pcopt2  23044  vmcn  27885  dipcn  27906  cvxsconn  31554
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