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Theorem cnmpt22f 23042
Description: The composition of continuous functions 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))
cnmpt21.a (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
cnmpt2t.b (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
cnmpt22f.f (πœ‘ β†’ 𝐹 ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁))
Assertion
Ref Expression
cnmpt22f (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴𝐹𝐡)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑁))
Distinct variable groups:   π‘₯,𝑦,𝐹   π‘₯,𝐿,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝑀,𝑦   π‘₯,𝑁,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦)   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)

Proof of Theorem cnmpt22f
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmpt21.j . 2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 cnmpt21.k . 2 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3 cnmpt21.a . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
4 cnmpt2t.b . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
5 cntop2 22608 . . . 4 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) β†’ 𝐿 ∈ Top)
63, 5syl 17 . . 3 (πœ‘ β†’ 𝐿 ∈ Top)
7 toptopon2 22283 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
86, 7sylib 217 . 2 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
9 cntop2 22608 . . . 4 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀) β†’ 𝑀 ∈ Top)
104, 9syl 17 . . 3 (πœ‘ β†’ 𝑀 ∈ Top)
11 toptopon2 22283 . . 3 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
1210, 11sylib 217 . 2 (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀))
13 txtopon 22958 . . . . . . 7 ((𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ 𝑀 ∈ (TopOnβ€˜βˆͺ 𝑀)) β†’ (𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(βˆͺ 𝐿 Γ— βˆͺ 𝑀)))
148, 12, 13syl2anc 585 . . . . . 6 (πœ‘ β†’ (𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(βˆͺ 𝐿 Γ— βˆͺ 𝑀)))
15 cnmpt22f.f . . . . . . . 8 (πœ‘ β†’ 𝐹 ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁))
16 cntop2 22608 . . . . . . . 8 (𝐹 ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁) β†’ 𝑁 ∈ Top)
1715, 16syl 17 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ Top)
18 toptopon2 22283 . . . . . . 7 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOnβ€˜βˆͺ 𝑁))
1917, 18sylib 217 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ (TopOnβ€˜βˆͺ 𝑁))
20 cnf2 22616 . . . . . 6 (((𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(βˆͺ 𝐿 Γ— βˆͺ 𝑀)) ∧ 𝑁 ∈ (TopOnβ€˜βˆͺ 𝑁) ∧ 𝐹 ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁)) β†’ 𝐹:(βˆͺ 𝐿 Γ— βˆͺ 𝑀)⟢βˆͺ 𝑁)
2114, 19, 15, 20syl3anc 1372 . . . . 5 (πœ‘ β†’ 𝐹:(βˆͺ 𝐿 Γ— βˆͺ 𝑀)⟢βˆͺ 𝑁)
2221ffnd 6670 . . . 4 (πœ‘ β†’ 𝐹 Fn (βˆͺ 𝐿 Γ— βˆͺ 𝑀))
23 fnov 7488 . . . 4 (𝐹 Fn (βˆͺ 𝐿 Γ— βˆͺ 𝑀) ↔ 𝐹 = (𝑧 ∈ βˆͺ 𝐿, 𝑀 ∈ βˆͺ 𝑀 ↦ (𝑧𝐹𝑀)))
2422, 23sylib 217 . . 3 (πœ‘ β†’ 𝐹 = (𝑧 ∈ βˆͺ 𝐿, 𝑀 ∈ βˆͺ 𝑀 ↦ (𝑧𝐹𝑀)))
2524, 15eqeltrrd 2835 . 2 (πœ‘ β†’ (𝑧 ∈ βˆͺ 𝐿, 𝑀 ∈ βˆͺ 𝑀 ↦ (𝑧𝐹𝑀)) ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁))
26 oveq12 7367 . 2 ((𝑧 = 𝐴 ∧ 𝑀 = 𝐡) β†’ (𝑧𝐹𝑀) = (𝐴𝐹𝐡))
271, 2, 3, 4, 8, 12, 25, 26cnmpt22 23041 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴𝐹𝐡)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4866   Γ— cxp 5632   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  Topctop 22258  TopOnctopon 22275   Cn ccn 22591   Γ—t ctx 22927
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-topgen 17330  df-top 22259  df-topon 22276  df-bases 22312  df-cn 22594  df-tx 22929
This theorem is referenced by:  cnmptcom  23045  cnmpt2plusg  23455  istgp2  23458  cnmpt2vsca  23562  cnmpt2ds  24222  divcn  24247  cnrehmeo  24332  htpycom  24355  htpyco1  24357  htpycc  24359  reparphti  24376  pcohtpylem  24398  cnmpt2ip  24628  cxpcn  26114  vmcn  29683  dipcn  29704  mndpluscn  32564  cvxsconn  33894
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