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Theorem cnmpt1k 22218
Description: The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptk1.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt1k.m (𝜑𝑀 ∈ (TopOn‘𝑊))
cnmpt1k.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))
cnmpt1k.b (𝜑 → (𝑦𝑌 ↦ (𝑧𝑍𝐵)) ∈ (𝐾 Cn (𝑀ko 𝐿)))
cnmpt1k.c (𝑧 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmpt1k (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀ko 𝐽)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑧,𝑍,𝑦   𝑧,𝐴   𝑥,𝐵   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑧,𝐶   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥)   𝐵(𝑦,𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)   𝑀(𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem cnmpt1k
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 cnmptk1.j . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmptk1.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
3 cnmpt1k.a . . . . . . 7 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))
4 cnf2 21785 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐴):𝑋𝑍)
51, 2, 3, 4syl3anc 1363 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑍)
6 eqid 2818 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6866 . . . . . 6 (∀𝑥𝑋 𝐴𝑍 ↔ (𝑥𝑋𝐴):𝑋𝑍)
85, 7sylibr 235 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑍)
98adantr 481 . . . 4 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴𝑍)
10 eqidd 2819 . . . 4 ((𝜑𝑦𝑌) → (𝑥𝑋𝐴) = (𝑥𝑋𝐴))
11 eqidd 2819 . . . 4 ((𝜑𝑦𝑌) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
12 cnmpt1k.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
139, 10, 11, 12fmptcof 6884 . . 3 ((𝜑𝑦𝑌) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
1413mpteq2dva 5152 . 2 (𝜑 → (𝑦𝑌 ↦ ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴))) = (𝑦𝑌 ↦ (𝑥𝑋𝐶)))
15 cnmptk1.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
16 cnmpt1k.b . . 3 (𝜑 → (𝑦𝑌 ↦ (𝑧𝑍𝐵)) ∈ (𝐾 Cn (𝑀ko 𝐿)))
17 topontop 21449 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
182, 17syl 17 . . . 4 (𝜑𝐿 ∈ Top)
19 cnmpt1k.m . . . . 5 (𝜑𝑀 ∈ (TopOn‘𝑊))
20 topontop 21449 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
2119, 20syl 17 . . . 4 (𝜑𝑀 ∈ Top)
22 eqid 2818 . . . . 5 (𝑀ko 𝐿) = (𝑀ko 𝐿)
2322xkotopon 22136 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2418, 21, 23syl2anc 584 . . 3 (𝜑 → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2521, 3xkoco1cn 22193 . . 3 (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥𝑋𝐴))) ∈ ((𝑀ko 𝐿) Cn (𝑀ko 𝐽)))
26 coeq1 5721 . . 3 (𝑤 = (𝑧𝑍𝐵) → (𝑤 ∘ (𝑥𝑋𝐴)) = ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)))
2715, 16, 24, 25, 26cnmpt11 22199 . 2 (𝜑 → (𝑦𝑌 ↦ ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴))) ∈ (𝐾 Cn (𝑀ko 𝐽)))
2814, 27eqeltrrd 2911 1 (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀ko 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cmpt 5137  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  Topctop 21429  TopOnctopon 21446   Cn ccn 21760  ko cxko 22097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-cmp 21923  df-xko 22099
This theorem is referenced by: (None)
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