Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt1k Structured version   Visualization version   GIF version

Theorem cnmpt1k 22284
 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 21851 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐴):𝑋𝑍)
51, 2, 3, 4syl3anc 1367 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑍)
6 eqid 2821 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6869 . . . . . 6 (∀𝑥𝑋 𝐴𝑍 ↔ (𝑥𝑋𝐴):𝑋𝑍)
85, 7sylibr 236 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑍)
98adantr 483 . . . 4 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴𝑍)
10 eqidd 2822 . . . 4 ((𝜑𝑦𝑌) → (𝑥𝑋𝐴) = (𝑥𝑋𝐴))
11 eqidd 2822 . . . 4 ((𝜑𝑦𝑌) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
12 cnmpt1k.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
139, 10, 11, 12fmptcof 6887 . . 3 ((𝜑𝑦𝑌) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
1413mpteq2dva 5154 . 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 21515 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
182, 17syl 17 . . . 4 (𝜑𝐿 ∈ Top)
19 cnmpt1k.m . . . . 5 (𝜑𝑀 ∈ (TopOn‘𝑊))
20 topontop 21515 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
2119, 20syl 17 . . . 4 (𝜑𝑀 ∈ Top)
22 eqid 2821 . . . . 5 (𝑀ko 𝐿) = (𝑀ko 𝐿)
2322xkotopon 22202 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2418, 21, 23syl2anc 586 . . 3 (𝜑 → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2521, 3xkoco1cn 22259 . . 3 (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥𝑋𝐴))) ∈ ((𝑀ko 𝐿) Cn (𝑀ko 𝐽)))
26 coeq1 5723 . . 3 (𝑤 = (𝑧𝑍𝐵) → (𝑤 ∘ (𝑥𝑋𝐴)) = ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)))
2715, 16, 24, 25, 26cnmpt11 22265 . 2 (𝜑 → (𝑦𝑌 ↦ ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴))) ∈ (𝐾 Cn (𝑀ko 𝐽)))
2814, 27eqeltrrd 2914 1 (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀ko 𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ↦ cmpt 5139   ∘ ccom 5554  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  Topctop 21495  TopOnctopon 21512   Cn ccn 21826   ↑ko cxko 22163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-fin 8507  df-fi 8869  df-rest 16690  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cn 21829  df-cmp 21989  df-xko 22165 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator