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Theorem xkopjcn 21269
 Description: Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both 𝑓 and 𝐴 as a function on (𝑆 ^ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 21297.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
xkopjcn.1 𝑋 = 𝑅
Assertion
Ref Expression
xkopjcn ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ ((𝑆 ^ko 𝑅) Cn 𝑆))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑆,𝑓   𝑓,𝑋

Proof of Theorem xkopjcn
StepHypRef Expression
1 eqid 2610 . . . . . 6 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
21xkotopon 21213 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
323adant3 1074 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4 xkopjcn.1 . . . . . . . . 9 𝑋 = 𝑅
54topopn 20536 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
653ad2ant1 1075 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑋𝑅)
7 fconst6g 6007 . . . . . . . 8 (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
873ad2ant2 1076 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9 pttop 21195 . . . . . . 7 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
106, 8, 9syl2anc 691 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11 eqid 2610 . . . . . . . . . 10 𝑆 = 𝑆
124, 11cnf 20860 . . . . . . . . 9 (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋 𝑆)
13 uniexg 6853 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑆 ∈ V)
14133ad2ant2 1076 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ V)
1514, 6elmapd 7758 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ ( 𝑆𝑚 𝑋) ↔ 𝑓:𝑋 𝑆))
1612, 15syl5ibr 235 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ ( 𝑆𝑚 𝑋)))
1716ssrdv 3574 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ ( 𝑆𝑚 𝑋))
18 simp2 1055 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ Top)
19 eqid 2610 . . . . . . . . 9 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2019, 11ptuniconst 21211 . . . . . . . 8 ((𝑋𝑅𝑆 ∈ Top) → ( 𝑆𝑚 𝑋) = (∏t‘(𝑋 × {𝑆})))
216, 18, 20syl2anc 691 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ( 𝑆𝑚 𝑋) = (∏t‘(𝑋 × {𝑆})))
2217, 21sseqtrd 3604 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆})))
23 eqid 2610 . . . . . . 7 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2423restuni 20776 . . . . . 6 (((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2510, 22, 24syl2anc 691 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2625fveq2d 6107 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
273, 26eleqtrd 2690 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆 ^ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
284, 19xkoptsub 21267 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))
29283adant3 1074 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))
30 eqid 2610 . . . 4 ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
3130cnss1 20890 . . 3 (((𝑆 ^ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))) ∧ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅)) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ^ko 𝑅) Cn 𝑆))
3227, 29, 31syl2anc 691 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ^ko 𝑅) Cn 𝑆))
3322resmptd 5371 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)))
34 simp3 1056 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
3523, 19ptpjcn 21224 . . . . . 6 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
366, 8, 34, 35syl3anc 1318 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37 fvconst2g 6372 . . . . . . 7 ((𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38373adant1 1072 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
3938oveq2d 6565 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4036, 39eleqtrd 2690 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4123cnrest 20899 . . . 4 (((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆) ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4240, 22, 41syl2anc 691 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4333, 42eqeltrrd 2689 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4432, 43sseldd 3569 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ ((𝑆 ^ko 𝑅) Cn 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   ↾t crest 15904  ∏tcpt 15922  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ^ko cxko 21174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cmp 21000  df-xko 21176 This theorem is referenced by:  cnmptkp  21293  xkofvcn  21297
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