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Theorem xkopjcn 21582
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 21610.) (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 2724 . . . . . 6 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
21xkotopon 21526 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
323adant3 1124 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4 xkopjcn.1 . . . . . . . . 9 𝑋 = 𝑅
54topopn 20834 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
653ad2ant1 1125 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑋𝑅)
7 fconst6g 6207 . . . . . . . 8 (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
873ad2ant2 1126 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9 pttop 21508 . . . . . . 7 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
106, 8, 9syl2anc 696 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11 eqid 2724 . . . . . . . . . 10 𝑆 = 𝑆
124, 11cnf 21173 . . . . . . . . 9 (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋 𝑆)
13 uniexg 7072 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑆 ∈ V)
14133ad2ant2 1126 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ V)
1514, 6elmapd 7988 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ ( 𝑆𝑚 𝑋) ↔ 𝑓:𝑋 𝑆))
1612, 15syl5ibr 236 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ ( 𝑆𝑚 𝑋)))
1716ssrdv 3715 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ ( 𝑆𝑚 𝑋))
18 simp2 1129 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ Top)
19 eqid 2724 . . . . . . . . 9 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2019, 11ptuniconst 21524 . . . . . . . 8 ((𝑋𝑅𝑆 ∈ Top) → ( 𝑆𝑚 𝑋) = (∏t‘(𝑋 × {𝑆})))
216, 18, 20syl2anc 696 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ( 𝑆𝑚 𝑋) = (∏t‘(𝑋 × {𝑆})))
2217, 21sseqtrd 3747 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆})))
23 eqid 2724 . . . . . . 7 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2423restuni 21089 . . . . . 6 (((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2510, 22, 24syl2anc 696 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2625fveq2d 6308 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
273, 26eleqtrd 2805 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆 ^ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
284, 19xkoptsub 21580 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))
29283adant3 1124 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))
30 eqid 2724 . . . 4 ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
3130cnss1 21203 . . 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 696 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ^ko 𝑅) Cn 𝑆))
3322resmptd 5562 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)))
34 simp3 1130 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
3523, 19ptpjcn 21537 . . . . . 6 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
366, 8, 34, 35syl3anc 1439 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37 fvconst2g 6583 . . . . . . 7 ((𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38373adant1 1122 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
3938oveq2d 6781 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4036, 39eleqtrd 2805 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4123cnrest 21212 . . . 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 696 . . 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 2804 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4432, 43sseldd 3710 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 1072   = wceq 1596  wcel 2103  Vcvv 3304  wss 3680  {csn 4285   cuni 4544  cmpt 4837   × cxp 5216  cres 5220  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  t crest 16204  tcpt 16222  Topctop 20821  TopOnctopon 20838   Cn ccn 21151   ^ko cxko 21487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fi 8433  df-rest 16206  df-topgen 16227  df-pt 16228  df-top 20822  df-topon 20839  df-bases 20873  df-cn 21154  df-cmp 21313  df-xko 21489
This theorem is referenced by:  cnmptkp  21606  xkofvcn  21610
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