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Theorem xkopjcn 21673
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 21701.) (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 2771 . . . . . 6 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
21xkotopon 21617 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
323adant3 1126 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4 xkopjcn.1 . . . . . . . . 9 𝑋 = 𝑅
54topopn 20924 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
653ad2ant1 1127 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑋𝑅)
7 fconst6g 6232 . . . . . . . 8 (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
873ad2ant2 1128 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9 pttop 21599 . . . . . . 7 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
106, 8, 9syl2anc 573 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11 eqid 2771 . . . . . . . . . 10 𝑆 = 𝑆
124, 11cnf 21264 . . . . . . . . 9 (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋 𝑆)
13 uniexg 7100 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑆 ∈ V)
14133ad2ant2 1128 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ V)
1514, 6elmapd 8021 . . . . . . . . 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 3758 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ ( 𝑆𝑚 𝑋))
18 simp2 1131 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ Top)
19 eqid 2771 . . . . . . . . 9 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2019, 11ptuniconst 21615 . . . . . . . 8 ((𝑋𝑅𝑆 ∈ Top) → ( 𝑆𝑚 𝑋) = (∏t‘(𝑋 × {𝑆})))
216, 18, 20syl2anc 573 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ( 𝑆𝑚 𝑋) = (∏t‘(𝑋 × {𝑆})))
2217, 21sseqtrd 3790 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆})))
23 eqid 2771 . . . . . . 7 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2423restuni 21180 . . . . . 6 (((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2510, 22, 24syl2anc 573 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2625fveq2d 6334 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
273, 26eleqtrd 2852 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆 ^ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
284, 19xkoptsub 21671 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))
29283adant3 1126 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))
30 eqid 2771 . . . 4 ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
3130cnss1 21294 . . 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 573 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ^ko 𝑅) Cn 𝑆))
3322resmptd 5591 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)))
34 simp3 1132 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
3523, 19ptpjcn 21628 . . . . . 6 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
366, 8, 34, 35syl3anc 1476 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37 fvconst2g 6609 . . . . . . 7 ((𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38373adant1 1124 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
3938oveq2d 6807 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4036, 39eleqtrd 2852 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4123cnrest 21303 . . . 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 573 . . 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 2851 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4432, 43sseldd 3753 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 1071   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  {csn 4316   cuni 4574  cmpt 4863   × cxp 5247  cres 5251  wf 6025  cfv 6029  (class class class)co 6791  𝑚 cmap 8007  t crest 16282  tcpt 16300  Topctop 20911  TopOnctopon 20928   Cn ccn 21242   ^ko cxko 21578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-ixp 8061  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-fi 8471  df-rest 16284  df-topgen 16305  df-pt 16306  df-top 20912  df-topon 20929  df-bases 20964  df-cn 21245  df-cmp 21404  df-xko 21580
This theorem is referenced by:  cnmptkp  21697  xkofvcn  21701
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