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Theorem ptcldmpt 22328
 Description: A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypotheses
Ref Expression
ptcldmpt.a (𝜑𝐴𝑉)
ptcldmpt.j ((𝜑𝑘𝐴) → 𝐽 ∈ Top)
ptcldmpt.c ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘𝐽))
Assertion
Ref Expression
ptcldmpt (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘𝐴𝐽))))
Distinct variable groups:   𝜑,𝑘   𝐴,𝑘
Allowed substitution hints:   𝐶(𝑘)   𝐽(𝑘)   𝑉(𝑘)

Proof of Theorem ptcldmpt
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2919 . . 3 𝑙𝐶
2 nfcsb1v 3831 . . 3 𝑘𝑙 / 𝑘𝐶
3 csbeq1a 3821 . . 3 (𝑘 = 𝑙𝐶 = 𝑙 / 𝑘𝐶)
41, 2, 3cbvixp 8509 . 2 X𝑘𝐴 𝐶 = X𝑙𝐴 𝑙 / 𝑘𝐶
5 ptcldmpt.a . . 3 (𝜑𝐴𝑉)
6 ptcldmpt.j . . . 4 ((𝜑𝑘𝐴) → 𝐽 ∈ Top)
76fmpttd 6876 . . 3 (𝜑 → (𝑘𝐴𝐽):𝐴⟶Top)
8 nfv 1915 . . . . 5 𝑘(𝜑𝑙𝐴)
9 nfcv 2919 . . . . . . 7 𝑘Clsd
10 nffvmpt1 6674 . . . . . . 7 𝑘((𝑘𝐴𝐽)‘𝑙)
119, 10nffv 6673 . . . . . 6 𝑘(Clsd‘((𝑘𝐴𝐽)‘𝑙))
122, 11nfel 2933 . . . . 5 𝑘𝑙 / 𝑘𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑙))
138, 12nfim 1897 . . . 4 𝑘((𝜑𝑙𝐴) → 𝑙 / 𝑘𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑙)))
14 eleq1w 2834 . . . . . 6 (𝑘 = 𝑙 → (𝑘𝐴𝑙𝐴))
1514anbi2d 631 . . . . 5 (𝑘 = 𝑙 → ((𝜑𝑘𝐴) ↔ (𝜑𝑙𝐴)))
16 2fveq3 6668 . . . . . 6 (𝑘 = 𝑙 → (Clsd‘((𝑘𝐴𝐽)‘𝑘)) = (Clsd‘((𝑘𝐴𝐽)‘𝑙)))
173, 16eleq12d 2846 . . . . 5 (𝑘 = 𝑙 → (𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑘)) ↔ 𝑙 / 𝑘𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑙))))
1815, 17imbi12d 348 . . . 4 (𝑘 = 𝑙 → (((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑘))) ↔ ((𝜑𝑙𝐴) → 𝑙 / 𝑘𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑙)))))
19 ptcldmpt.c . . . . 5 ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘𝐽))
20 simpr 488 . . . . . . 7 ((𝜑𝑘𝐴) → 𝑘𝐴)
21 eqid 2758 . . . . . . . 8 (𝑘𝐴𝐽) = (𝑘𝐴𝐽)
2221fvmpt2 6775 . . . . . . 7 ((𝑘𝐴𝐽 ∈ Top) → ((𝑘𝐴𝐽)‘𝑘) = 𝐽)
2320, 6, 22syl2anc 587 . . . . . 6 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐽)‘𝑘) = 𝐽)
2423fveq2d 6667 . . . . 5 ((𝜑𝑘𝐴) → (Clsd‘((𝑘𝐴𝐽)‘𝑘)) = (Clsd‘𝐽))
2519, 24eleqtrrd 2855 . . . 4 ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑘)))
2613, 18, 25chvarfv 2240 . . 3 ((𝜑𝑙𝐴) → 𝑙 / 𝑘𝐶 ∈ (Clsd‘((𝑘𝐴𝐽)‘𝑙)))
275, 7, 26ptcld 22327 . 2 (𝜑X𝑙𝐴 𝑙 / 𝑘𝐶 ∈ (Clsd‘(∏t‘(𝑘𝐴𝐽))))
284, 27eqeltrid 2856 1 (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘𝐴𝐽))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⦋csb 3807   ↦ cmpt 5116  ‘cfv 6340  Xcixp 8492  ∏tcpt 16784  Topctop 21607  Clsdccld 21730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7586  df-1o 8118  df-er 8305  df-ixp 8493  df-en 8541  df-fin 8544  df-fi 8921  df-topgen 16789  df-pt 16790  df-top 21608  df-bases 21660  df-cld 21733 This theorem is referenced by:  ptclsg  22329  kelac1  40425
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