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Theorem ptcld 21339
Description: A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypotheses
Ref Expression
ptcld.a (𝜑𝐴𝑉)
ptcld.f (𝜑𝐹:𝐴⟶Top)
ptcld.c ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘(𝐹𝑘)))
Assertion
Ref Expression
ptcld (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))
Distinct variable groups:   𝜑,𝑘   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem ptcld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ptcld.c . . . . 5 ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘(𝐹𝑘)))
2 eqid 2621 . . . . . 6 (𝐹𝑘) = (𝐹𝑘)
32cldss 20756 . . . . 5 (𝐶 ∈ (Clsd‘(𝐹𝑘)) → 𝐶 (𝐹𝑘))
41, 3syl 17 . . . 4 ((𝜑𝑘𝐴) → 𝐶 (𝐹𝑘))
54ralrimiva 2961 . . 3 (𝜑 → ∀𝑘𝐴 𝐶 (𝐹𝑘))
6 boxriin 7902 . . 3 (∀𝑘𝐴 𝐶 (𝐹𝑘) → X𝑘𝐴 𝐶 = (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
75, 6syl 17 . 2 (𝜑X𝑘𝐴 𝐶 = (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
8 ptcld.a . . . . 5 (𝜑𝐴𝑉)
9 ptcld.f . . . . 5 (𝜑𝐹:𝐴⟶Top)
10 eqid 2621 . . . . . 6 (∏t𝐹) = (∏t𝐹)
1110ptuni 21320 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
128, 9, 11syl2anc 692 . . . 4 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
1312ineq1d 3796 . . 3 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
14 pttop 21308 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
158, 9, 14syl2anc 692 . . . 4 (𝜑 → (∏t𝐹) ∈ Top)
16 sseq1 3610 . . . . . . . . . . 11 (𝐶 = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → (𝐶 (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
17 sseq1 3610 . . . . . . . . . . 11 ( (𝐹𝑘) = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → ( (𝐹𝑘) ⊆ (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
18 simpl 473 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ 𝑘 = 𝑥) → 𝐶 (𝐹𝑘))
19 ssid 3608 . . . . . . . . . . . 12 (𝐹𝑘) ⊆ (𝐹𝑘)
2019a1i 11 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ ¬ 𝑘 = 𝑥) → (𝐹𝑘) ⊆ (𝐹𝑘))
2116, 17, 18, 20ifbothda 4100 . . . . . . . . . 10 (𝐶 (𝐹𝑘) → if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
2221ralimi 2947 . . . . . . . . 9 (∀𝑘𝐴 𝐶 (𝐹𝑘) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
23 ss2ixp 7873 . . . . . . . . 9 (∀𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
245, 22, 233syl 18 . . . . . . . 8 (𝜑X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
2524adantr 481 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
2612adantr 481 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
2725, 26sseqtrd 3625 . . . . . 6 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹))
2812eqcomd 2627 . . . . . . . . . 10 (𝜑 (∏t𝐹) = X𝑘𝐴 (𝐹𝑘))
2928difeq1d 3710 . . . . . . . . 9 (𝜑 → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
3029adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
31 simpr 477 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
325adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑘𝐴 𝐶 (𝐹𝑘))
33 boxcutc 7903 . . . . . . . . 9 ((𝑥𝐴 ∧ ∀𝑘𝐴 𝐶 (𝐹𝑘)) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
3431, 32, 33syl2anc 692 . . . . . . . 8 ((𝜑𝑥𝐴) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
35 ixpeq2 7874 . . . . . . . . . 10 (∀𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
36 fveq2 6153 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
3736unieqd 4417 . . . . . . . . . . . . 13 (𝑘 = 𝑥 (𝐹𝑘) = (𝐹𝑥))
38 csbeq1a 3527 . . . . . . . . . . . . 13 (𝑘 = 𝑥𝐶 = 𝑥 / 𝑘𝐶)
3937, 38difeq12d 3712 . . . . . . . . . . . 12 (𝑘 = 𝑥 → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
4039adantl 482 . . . . . . . . . . 11 ((𝑘𝐴𝑘 = 𝑥) → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
4140ifeq1da 4093 . . . . . . . . . 10 (𝑘𝐴 → if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
4235, 41mprg 2921 . . . . . . . . 9 X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘))
4342a1i 11 . . . . . . . 8 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
4430, 34, 433eqtrd 2659 . . . . . . 7 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
458adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐴𝑉)
469adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐹:𝐴⟶Top)
471ralrimiva 2961 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)))
48 nfv 1840 . . . . . . . . . . . 12 𝑥 𝐶 ∈ (Clsd‘(𝐹𝑘))
49 nfcsb1v 3534 . . . . . . . . . . . . 13 𝑘𝑥 / 𝑘𝐶
5049nfel1 2775 . . . . . . . . . . . 12 𝑘𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))
5136fveq2d 6157 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → (Clsd‘(𝐹𝑘)) = (Clsd‘(𝐹𝑥)))
5238, 51eleq12d 2692 . . . . . . . . . . . 12 (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))))
5348, 50, 52cbvral 3158 . . . . . . . . . . 11 (∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5447, 53sylib 208 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5554r19.21bi 2927 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
56 eqid 2621 . . . . . . . . . 10 (𝐹𝑥) = (𝐹𝑥)
5756cldopn 20758 . . . . . . . . 9 (𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5855, 57syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5945, 46, 58ptopn2 21310 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)) ∈ (∏t𝐹))
6044, 59eqeltrd 2698 . . . . . 6 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))
61 eqid 2621 . . . . . . . . 9 (∏t𝐹) = (∏t𝐹)
6261iscld 20754 . . . . . . . 8 ((∏t𝐹) ∈ Top → (X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)) ↔ (X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹) ∧ ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))))
6315, 62syl 17 . . . . . . 7 (𝜑 → (X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)) ↔ (X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹) ∧ ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))))
6463adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → (X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)) ↔ (X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹) ∧ ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))))
6527, 60, 64mpbir2and 956 . . . . 5 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6665ralrimiva 2961 . . . 4 (𝜑 → ∀𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6761riincld 20771 . . . 4 (((∏t𝐹) ∈ Top ∧ ∀𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹))) → ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
6815, 66, 67syl2anc 692 . . 3 (𝜑 → ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
6913, 68eqeltrd 2698 . 2 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
707, 69eqeltrd 2698 1 (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  csb 3518  cdif 3556  cin 3558  wss 3559  ifcif 4063   cuni 4407   ciin 4491  wf 5848  cfv 5852  Xcixp 7860  tcpt 16031  Topctop 20630  Clsdccld 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-ixp 7861  df-en 7908  df-fin 7911  df-fi 8269  df-topgen 16036  df-pt 16037  df-top 20631  df-bases 20674  df-cld 20746
This theorem is referenced by:  ptcldmpt  21340
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