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Theorem ptcld 21464
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 2651 . . . . . 6 (𝐹𝑘) = (𝐹𝑘)
32cldss 20881 . . . . 5 (𝐶 ∈ (Clsd‘(𝐹𝑘)) → 𝐶 (𝐹𝑘))
41, 3syl 17 . . . 4 ((𝜑𝑘𝐴) → 𝐶 (𝐹𝑘))
54ralrimiva 2995 . . 3 (𝜑 → ∀𝑘𝐴 𝐶 (𝐹𝑘))
6 boxriin 7992 . . 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 2651 . . . . . 6 (∏t𝐹) = (∏t𝐹)
1110ptuni 21445 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
128, 9, 11syl2anc 694 . . . 4 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
1312ineq1d 3846 . . 3 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
14 pttop 21433 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
158, 9, 14syl2anc 694 . . . 4 (𝜑 → (∏t𝐹) ∈ Top)
16 sseq1 3659 . . . . . . . . . . 11 (𝐶 = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → (𝐶 (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
17 sseq1 3659 . . . . . . . . . . 11 ( (𝐹𝑘) = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → ( (𝐹𝑘) ⊆ (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
18 simpl 472 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ 𝑘 = 𝑥) → 𝐶 (𝐹𝑘))
19 ssid 3657 . . . . . . . . . . . 12 (𝐹𝑘) ⊆ (𝐹𝑘)
2019a1i 11 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ ¬ 𝑘 = 𝑥) → (𝐹𝑘) ⊆ (𝐹𝑘))
2116, 17, 18, 20ifbothda 4156 . . . . . . . . . 10 (𝐶 (𝐹𝑘) → if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
2221ralimi 2981 . . . . . . . . 9 (∀𝑘𝐴 𝐶 (𝐹𝑘) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
23 ss2ixp 7963 . . . . . . . . 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 480 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
2612adantr 480 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
2725, 26sseqtrd 3674 . . . . . 6 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹))
2812eqcomd 2657 . . . . . . . . . 10 (𝜑 (∏t𝐹) = X𝑘𝐴 (𝐹𝑘))
2928difeq1d 3760 . . . . . . . . 9 (𝜑 → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
3029adantr 480 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
31 simpr 476 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
325adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑘𝐴 𝐶 (𝐹𝑘))
33 boxcutc 7993 . . . . . . . . 9 ((𝑥𝐴 ∧ ∀𝑘𝐴 𝐶 (𝐹𝑘)) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
3431, 32, 33syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝐴) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
35 ixpeq2 7964 . . . . . . . . . 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 6229 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
3736unieqd 4478 . . . . . . . . . . . . 13 (𝑘 = 𝑥 (𝐹𝑘) = (𝐹𝑥))
38 csbeq1a 3575 . . . . . . . . . . . . 13 (𝑘 = 𝑥𝐶 = 𝑥 / 𝑘𝐶)
3937, 38difeq12d 3762 . . . . . . . . . . . 12 (𝑘 = 𝑥 → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
4039adantl 481 . . . . . . . . . . 11 ((𝑘𝐴𝑘 = 𝑥) → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
4140ifeq1da 4149 . . . . . . . . . 10 (𝑘𝐴 → if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
4235, 41mprg 2955 . . . . . . . . 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 2689 . . . . . . 7 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
458adantr 480 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐴𝑉)
469adantr 480 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐹:𝐴⟶Top)
471ralrimiva 2995 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)))
48 nfv 1883 . . . . . . . . . . . 12 𝑥 𝐶 ∈ (Clsd‘(𝐹𝑘))
49 nfcsb1v 3582 . . . . . . . . . . . . 13 𝑘𝑥 / 𝑘𝐶
5049nfel1 2808 . . . . . . . . . . . 12 𝑘𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))
5136fveq2d 6233 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → (Clsd‘(𝐹𝑘)) = (Clsd‘(𝐹𝑥)))
5238, 51eleq12d 2724 . . . . . . . . . . . 12 (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))))
5348, 50, 52cbvral 3197 . . . . . . . . . . 11 (∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5447, 53sylib 208 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5554r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
56 eqid 2651 . . . . . . . . . 10 (𝐹𝑥) = (𝐹𝑥)
5756cldopn 20883 . . . . . . . . 9 (𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5855, 57syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5945, 46, 58ptopn2 21435 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)) ∈ (∏t𝐹))
6044, 59eqeltrd 2730 . . . . . 6 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))
61 eqid 2651 . . . . . . . . 9 (∏t𝐹) = (∏t𝐹)
6261iscld 20879 . . . . . . . 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 480 . . . . . 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 977 . . . . 5 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6665ralrimiva 2995 . . . 4 (𝜑 → ∀𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6761riincld 20896 . . . 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 694 . . 3 (𝜑 → ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
6913, 68eqeltrd 2730 . 2 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
707, 69eqeltrd 2730 1 (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  csb 3566  cdif 3604  cin 3606  wss 3607  ifcif 4119   cuni 4468   ciin 4553  wf 5922  cfv 5926  Xcixp 7950  tcpt 16146  Topctop 20746  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-ixp 7951  df-en 7998  df-fin 8001  df-fi 8358  df-topgen 16151  df-pt 16152  df-top 20747  df-bases 20798  df-cld 20871
This theorem is referenced by:  ptcldmpt  21465
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