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Theorem ptcld 22213
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 2819 . . . . . 6 (𝐹𝑘) = (𝐹𝑘)
32cldss 21629 . . . . 5 (𝐶 ∈ (Clsd‘(𝐹𝑘)) → 𝐶 (𝐹𝑘))
41, 3syl 17 . . . 4 ((𝜑𝑘𝐴) → 𝐶 (𝐹𝑘))
54ralrimiva 3180 . . 3 (𝜑 → ∀𝑘𝐴 𝐶 (𝐹𝑘))
6 boxriin 8496 . . 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 2819 . . . . . 6 (∏t𝐹) = (∏t𝐹)
1110ptuni 22194 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
128, 9, 11syl2anc 586 . . . 4 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
1312ineq1d 4186 . . 3 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
14 pttop 22182 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
158, 9, 14syl2anc 586 . . . 4 (𝜑 → (∏t𝐹) ∈ Top)
16 sseq1 3990 . . . . . . . . . . 11 (𝐶 = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → (𝐶 (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
17 sseq1 3990 . . . . . . . . . . 11 ( (𝐹𝑘) = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → ( (𝐹𝑘) ⊆ (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
18 simpl 485 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ 𝑘 = 𝑥) → 𝐶 (𝐹𝑘))
19 ssidd 3988 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ ¬ 𝑘 = 𝑥) → (𝐹𝑘) ⊆ (𝐹𝑘))
2016, 17, 18, 19ifbothda 4502 . . . . . . . . . 10 (𝐶 (𝐹𝑘) → if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
2120ralimi 3158 . . . . . . . . 9 (∀𝑘𝐴 𝐶 (𝐹𝑘) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
22 ss2ixp 8466 . . . . . . . . 9 (∀𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
235, 21, 223syl 18 . . . . . . . 8 (𝜑X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
2423adantr 483 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
2512adantr 483 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
2624, 25sseqtrd 4005 . . . . . 6 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹))
2712eqcomd 2825 . . . . . . . . . 10 (𝜑 (∏t𝐹) = X𝑘𝐴 (𝐹𝑘))
2827difeq1d 4096 . . . . . . . . 9 (𝜑 → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
2928adantr 483 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
30 simpr 487 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
315adantr 483 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑘𝐴 𝐶 (𝐹𝑘))
32 boxcutc 8497 . . . . . . . . 9 ((𝑥𝐴 ∧ ∀𝑘𝐴 𝐶 (𝐹𝑘)) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
3330, 31, 32syl2anc 586 . . . . . . . 8 ((𝜑𝑥𝐴) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
34 ixpeq2 8467 . . . . . . . . . 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)))
35 fveq2 6663 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
3635unieqd 4840 . . . . . . . . . . . . 13 (𝑘 = 𝑥 (𝐹𝑘) = (𝐹𝑥))
37 csbeq1a 3895 . . . . . . . . . . . . 13 (𝑘 = 𝑥𝐶 = 𝑥 / 𝑘𝐶)
3836, 37difeq12d 4098 . . . . . . . . . . . 12 (𝑘 = 𝑥 → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
3938adantl 484 . . . . . . . . . . 11 ((𝑘𝐴𝑘 = 𝑥) → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
4039ifeq1da 4495 . . . . . . . . . 10 (𝑘𝐴 → if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
4134, 40mprg 3150 . . . . . . . . 9 X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘))
4241a1i 11 . . . . . . . 8 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
4329, 33, 423eqtrd 2858 . . . . . . 7 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
448adantr 483 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐴𝑉)
459adantr 483 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐹:𝐴⟶Top)
461ralrimiva 3180 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)))
47 nfv 1908 . . . . . . . . . . . 12 𝑥 𝐶 ∈ (Clsd‘(𝐹𝑘))
48 nfcsb1v 3905 . . . . . . . . . . . . 13 𝑘𝑥 / 𝑘𝐶
4948nfel1 2992 . . . . . . . . . . . 12 𝑘𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))
50 2fveq3 6668 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → (Clsd‘(𝐹𝑘)) = (Clsd‘(𝐹𝑥)))
5137, 50eleq12d 2905 . . . . . . . . . . . 12 (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))))
5247, 49, 51cbvralw 3440 . . . . . . . . . . 11 (∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5346, 52sylib 220 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5453r19.21bi 3206 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
55 eqid 2819 . . . . . . . . . 10 (𝐹𝑥) = (𝐹𝑥)
5655cldopn 21631 . . . . . . . . 9 (𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5754, 56syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5844, 45, 57ptopn2 22184 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)) ∈ (∏t𝐹))
5943, 58eqeltrd 2911 . . . . . 6 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))
60 eqid 2819 . . . . . . . . 9 (∏t𝐹) = (∏t𝐹)
6160iscld 21627 . . . . . . . 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))))
6215, 61syl 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))))
6362adantr 483 . . . . . 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))))
6426, 59, 63mpbir2and 711 . . . . 5 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6564ralrimiva 3180 . . . 4 (𝜑 → ∀𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6660riincld 21644 . . . 4 (((∏t𝐹) ∈ Top ∧ ∀𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹))) → ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
6715, 65, 66syl2anc 586 . . 3 (𝜑 → ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
6813, 67eqeltrd 2911 . 2 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
697, 68eqeltrd 2911 1 (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wral 3136  csb 3881  cdif 3931  cin 3933  wss 3934  ifcif 4465   cuni 4830   ciin 4911  wf 6344  cfv 6348  Xcixp 8453  tcpt 16704  Topctop 21493  Clsdccld 21616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-ixp 8454  df-en 8502  df-fin 8505  df-fi 8867  df-topgen 16709  df-pt 16710  df-top 21494  df-bases 21546  df-cld 21619
This theorem is referenced by:  ptcldmpt  22214
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