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Theorem ptcld 22964
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 2736 . . . . . 6 (𝐹𝑘) = (𝐹𝑘)
32cldss 22380 . . . . 5 (𝐶 ∈ (Clsd‘(𝐹𝑘)) → 𝐶 (𝐹𝑘))
41, 3syl 17 . . . 4 ((𝜑𝑘𝐴) → 𝐶 (𝐹𝑘))
54ralrimiva 3143 . . 3 (𝜑 → ∀𝑘𝐴 𝐶 (𝐹𝑘))
6 boxriin 8878 . . 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 2736 . . . . . 6 (∏t𝐹) = (∏t𝐹)
1110ptuni 22945 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
128, 9, 11syl2anc 584 . . . 4 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
1312ineq1d 4171 . . 3 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
14 pttop 22933 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
158, 9, 14syl2anc 584 . . . 4 (𝜑 → (∏t𝐹) ∈ Top)
16 sseq1 3969 . . . . . . . . . . 11 (𝐶 = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → (𝐶 (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
17 sseq1 3969 . . . . . . . . . . 11 ( (𝐹𝑘) = if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) → ( (𝐹𝑘) ⊆ (𝐹𝑘) ↔ if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘)))
18 simpl 483 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ 𝑘 = 𝑥) → 𝐶 (𝐹𝑘))
19 ssidd 3967 . . . . . . . . . . 11 ((𝐶 (𝐹𝑘) ∧ ¬ 𝑘 = 𝑥) → (𝐹𝑘) ⊆ (𝐹𝑘))
2016, 17, 18, 19ifbothda 4524 . . . . . . . . . 10 (𝐶 (𝐹𝑘) → if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
2120ralimi 3086 . . . . . . . . 9 (∀𝑘𝐴 𝐶 (𝐹𝑘) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (𝐹𝑘))
22 ss2ixp 8848 . . . . . . . . 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 481 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ X𝑘𝐴 (𝐹𝑘))
2512adantr 481 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
2624, 25sseqtrd 3984 . . . . . 6 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ⊆ (∏t𝐹))
2712eqcomd 2742 . . . . . . . . . 10 (𝜑 (∏t𝐹) = X𝑘𝐴 (𝐹𝑘))
2827difeq1d 4081 . . . . . . . . 9 (𝜑 → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
2928adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))))
30 simpr 485 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
315adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑘𝐴 𝐶 (𝐹𝑘))
32 boxcutc 8879 . . . . . . . . 9 ((𝑥𝐴 ∧ ∀𝑘𝐴 𝐶 (𝐹𝑘)) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
3330, 31, 32syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝐴) → (X𝑘𝐴 (𝐹𝑘) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)))
34 ixpeq2 8849 . . . . . . . . . 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 6842 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
3635unieqd 4879 . . . . . . . . . . . . 13 (𝑘 = 𝑥 (𝐹𝑘) = (𝐹𝑥))
37 csbeq1a 3869 . . . . . . . . . . . . 13 (𝑘 = 𝑥𝐶 = 𝑥 / 𝑘𝐶)
3836, 37difeq12d 4083 . . . . . . . . . . . 12 (𝑘 = 𝑥 → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
3938adantl 482 . . . . . . . . . . 11 ((𝑘𝐴𝑘 = 𝑥) → ( (𝐹𝑘) ∖ 𝐶) = ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶))
4039ifeq1da 4517 . . . . . . . . . 10 (𝑘𝐴 → if(𝑘 = 𝑥, ( (𝐹𝑘) ∖ 𝐶), (𝐹𝑘)) = if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
4134, 40mprg 3070 . . . . . . . . 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 2780 . . . . . . 7 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) = X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)))
448adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐴𝑉)
459adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐹:𝐴⟶Top)
461ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)))
47 nfv 1917 . . . . . . . . . . . 12 𝑥 𝐶 ∈ (Clsd‘(𝐹𝑘))
48 nfcsb1v 3880 . . . . . . . . . . . . 13 𝑘𝑥 / 𝑘𝐶
4948nfel1 2923 . . . . . . . . . . . 12 𝑘𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))
50 2fveq3 6847 . . . . . . . . . . . . 13 (𝑘 = 𝑥 → (Clsd‘(𝐹𝑘)) = (Clsd‘(𝐹𝑥)))
5137, 50eleq12d 2832 . . . . . . . . . . . 12 (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥))))
5247, 49, 51cbvralw 3289 . . . . . . . . . . 11 (∀𝑘𝐴 𝐶 ∈ (Clsd‘(𝐹𝑘)) ↔ ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5346, 52sylib 217 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
5453r19.21bi 3234 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)))
55 eqid 2736 . . . . . . . . . 10 (𝐹𝑥) = (𝐹𝑥)
5655cldopn 22382 . . . . . . . . 9 (𝑥 / 𝑘𝐶 ∈ (Clsd‘(𝐹𝑥)) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5754, 56syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶) ∈ (𝐹𝑥))
5844, 45, 57ptopn2 22935 . . . . . . 7 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, ( (𝐹𝑥) ∖ 𝑥 / 𝑘𝐶), (𝐹𝑘)) ∈ (∏t𝐹))
5943, 58eqeltrd 2838 . . . . . 6 ((𝜑𝑥𝐴) → ( (∏t𝐹) ∖ X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (∏t𝐹))
60 eqid 2736 . . . . . . . . 9 (∏t𝐹) = (∏t𝐹)
6160iscld 22378 . . . . . . . 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 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))))
6426, 59, 63mpbir2and 711 . . . . 5 ((𝜑𝑥𝐴) → X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6564ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘)) ∈ (Clsd‘(∏t𝐹)))
6660riincld 22395 . . . 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 584 . . 3 (𝜑 → ( (∏t𝐹) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
6813, 67eqeltrd 2838 . 2 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ 𝑥𝐴 X𝑘𝐴 if(𝑘 = 𝑥, 𝐶, (𝐹𝑘))) ∈ (Clsd‘(∏t𝐹)))
697, 68eqeltrd 2838 1 (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  csb 3855  cdif 3907  cin 3909  wss 3910  ifcif 4486   cuni 4865   ciin 4955  wf 6492  cfv 6496  Xcixp 8835  tcpt 17320  Topctop 22242  Clsdccld 22367
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-er 8648  df-ixp 8836  df-en 8884  df-fin 8887  df-fi 9347  df-topgen 17325  df-pt 17326  df-top 22243  df-bases 22296  df-cld 22370
This theorem is referenced by:  ptcldmpt  22965
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