Theorem ptcldmpt 23546

Description: A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Hypotheses

Ref	Expression
ptcldmpt.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ptcldmpt.j	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ Top)
ptcldmpt.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘𝐽))

Assertion

Ref	Expression
ptcldmpt	⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))

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.

Step	Hyp	Ref	Expression
1		nfcv 2899	. . 3 ⊢ Ⅎ𝑙𝐶
2		nfcsb1v 3919	. . 3 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶
3		csbeq1a 3908	. . 3 ⊢ (𝑘 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑘⦌𝐶)
4	1, 2, 3	cbvixp 8941	. 2 ⊢ X𝑘 ∈ 𝐴 𝐶 = X𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐶
5		ptcldmpt.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		ptcldmpt.j	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ Top)
7	6	fmpttd 7130	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐽):𝐴⟶Top)
8		nfv 1909	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ 𝐴)
9		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑘Clsd
10		nffvmpt1 6913	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)
11	9, 10	nffv 6912	. . . . . 6 ⊢ Ⅎ𝑘(Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
12	2, 11	nfel 2914	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
13	8, 12	nfim 1891	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
14		eleq1w 2812	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴))
15	14	anbi2d 628	. . . . 5 ⊢ (𝑘 = 𝑙 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)))
16		2fveq3 6907	. . . . . 6 ⊢ (𝑘 = 𝑙 → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
17	3, 16	eleq12d 2823	. . . . 5 ⊢ (𝑘 = 𝑙 → (𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) ↔ ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))))
18	15, 17	imbi12d 343	. . . 4 ⊢ (𝑘 = 𝑙 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘))) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))))
19		ptcldmpt.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘𝐽))
20		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
21		eqid 2728	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐽) = (𝑘 ∈ 𝐴 ↦ 𝐽)
22	21	fvmpt2 7021	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐽 ∈ Top) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
23	20, 6, 22	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
24	23	fveq2d 6906	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘𝐽))
25	19, 24	eleqtrrd 2832	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)))
26	13, 18, 25	chvarfv 2228	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
27	5, 7, 26	ptcld 23545	. 2 ⊢ (𝜑 → X𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))
28	4, 27	eqeltrid 2833	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⦋csb 3894   ↦ cmpt 5235  ‘cfv 6553  Xcixp 8924  ∏_tcpt 17429  Topctop 22823  Clsdccld 22948

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7879  df-1o 8495  df-er 8733  df-ixp 8925  df-en 8973  df-fin 8976  df-fi 9444  df-topgen 17434  df-pt 17435  df-top 22824  df-bases 22877  df-cld 22951

This theorem is referenced by:  ptclsg  23547  kelac1  42536