Theorem ptcldmpt 23638

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 2903	. . 3 ⊢ Ⅎ𝑙𝐶
2		nfcsb1v 3933	. . 3 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶
3		csbeq1a 3922	. . 3 ⊢ (𝑘 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑘⦌𝐶)
4	1, 2, 3	cbvixp 8953	. 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 7135	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐽):𝐴⟶Top)
8		nfv 1912	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ 𝐴)
9		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑘Clsd
10		nffvmpt1 6918	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)
11	9, 10	nffv 6917	. . . . . 6 ⊢ Ⅎ𝑘(Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
12	2, 11	nfel 2918	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
13	8, 12	nfim 1894	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
14		eleq1w 2822	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴))
15	14	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝑙 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)))
16		2fveq3 6912	. . . . . 6 ⊢ (𝑘 = 𝑙 → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
17	3, 16	eleq12d 2833	. . . . 5 ⊢ (𝑘 = 𝑙 → (𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) ↔ ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))))
18	15, 17	imbi12d 344	. . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
21		eqid 2735	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐽) = (𝑘 ∈ 𝐴 ↦ 𝐽)
22	21	fvmpt2 7027	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐽 ∈ Top) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
23	20, 6, 22	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
24	23	fveq2d 6911	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘𝐽))
25	19, 24	eleqtrrd 2842	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)))
26	13, 18, 25	chvarfv 2238	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
27	5, 7, 26	ptcld 23637	. 2 ⊢ (𝜑 → X𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))
28	4, 27	eqeltrid 2843	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ⦋csb 3908   ↦ cmpt 5231  ‘cfv 6563  Xcixp 8936  ∏_tcpt 17485  Topctop 22915  Clsdccld 23040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-ixp 8937  df-en 8985  df-fin 8988  df-fi 9449  df-topgen 17490  df-pt 17491  df-top 22916  df-bases 22969  df-cld 23043

This theorem is referenced by:  ptclsg  23639  kelac1  43052