Theorem ptcldmpt 23579

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 3862	. . 3 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶
3		csbeq1a 3852	. . 3 ⊢ (𝑘 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑘⦌𝐶)
4	1, 2, 3	cbvixp 8862	. 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 7068	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐽):𝐴⟶Top)
8		nfv 1916	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ 𝐴)
9		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑘Clsd
10		nffvmpt1 6852	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)
11	9, 10	nffv 6851	. . . . . 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 1898	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
14		eleq1w 2820	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴))
15	14	anbi2d 631	. . . . 5 ⊢ (𝑘 = 𝑙 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)))
16		2fveq3 6846	. . . . . 6 ⊢ (𝑘 = 𝑙 → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
17	3, 16	eleq12d 2831	. . . . 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 2737	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐽) = (𝑘 ∈ 𝐴 ↦ 𝐽)
22	21	fvmpt2 6960	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐽 ∈ Top) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
23	20, 6, 22	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
24	23	fveq2d 6845	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘𝐽))
25	19, 24	eleqtrrd 2840	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)))
26	13, 18, 25	chvarfv 2248	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
27	5, 7, 26	ptcld 23578	. 2 ⊢ (𝜑 → X𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))
28	4, 27	eqeltrid 2841	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⦋csb 3838   ↦ cmpt 5167  ‘cfv 6499  Xcixp 8845  ∏_tcpt 17401  Topctop 22858  Clsdccld 22981

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7818  df-1o 8405  df-2o 8406  df-ixp 8846  df-en 8894  df-fin 8897  df-fi 9324  df-topgen 17406  df-pt 17407  df-top 22859  df-bases 22911  df-cld 22984

This theorem is referenced by:  ptclsg  23580  kelac1  43491