Theorem ptcldmpt 23643

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 2908	. . 3 ⊢ Ⅎ𝑙𝐶
2		nfcsb1v 3946	. . 3 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶
3		csbeq1a 3935	. . 3 ⊢ (𝑘 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑘⦌𝐶)
4	1, 2, 3	cbvixp 8972	. 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 7149	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐽):𝐴⟶Top)
8		nfv 1913	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ 𝐴)
9		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑘Clsd
10		nffvmpt1 6931	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)
11	9, 10	nffv 6930	. . . . . 6 ⊢ Ⅎ𝑘(Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
12	2, 11	nfel 2923	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
13	8, 12	nfim 1895	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
14		eleq1w 2827	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴))
15	14	anbi2d 629	. . . . 5 ⊢ (𝑘 = 𝑙 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)))
16		2fveq3 6925	. . . . . 6 ⊢ (𝑘 = 𝑙 → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
17	3, 16	eleq12d 2838	. . . . 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 2740	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐽) = (𝑘 ∈ 𝐴 ↦ 𝐽)
22	21	fvmpt2 7040	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐽 ∈ Top) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
23	20, 6, 22	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
24	23	fveq2d 6924	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘𝐽))
25	19, 24	eleqtrrd 2847	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)))
26	13, 18, 25	chvarfv 2241	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
27	5, 7, 26	ptcld 23642	. 2 ⊢ (𝜑 → X𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))
28	4, 27	eqeltrid 2848	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⦋csb 3921   ↦ cmpt 5249  ‘cfv 6573  Xcixp 8955  ∏_tcpt 17498  Topctop 22920  Clsdccld 23045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-2o 8523  df-ixp 8956  df-en 9004  df-fin 9007  df-fi 9480  df-topgen 17503  df-pt 17504  df-top 22921  df-bases 22974  df-cld 23048

This theorem is referenced by:  ptclsg  23644  kelac1  43020