Theorem ptcldmpt 23782

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 2924	. . 3 ⊢ Ⅎ𝑙𝐶
2		nfcsb1v 3876	. . 3 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶
3		csbeq1a 3866	. . 3 ⊢ (𝑘 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑘⦌𝐶)
4	1, 2, 3	cbvixp 8910	. 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 7110	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐽):𝐴⟶Top)
8		nfv 1943	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ 𝐴)
9		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑘Clsd
10		nffvmpt1 6892	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)
11	9, 10	nffv 6891	. . . . . 6 ⊢ Ⅎ𝑘(Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
12	2, 11	nfel 2938	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))
13	8, 12	nfim 1925	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
14		eleq1w 2845	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴))
15	14	anbi2d 641	. . . . 5 ⊢ (𝑘 = 𝑙 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)))
16		2fveq3 6886	. . . . . 6 ⊢ (𝑘 = 𝑙 → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
17	3, 16	eleq12d 2856	. . . . 5 ⊢ (𝑘 = 𝑙 → (𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) ↔ ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙))))
18	15, 17	imbi12d 347	. . . 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 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
21		eqid 2762	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐽) = (𝑘 ∈ 𝐴 ↦ 𝐽)
22	21	fvmpt2 7001	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐽 ∈ Top) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
23	20, 6, 22	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘) = 𝐽)
24	23	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)) = (Clsd‘𝐽))
25	19, 24	eleqtrrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑘)))
26	13, 18, 25	chvarfv 2275	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘((𝑘 ∈ 𝐴 ↦ 𝐽)‘𝑙)))
27	5, 7, 26	ptcld 23781	. 2 ⊢ (𝜑 → X𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))
28	4, 27	eqeltrid 2866	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ⦋csb 3852   ↦ cmpt 5191  ‘cfv 6536  Xcixp 8893  ∏_tcpt 17497  Topctop 23061  Clsdccld 23184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7861  df-1o 8451  df-2o 8452  df-ixp 8894  df-en 8942  df-fin 8945  df-fi 9369  df-topgen 17502  df-pt 17503  df-top 23062  df-bases 23114  df-cld 23187

This theorem is used by:  ptclsg  23783  kelac1  43818