Theorem csubspset 17208

Description: The set of closed subspaces of a pre-Hilbert space.

Hypotheses

Ref	Expression
csubspset.v	|- V = (vbase` H)
csubspset.o	|- O = (ocv` H)
csubspset.c	|- C = (CSubSp` H)

Assertion

Ref	Expression
csubspset	|- (H e. A -> C = {s | (s C_ V /\ s = (O` (O` s)))})

Distinct variable groups:   H,s   V,s

Proof of Theorem csubspset

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (H e. A -> H e. _V)
2		fveq2 4681	. . . . . . . 8 |- (h = H -> (vbase` h) = (vbase` H))
3		csubspset.v	. . . . . . . 8 |- V = (vbase` H)
4	2, 3	syl6eqr 1946	. . . . . . 7 |- (h = H -> (vbase` h) = V)
5	4	sseq2d 2645	. . . . . 6 |- (h = H -> (s C_ (vbase` h) <-> s C_ V))
6		fveq2 4681	. . . . . . . . 9 |- (h = H -> (ocv` h) = (ocv` H))
7		csubspset.o	. . . . . . . . 9 |- O = (ocv` H)
8	6, 7	syl6eqr 1946	. . . . . . . 8 |- (h = H -> (ocv` h) = O)
9	8	fveq1d 4683	. . . . . . . 8 |- (h = H -> ((ocv` h)` s) = (O` s))
10	8, 9	fveq12d 10152	. . . . . . 7 |- (h = H -> ((ocv` h)` ((ocv` h)` s)) = (O` (O` s)))
11	10	eqeq2d 1895	. . . . . 6 |- (h = H -> (s = ((ocv` h)` ((ocv` h)` s)) <-> s = (O` (O` s))))
12	5, 11	anbi12d 690	. . . . 5 |- (h = H -> ((s C_ (vbase` h) /\ s = ((ocv` h)` ((ocv` h)` s))) <-> (s C_ V /\ s = (O` (O` s)))))
13	12	abbidv 2008	. . . 4 |- (h = H -> {s | (s C_ (vbase` h) /\ s = ((ocv` h)` ((ocv` h)` s)))} = {s | (s C_ V /\ s = (O` (O` s)))})
14		df-csubsp 17202	. . . 4 |- CSubSp = (h e. _V |-> {s | (s C_ (vbase` h) /\ s = ((ocv` h)` ((ocv` h)` s)))})
15		fvex 4689	. . . . . . 7 |- (vbase` H) e. _V
16	3, 15	eqeltri 1967	. . . . . 6 |- V e. _V
17	16	pwex 3487	. . . . 5 |- ~PV e. _V
18		visset 2295	. . . . . . . . 9 |- s e. _V
19	18	elpw 3037	. . . . . . . 8 |- (s e. ~PV <-> s C_ V)
20	19	anbi1i 539	. . . . . . 7 |- ((s e. ~PV /\ s = (O` (O` s))) <-> (s C_ V /\ s = (O` (O` s))))
21	20	abbii 2006	. . . . . 6 |- {s | (s e. ~PV /\ s = (O` (O` s)))} = {s | (s C_ V /\ s = (O` (O` s)))}
22		ssab2 2691	. . . . . 6 |- {s | (s e. ~PV /\ s = (O` (O` s)))} C_ ~PV
23	21, 22	eqsstr3i 2648	. . . . 5 |- {s | (s C_ V /\ s = (O` (O` s)))} C_ ~PV
24	17, 23	ssexi 3456	. . . 4 |- {s | (s C_ V /\ s = (O` (O` s)))} e. _V
25	13, 14, 24	fvmpt 5015	. . 3 |- (H e. _V -> (CSubSp` H) = {s | (s C_ V /\ s = (O` (O` s)))})
26		csubspset.c	. . 3 |- C = (CSubSp` H)
27	25, 26	syl5eq 1940	. 2 |- (H e. _V -> C = {s | (s C_ V /\ s = (O` (O` s)))})
28	1, 27	syl 12	1 |- (H e. A -> C = {s | (s C_ V /\ s = (O` (O` s)))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  ` cfv 3998  vbasecvbase 17180  ocvcocv 17197  CSubSpccsubsp 17198

This theorem is referenced by:  iscsubsp 17209

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-mpt 5006  df-csubsp 17202

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