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Theorem psubclsetN 30734
 Description: The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a
psubclset.p
psubclset.c
Assertion
Ref Expression
psubclsetN
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem psubclsetN
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2965 . 2
2 psubclset.c . . 3
3 fveq2 5729 . . . . . . . 8
4 psubclset.a . . . . . . . 8
53, 4syl6eqr 2487 . . . . . . 7
65sseq2d 3377 . . . . . 6
7 fveq2 5729 . . . . . . . . 9
8 psubclset.p . . . . . . . . 9
97, 8syl6eqr 2487 . . . . . . . 8
109fveq1d 5731 . . . . . . . 8
119, 10fveq12d 5735 . . . . . . 7
1211eqeq1d 2445 . . . . . 6
136, 12anbi12d 693 . . . . 5
1413abbidv 2551 . . . 4
15 df-psubclN 30733 . . . 4
16 fvex 5743 . . . . . . 7
174, 16eqeltri 2507 . . . . . 6
1817pwex 4383 . . . . 5
19 df-pw 3802 . . . . . . . . 9
2019abeq2i 2544 . . . . . . . 8
2120anbi1i 678 . . . . . . 7
2221abbii 2549 . . . . . 6
23 ssab2 3428 . . . . . 6
2422, 23eqsstr3i 3380 . . . . 5
2518, 24ssexi 4349 . . . 4
2614, 15, 25fvmpt 5807 . . 3
272, 26syl5eq 2481 . 2
281, 27syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cab 2423  cvv 2957   wss 3321  cpw 3800  cfv 5455  catm 30062  cpolN 30700  cpscN 30732 This theorem is referenced by:  ispsubclN  30735 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-psubclN 30733
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