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Theorem lcdval 32314
 Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
Hypotheses
Ref Expression
lcdval.h
lcdval.o
lcdval.c LCDual
lcdval.u
lcdval.f LFnl
lcdval.l LKer
lcdval.d LDual
lcdval.k
Assertion
Ref Expression
lcdval s
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   ()   ()   ()   ()

Proof of Theorem lcdval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lcdval.k . 2
2 lcdval.c . . . 4 LCDual
3 lcdval.h . . . . . 6
43lcdfval 32313 . . . . 5 LCDual LDuals LFnl LKer LKer
54fveq1d 5722 . . . 4 LCDual LDuals LFnl LKer LKer
62, 5syl5eq 2479 . . 3 LDuals LFnl LKer LKer
7 fveq2 5720 . . . . . . . 8
8 lcdval.u . . . . . . . 8
97, 8syl6eqr 2485 . . . . . . 7
109fveq2d 5724 . . . . . 6 LDual LDual
11 lcdval.d . . . . . 6 LDual
1210, 11syl6eqr 2485 . . . . 5 LDual
139fveq2d 5724 . . . . . . 7 LFnl LFnl
14 lcdval.f . . . . . . 7 LFnl
1513, 14syl6eqr 2485 . . . . . 6 LFnl
16 fveq2 5720 . . . . . . . . 9
17 lcdval.o . . . . . . . . 9
1816, 17syl6eqr 2485 . . . . . . . 8
199fveq2d 5724 . . . . . . . . . . 11 LKer LKer
20 lcdval.l . . . . . . . . . . 11 LKer
2119, 20syl6eqr 2485 . . . . . . . . . 10 LKer
2221fveq1d 5722 . . . . . . . . 9 LKer
2318, 22fveq12d 5726 . . . . . . . 8 LKer
2418, 23fveq12d 5726 . . . . . . 7 LKer
2524, 22eqeq12d 2449 . . . . . 6 LKer LKer
2615, 25rabeqbidv 2943 . . . . 5 LFnl LKer LKer
2712, 26oveq12d 6091 . . . 4 LDuals LFnl LKer LKer s
28 eqid 2435 . . . 4 LDuals LFnl LKer LKer LDuals LFnl LKer LKer
29 ovex 6098 . . . 4 s
3027, 28, 29fvmpt 5798 . . 3 LDuals LFnl LKer LKer s
316, 30sylan9eq 2487 . 2 s
321, 31syl 16 1 s
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  crab 2701   cmpt 4258  cfv 5446  (class class class)co 6073   ↾s cress 13462  LFnlclfn 29782  LKerclk 29810  LDualcld 29848  clh 30708  cdvh 31803  coch 32072  LCDualclcd 32311 This theorem is referenced by:  lcdval2  32315  lcdlvec  32316  lcdvadd  32322  lcdsca  32324  lcdvs  32328  lcd0v  32336  lcdlsp  32346 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-lcdual 32312
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