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Theorem lcdfval 32313
 Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
Hypothesis
Ref Expression
lcdval.h
Assertion
Ref Expression
lcdfval LCDual LDuals LFnl LKer LKer
Distinct variable groups:   ,   ,,
Allowed substitution hints:   ()   (,)

Proof of Theorem lcdfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2956 . 2
2 fveq2 5720 . . . . 5
3 lcdval.h . . . . 5
42, 3syl6eqr 2485 . . . 4
5 fveq2 5720 . . . . . . 7
65fveq1d 5722 . . . . . 6
76fveq2d 5724 . . . . 5 LDual LDual
86fveq2d 5724 . . . . . 6 LFnl LFnl
9 fveq2 5720 . . . . . . . . 9
109fveq1d 5722 . . . . . . . 8
116fveq2d 5724 . . . . . . . . . 10 LKer LKer
1211fveq1d 5722 . . . . . . . . 9 LKer LKer
1310, 12fveq12d 5726 . . . . . . . 8 LKer LKer
1410, 13fveq12d 5726 . . . . . . 7 LKer LKer
1514, 12eqeq12d 2449 . . . . . 6 LKer LKer LKer LKer
168, 15rabeqbidv 2943 . . . . 5 LFnl LKer LKer LFnl LKer LKer
177, 16oveq12d 6091 . . . 4 LDuals LFnl LKer LKer LDuals LFnl LKer LKer
184, 17mpteq12dv 4279 . . 3 LDuals LFnl LKer LKer LDuals LFnl LKer LKer
19 df-lcdual 32312 . . 3 LCDual LDuals LFnl LKer LKer
20 fvex 5734 . . . . 5
213, 20eqeltri 2505 . . . 4
2221mptex 5958 . . 3 LDuals LFnl LKer LKer
2318, 19, 22fvmpt 5798 . 2 LCDual LDuals LFnl LKer LKer
241, 23syl 16 1 LCDual LDuals LFnl LKer LKer
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  crab 2701  cvv 2948   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:  lcdval  32314 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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