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Theorem lflnegl 29811
 Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29881, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
lflnegcl.v
lflnegcl.r Scalar
lflnegcl.i
lflnegcl.n
lflnegcl.f LFnl
lflnegcl.w
lflnegcl.g
lflnegl.p
lflnegl.o
Assertion
Ref Expression
lflnegl
Distinct variable groups:   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem lflnegl
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lflnegcl.v . . . 4
2 fvex 5734 . . . 4
31, 2eqeltri 2505 . . 3
43a1i 11 . 2
5 lflnegcl.w . . 3
6 lflnegcl.g . . 3
7 lflnegcl.r . . . 4 Scalar
8 eqid 2435 . . . 4
9 lflnegcl.f . . . 4 LFnl
107, 8, 1, 9lflf 29798 . . 3
115, 6, 10syl2anc 643 . 2
12 lflnegl.o . . . 4
13 fvex 5734 . . . 4
1412, 13eqeltri 2505 . . 3
1514a1i 11 . 2
16 lflnegcl.i . . . 4
177lmodrng 15950 . . . . 5
18 rnggrp 15661 . . . . 5
195, 17, 183syl 19 . . . 4
208, 16, 19grpinvf1o 14853 . . 3
21 f1of 5666 . . 3
2220, 21syl 16 . 2
23 lflnegcl.n . . 3
2423a1i 11 . 2
25 lflnegl.p . . . 4
268, 25, 12, 16grplinv 14843 . . 3
2719, 26sylan 458 . 2
284, 11, 15, 22, 24, 27caofinvl 6323 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cvv 2948  csn 3806   cmpt 4258   cxp 4868  wf 5442  wf1o 5445  cfv 5446  (class class class)co 6073   cof 6295  cbs 13461   cplusg 13521  Scalarcsca 13524  c0g 13715  cgrp 14677  cminusg 14678  crg 15652  clmod 15942  LFnlclfn 29792 This theorem is referenced by:  ldualgrplem  29880 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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693 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-rmo 2705  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-pw 3793  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-oprab 6077  df-mpt2 6078  df-of 6297  df-riota 6541  df-map 7012  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-rng 15655  df-lmod 15944  df-lfl 29793
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