Theorem lflnegl 34363

Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 34433, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)

Hypotheses

Ref	Expression
lflnegcl.v	|- V = ( Base ` W )
lflnegcl.r	|- R = (Scalar ` W )
lflnegcl.i	|- I = ( invg ` R )
lflnegcl.n	|- N = ( x e. V |-> ( I ` ( G ` x ) ) )
lflnegcl.f	|- F = (LFnl ` W )
lflnegcl.w	|- ( ph -> W e. LMod )
lflnegcl.g	|- ( ph -> G e. F )
lflnegl.p	|- .+ = ( +g ` R )
lflnegl.o	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
lflnegl	|- ( ph -> ( N oF .+ G ) = ( V X. { .0. } ) )

Distinct variable groups:   x, G   x, I   x, R   x, V   x, W   ph, x

Allowed substitution hints:   .+ ( x)   F( x)   N( x)   .0. ( x)

Proof of Theorem lflnegl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lflnegcl.v	. . . 4 |- V = ( Base ` W )
2		fvex 6201	. . . 4 |- ( Base ` W ) e. _V
3	1, 2	eqeltri 2697	. . 3 |- V e. _V
4	3	a1i 11	. 2 |- ( ph -> V e. _V )
5		lflnegcl.w	. . 3 |- ( ph -> W e. LMod )
6		lflnegcl.g	. . 3 |- ( ph -> G e. F )
7		lflnegcl.r	. . . 4 |- R = (Scalar ` W )
8		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
9		lflnegcl.f	. . . 4 |- F = (LFnl ` W )
10	7, 8, 1, 9	lflf 34350	. . 3 |- ( ( W e. LMod /\ G e. F ) -> G : V --> ( Base ` R ) )
11	5, 6, 10	syl2anc 693	. 2 |- ( ph -> G : V --> ( Base ` R ) )
12		lflnegl.o	. . . 4 |- .0. = ( 0g ` R )
13		fvex 6201	. . . 4 |- ( 0g ` R ) e. _V
14	12, 13	eqeltri 2697	. . 3 |- .0. e. _V
15	14	a1i 11	. 2 |- ( ph -> .0. e. _V )
16		lflnegcl.i	. . . 4 |- I = ( invg ` R )
17	7	lmodring 18871	. . . . 5 |- ( W e. LMod -> R e. Ring )
18		ringgrp 18552	. . . . 5 |- ( R e. Ring -> R e. Grp )
19	5, 17, 18	3syl 18	. . . 4 |- ( ph -> R e. Grp )
20	8, 16, 19	grpinvf1o 17485	. . 3 |- ( ph -> I : ( Base ` R ) -1-1-onto-> ( Base ` R ) )
21		f1of 6137	. . 3 |- ( I : ( Base ` R ) -1-1-onto-> ( Base ` R ) -> I : ( Base ` R ) --> ( Base ` R ) )
22	20, 21	syl 17	. 2 |- ( ph -> I : ( Base ` R ) --> ( Base ` R ) )
23		lflnegcl.n	. . 3 |- N = ( x e. V |-> ( I ` ( G ` x ) ) )
24	23	a1i 11	. 2 |- ( ph -> N = ( x e. V |-> ( I ` ( G ` x ) ) ) )
25		lflnegl.p	. . . 4 |- .+ = ( +g ` R )
26	8, 25, 12, 16	grplinv 17468	. . 3 |- ( ( R e. Grp /\ y e. ( Base ` R ) ) -> ( ( I ` y ) .+ y ) = .0. )
27	19, 26	sylan 488	. 2 |- ( ( ph /\ y e. ( Base ` R ) ) -> ( ( I ` y ) .+ y ) = .0. )
28	4, 11, 15, 22, 24, 27	caofinvl 6924	1 |- ( ph -> ( N oF .+ G ) = ( V X. { .0. } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   Ringcrg 18547   LModclmod 18863  LFnlclfn 34344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ring 18549  df-lmod 18865  df-lfl 34345

This theorem is referenced by:  ldualgrplem  34432