Theorem lincext2 42244

Description: Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

Ref	Expression
lincext.b	|- B = ( Base ` M )
lincext.r	|- R = (Scalar ` M )
lincext.e	|- E = ( Base ` R )
lincext.0	|- .0. = ( 0g ` R )
lincext.z	|- Z = ( 0g ` M )
lincext.n	|- N = ( invg ` R )
lincext.f	|- F = ( z e. S |-> if ( z = X , ( N ` Y ) , ( G ` z ) ) )

Assertion

Ref	Expression
lincext2	|- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> F finSupp .0. )

Distinct variable groups:   z, B   z, E   z, G   z, M   z, S   z, X   z, Y

Allowed substitution hints:   R( z)   F( z)   N( z)   .0. ( z)   Z( z)

Proof of Theorem lincext2

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . 6 |- ( N ` Y ) e. _V
2		fvex 6201	. . . . . 6 |- ( G ` z ) e. _V
3	1, 2	ifex 4156	. . . . 5 |- if ( z = X , ( N ` Y ) , ( G ` z ) ) e. _V
4		lincext.f	. . . . 5 |- F = ( z e. S |-> if ( z = X , ( N ` Y ) , ( G ` z ) ) )
5	3, 4	dmmpti 6023	. . . 4 |- dom F = S
6	5	difeq1i 3724	. . 3 |- ( dom F \ ( S \ { X } ) ) = ( S \ ( S \ { X } ) )
7		snssi 4339	. . . . . . 7 |- ( X e. S -> { X } C_ S )
8	7	3ad2ant2 1083	. . . . . 6 |- ( ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) -> { X } C_ S )
9	8	3ad2ant2 1083	. . . . 5 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> { X } C_ S )
10		dfss4 3858	. . . . 5 |- ( { X } C_ S <-> ( S \ ( S \ { X } ) ) = { X } )
11	9, 10	sylib 208	. . . 4 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> ( S \ ( S \ { X } ) ) = { X } )
12		snfi 8038	. . . 4 |- { X } e. Fin
13	11, 12	syl6eqel 2709	. . 3 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> ( S \ ( S \ { X } ) ) e. Fin )
14	6, 13	syl5eqel 2705	. 2 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> ( dom F \ ( S \ { X } ) ) e. Fin )
15		lincext.b	. . . 4 |- B = ( Base ` M )
16		lincext.r	. . . 4 |- R = (Scalar ` M )
17		lincext.e	. . . 4 |- E = ( Base ` R )
18		lincext.0	. . . 4 |- .0. = ( 0g ` R )
19		lincext.z	. . . 4 |- Z = ( 0g ` M )
20		lincext.n	. . . 4 |- N = ( invg ` R )
21	15, 16, 17, 18, 19, 20, 4	lincext1 42243	. . 3 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) ) -> F e. ( E ^m S ) )
22	21	3adant3 1081	. 2 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> F e. ( E ^m S ) )
23		elmapfun 7881	. . 3 |- ( F e. ( E ^m S ) -> Fun F )
24	22, 23	syl 17	. 2 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> Fun F )
25		elmapi 7879	. . . . 5 |- ( G e. ( E ^m ( S \ { X } ) ) -> G : ( S \ { X } ) --> E )
26	4	fdmdifeqresdif 42120	. . . . 5 |- ( G : ( S \ { X } ) --> E -> G = ( F |` ( S \ { X } ) ) )
27	25, 26	syl 17	. . . 4 |- ( G e. ( E ^m ( S \ { X } ) ) -> G = ( F |` ( S \ { X } ) ) )
28	27	3ad2ant3 1084	. . 3 |- ( ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) -> G = ( F |` ( S \ { X } ) ) )
29	28	3ad2ant2 1083	. 2 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> G = ( F |` ( S \ { X } ) ) )
30		simp3 1063	. 2 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> G finSupp .0. )
31		fvex 6201	. . . 4 |- ( 0g ` R ) e. _V
32	18, 31	eqeltri 2697	. . 3 |- .0. e. _V
33	32	a1i 11	. 2 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> .0. e. _V )
34	14, 22, 24, 29, 30, 33	resfsupp 8302	1 |- ( ( ( M e. LMod /\ S e. ~P B ) /\ ( Y e. E /\ X e. S /\ G e. ( E ^m ( S \ { X } ) ) ) /\ G finSupp .0. ) -> F finSupp .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   invgcminusg 17423   LModclmod 18863

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fsupp 8276  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ring 18549  df-lmod 18865

This theorem is referenced by:  lincext3  42245  lindslinindsimp1  42246  islindeps2  42272