Theorem lindslinindimp2lem3 42249

Description: Lemma 3 for lindslinindsimp2 42252. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

Ref	Expression
lindslinind.r	|- R = (Scalar ` M )
lindslinind.b	|- B = ( Base ` R )
lindslinind.0	|- .0. = ( 0g ` R )
lindslinind.z	|- Z = ( 0g ` M )
lindslinind.y	|- Y = ( ( invg ` R ) ` ( f ` x ) )
lindslinind.g	|- G = ( f |` ( S \ { x } ) )

Assertion

Ref	Expression
lindslinindimp2lem3	|- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. ) ) -> G finSupp .0. )

Distinct variable groups:   B, f   f, M   R, f, x   S, f, x   f, Z   .0. , f, x

Allowed substitution hints:   B( x)   G( x, f)   M( x)   V( x, f)   Y( x, f)   Z( x)

Proof of Theorem lindslinindimp2lem3

Step	Hyp	Ref	Expression
1		lindslinind.g	. 2 |- G = ( f |` ( S \ { x } ) )
2		simp3r 1090	. . 3 |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. ) ) -> f finSupp .0. )
3		lindslinind.0	. . . . 5 |- .0. = ( 0g ` R )
4		fvex 6201	. . . . 5 |- ( 0g ` R ) e. _V
5	3, 4	eqeltri 2697	. . . 4 |- .0. e. _V
6	5	a1i 11	. . 3 |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. ) ) -> .0. e. _V )
7	2, 6	fsuppres 8300	. 2 |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. ) ) -> ( f |` ( S \ { x } ) ) finSupp .0. )
8	1, 7	syl5eqbr 4688	1 |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S ) /\ ( f e. ( B ^m S ) /\ f finSupp .0. ) ) -> G finSupp .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   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-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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  lindslinindimp2lem4  42250