Theorem lindslinindimp2lem1 42247

Description: Lemma 1 for lindslinindsimp2 42252. (Contributed by AV, 25-Apr-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
lindslinindimp2lem1	|- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S /\ f e. ( B ^m S ) ) ) -> Y e. B )

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 lindslinindimp2lem1

Step	Hyp	Ref	Expression
1		lindslinind.y	. 2 |- Y = ( ( invg ` R ) ` ( f ` x ) )
2		lindslinind.r	. . . . 5 |- R = (Scalar ` M )
3	2	lmodfgrp 18872	. . . 4 |- ( M e. LMod -> R e. Grp )
4	3	adantl 482	. . 3 |- ( ( S e. V /\ M e. LMod ) -> R e. Grp )
5		elmapi 7879	. . . . . 6 |- ( f e. ( B ^m S ) -> f : S --> B )
6		ffvelrn 6357	. . . . . . . 8 |- ( ( f : S --> B /\ x e. S ) -> ( f ` x ) e. B )
7	6	a1d 25	. . . . . . 7 |- ( ( f : S --> B /\ x e. S ) -> ( S C_ ( Base ` M ) -> ( f ` x ) e. B ) )
8	7	ex 450	. . . . . 6 |- ( f : S --> B -> ( x e. S -> ( S C_ ( Base ` M ) -> ( f ` x ) e. B ) ) )
9	5, 8	syl 17	. . . . 5 |- ( f e. ( B ^m S ) -> ( x e. S -> ( S C_ ( Base ` M ) -> ( f ` x ) e. B ) ) )
10	9	com13 88	. . . 4 |- ( S C_ ( Base ` M ) -> ( x e. S -> ( f e. ( B ^m S ) -> ( f ` x ) e. B ) ) )
11	10	3imp 1256	. . 3 |- ( ( S C_ ( Base ` M ) /\ x e. S /\ f e. ( B ^m S ) ) -> ( f ` x ) e. B )
12		lindslinind.b	. . . 4 |- B = ( Base ` R )
13		eqid 2622	. . . 4 |- ( invg ` R ) = ( invg ` R )
14	12, 13	grpinvcl 17467	. . 3 |- ( ( R e. Grp /\ ( f ` x ) e. B ) -> ( ( invg ` R ) ` ( f ` x ) ) e. B )
15	4, 11, 14	syl2an 494	. 2 |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S /\ f e. ( B ^m S ) ) ) -> ( ( invg ` R ) ` ( f ` x ) ) e. B )
16	1, 15	syl5eqel 2705	1 |- ( ( ( S e. V /\ M e. LMod ) /\ ( S C_ ( Base ` M ) /\ x e. S /\ f e. ( B ^m S ) ) ) -> Y e. B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   {csn 4177   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   Grpcgrp 17422   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-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-1st 7168  df-2nd 7169  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

This theorem is referenced by: (None)