Theorem snlindsntorlem 31009

Description: Lemma for snlindsntor 31010. (Contributed by AV, 15-Apr-2019.)

Hypotheses

Ref	Expression
snlindsntor.b	|- B = ( Base ` M )
snlindsntor.r	|- R = (Scalar ` M )
snlindsntor.s	|- S = ( Base ` R )
snlindsntor.0	|- .0. = ( 0g ` R )
snlindsntor.z	|- Z = ( 0g ` M )
snlindsntor.t	|- .x. = ( .s ` M )

Assertion

Ref	Expression
snlindsntorlem	|- ( ( M e. LMod /\ X e. B ) -> ( A. f e. ( S ^m { X } ) ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) -> A. s e. S ( ( s .x. X ) = Z -> s = .0. ) ) )

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

Allowed substitution hints:   R( f, s)

Proof of Theorem snlindsntorlem

Step	Hyp	Ref	Expression
1		eqidd 2444	. . . . . . . 8 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> { <. X , s >. } = { <. X , s >. } )
2		fsng 5887	. . . . . . . . 9 |- ( ( X e. B /\ s e. S ) -> ( { <. X , s >. } : { X } --> { s } <-> { <. X , s >. } = { <. X , s >. } ) )
3	2	adantll 713	. . . . . . . 8 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( { <. X , s >. } : { X } --> { s } <-> { <. X , s >. } = { <. X , s >. } ) )
4	1, 3	mpbird 232	. . . . . . 7 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> { <. X , s >. } : { X } --> { s } )
5		snssi 4022	. . . . . . . 8 |- ( s e. S -> { s } C_ S )
6	5	adantl 466	. . . . . . 7 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> { s } C_ S )
7		fss 5572	. . . . . . 7 |- ( ( { <. X , s >. } : { X } --> { s } /\ { s } C_ S ) -> { <. X , s >. } : { X } --> S )
8	4, 6, 7	syl2anc 661	. . . . . 6 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> { <. X , s >. } : { X } --> S )
9		snlindsntor.s	. . . . . . . . 9 |- S = ( Base ` R )
10		fvex 5706	. . . . . . . . 9 |- ( Base ` R ) e. _V
11	9, 10	eqeltri 2513	. . . . . . . 8 |- S e. _V
12		snex 4538	. . . . . . . 8 |- { X } e. _V
13	11, 12	pm3.2i 455	. . . . . . 7 |- ( S e. _V /\ { X } e. _V )
14		elmapg 7232	. . . . . . 7 |- ( ( S e. _V /\ { X } e. _V ) -> ( { <. X , s >. } e. ( S ^m { X } ) <-> { <. X , s >. } : { X } --> S ) )
15	13, 14	mp1i 12	. . . . . 6 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( { <. X , s >. } e. ( S ^m { X } ) <-> { <. X , s >. } : { X } --> S ) )
16	8, 15	mpbird 232	. . . . 5 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> { <. X , s >. } e. ( S ^m { X } ) )
17		oveq1 6103	. . . . . . . . 9 |- ( f = { <. X , s >. } -> ( f ( linC ` M ) { X } ) = ( { <. X , s >. } ( linC ` M ) { X } ) )
18	17	eqeq1d 2451	. . . . . . . 8 |- ( f = { <. X , s >. } -> ( ( f ( linC ` M ) { X } ) = Z <-> ( { <. X , s >. } ( linC ` M ) { X } ) = Z ) )
19		fveq1 5695	. . . . . . . . 9 |- ( f = { <. X , s >. } -> ( f ` X ) = ( { <. X , s >. } ` X ) )
20	19	eqeq1d 2451	. . . . . . . 8 |- ( f = { <. X , s >. } -> ( ( f ` X ) = .0. <-> ( { <. X , s >. } ` X ) = .0. ) )
21	18, 20	imbi12d 320	. . . . . . 7 |- ( f = { <. X , s >. } -> ( ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) <-> ( ( { <. X , s >. } ( linC ` M ) { X } ) = Z -> ( { <. X , s >. } ` X ) = .0. ) ) )
22		snlindsntor.b	. . . . . . . . . . 11 |- B = ( Base ` M )
23		snlindsntor.r	. . . . . . . . . . 11 |- R = (Scalar ` M )
24		snlindsntor.t	. . . . . . . . . . 11 |- .x. = ( .s ` M )
25		eqid 2443	. . . . . . . . . . 11 |- { <. X , s >. } = { <. X , s >. }
26	22, 23, 9, 24, 25	lincvalsn 30956	. . . . . . . . . 10 |- ( ( M e. LMod /\ X e. B /\ s e. S ) -> ( { <. X , s >. } ( linC ` M ) { X } ) = ( s .x. X ) )
27	26	3expa 1187	. . . . . . . . 9 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( { <. X , s >. } ( linC ` M ) { X } ) = ( s .x. X ) )
28	27	eqeq1d 2451	. . . . . . . 8 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( ( { <. X , s >. } ( linC ` M ) { X } ) = Z <-> ( s .x. X ) = Z ) )
29		fvsng 5917	. . . . . . . . . 10 |- ( ( X e. B /\ s e. S ) -> ( { <. X , s >. } ` X ) = s )
30	29	adantll 713	. . . . . . . . 9 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( { <. X , s >. } ` X ) = s )
31	30	eqeq1d 2451	. . . . . . . 8 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( ( { <. X , s >. } ` X ) = .0. <-> s = .0. ) )
32	28, 31	imbi12d 320	. . . . . . 7 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( ( ( { <. X , s >. } ( linC ` M ) { X } ) = Z -> ( { <. X , s >. } ` X ) = .0. ) <-> ( ( s .x. X ) = Z -> s = .0. ) ) )
33	21, 32	sylan9bbr 700	. . . . . 6 |- ( ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) /\ f = { <. X , s >. } ) -> ( ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) <-> ( ( s .x. X ) = Z -> s = .0. ) ) )
34	33	biimpd 207	. . . . 5 |- ( ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) /\ f = { <. X , s >. } ) -> ( ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) -> ( ( s .x. X ) = Z -> s = .0. ) ) )
35	16, 34	rspcimdv 3079	. . . 4 |- ( ( ( M e. LMod /\ X e. B ) /\ s e. S ) -> ( A. f e. ( S ^m { X } ) ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) -> ( ( s .x. X ) = Z -> s = .0. ) ) )
36	35	impancom 440	. . 3 |- ( ( ( M e. LMod /\ X e. B ) /\ A. f e. ( S ^m { X } ) ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) ) -> ( s e. S -> ( ( s .x. X ) = Z -> s = .0. ) ) )
37	36	ralrimiv 2803	. 2 |- ( ( ( M e. LMod /\ X e. B ) /\ A. f e. ( S ^m { X } ) ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) ) -> A. s e. S ( ( s .x. X ) = Z -> s = .0. ) )
38	37	ex 434	1 |- ( ( M e. LMod /\ X e. B ) -> ( A. f e. ( S ^m { X } ) ( ( f ( linC ` M ) { X } ) = Z -> ( f ` X ) = .0. ) -> A. s e. S ( ( s .x. X ) = Z -> s = .0. ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2720   _Vcvv 2977   C_ wss 3333   {csn 3882   <.cop 3888   -->wf 5419   ` cfv 5423  (class class class)co 6096   ^m cmap 7219   Basecbs 14179  Scalarcsca 14246   .scvsca 14247   0gc0g 14383   LModclmod 16953   linC clinc 30943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-0g 14385  df-gsum 14386  df-mnd 15420  df-grp 15550  df-mulg 15553  df-cntz 15840  df-lmod 16955  df-linc 30945

This theorem is referenced by:  snlindsntor  31010