Theorem kercvrlsm 37653

Description: The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypotheses

Ref	Expression
kercvrlsm.u	|- U = ( LSubSp ` S )
kercvrlsm.p	|- .(+) = ( LSSum ` S )
kercvrlsm.z	|- .0. = ( 0g ` T )
kercvrlsm.k	|- K = ( `' F " { .0. } )
kercvrlsm.b	|- B = ( Base ` S )
kercvrlsm.f	|- ( ph -> F e. ( S LMHom T ) )
kercvrlsm.d	|- ( ph -> D e. U )
kercvrlsm.cv	|- ( ph -> ( F " D ) = ran F )

Assertion

Ref	Expression
kercvrlsm	|- ( ph -> ( K .(+) D ) = B )

Proof of Theorem kercvrlsm

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kercvrlsm.f	. . . . 5 |- ( ph -> F e. ( S LMHom T ) )
2		lmhmlmod1 19033	. . . . 5 |- ( F e. ( S LMHom T ) -> S e. LMod )
3	1, 2	syl 17	. . . 4 |- ( ph -> S e. LMod )
4		kercvrlsm.k	. . . . . 6 |- K = ( `' F " { .0. } )
5		kercvrlsm.z	. . . . . 6 |- .0. = ( 0g ` T )
6		kercvrlsm.u	. . . . . 6 |- U = ( LSubSp ` S )
7	4, 5, 6	lmhmkerlss 19051	. . . . 5 |- ( F e. ( S LMHom T ) -> K e. U )
8	1, 7	syl 17	. . . 4 |- ( ph -> K e. U )
9		kercvrlsm.d	. . . 4 |- ( ph -> D e. U )
10		kercvrlsm.p	. . . . 5 |- .(+) = ( LSSum ` S )
11	6, 10	lsmcl 19083	. . . 4 |- ( ( S e. LMod /\ K e. U /\ D e. U ) -> ( K .(+) D ) e. U )
12	3, 8, 9, 11	syl3anc 1326	. . 3 |- ( ph -> ( K .(+) D ) e. U )
13		kercvrlsm.b	. . . 4 |- B = ( Base ` S )
14	13, 6	lssss 18937	. . 3 |- ( ( K .(+) D ) e. U -> ( K .(+) D ) C_ B )
15	12, 14	syl 17	. 2 |- ( ph -> ( K .(+) D ) C_ B )
16		eqid 2622	. . . . . . . . . . 11 |- ( Base ` T ) = ( Base ` T )
17	13, 16	lmhmf 19034	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> F : B --> ( Base ` T ) )
18	1, 17	syl 17	. . . . . . . . 9 |- ( ph -> F : B --> ( Base ` T ) )
19		ffn 6045	. . . . . . . . 9 |- ( F : B --> ( Base ` T ) -> F Fn B )
20	18, 19	syl 17	. . . . . . . 8 |- ( ph -> F Fn B )
21		fnfvelrn 6356	. . . . . . . 8 |- ( ( F Fn B /\ a e. B ) -> ( F ` a ) e. ran F )
22	20, 21	sylan 488	. . . . . . 7 |- ( ( ph /\ a e. B ) -> ( F ` a ) e. ran F )
23		kercvrlsm.cv	. . . . . . . 8 |- ( ph -> ( F " D ) = ran F )
24	23	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. B ) -> ( F " D ) = ran F )
25	22, 24	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ a e. B ) -> ( F ` a ) e. ( F " D ) )
26	20	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. B ) -> F Fn B )
27	13, 6	lssss 18937	. . . . . . . . 9 |- ( D e. U -> D C_ B )
28	9, 27	syl 17	. . . . . . . 8 |- ( ph -> D C_ B )
29	28	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. B ) -> D C_ B )
30		fvelimab 6253	. . . . . . 7 |- ( ( F Fn B /\ D C_ B ) -> ( ( F ` a ) e. ( F " D ) <-> E. b e. D ( F ` b ) = ( F ` a ) ) )
31	26, 29, 30	syl2anc 693	. . . . . 6 |- ( ( ph /\ a e. B ) -> ( ( F ` a ) e. ( F " D ) <-> E. b e. D ( F ` b ) = ( F ` a ) ) )
32	25, 31	mpbid 222	. . . . 5 |- ( ( ph /\ a e. B ) -> E. b e. D ( F ` b ) = ( F ` a ) )
33		lmodgrp 18870	. . . . . . . . . . . . 13 |- ( S e. LMod -> S e. Grp )
34	3, 33	syl 17	. . . . . . . . . . . 12 |- ( ph -> S e. Grp )
35	34	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> S e. Grp )
36		simprl 794	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> a e. B )
37	28	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ b e. D ) -> b e. B )
38	37	adantrl 752	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> b e. B )
39		eqid 2622	. . . . . . . . . . . 12 |- ( +g ` S ) = ( +g ` S )
40		eqid 2622	. . . . . . . . . . . 12 |- ( -g ` S ) = ( -g ` S )
41	13, 39, 40	grpnpcan 17507	. . . . . . . . . . 11 |- ( ( S e. Grp /\ a e. B /\ b e. B ) -> ( ( a ( -g ` S ) b ) ( +g ` S ) b ) = a )
42	35, 36, 38, 41	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> ( ( a ( -g ` S ) b ) ( +g ` S ) b ) = a )
43	42	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> ( ( a ( -g ` S ) b ) ( +g ` S ) b ) = a )
44	3	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> S e. LMod )
45	13, 6	lssss 18937	. . . . . . . . . . . 12 |- ( K e. U -> K C_ B )
46	8, 45	syl 17	. . . . . . . . . . 11 |- ( ph -> K C_ B )
47	46	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> K C_ B )
48	28	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> D C_ B )
49		eqcom 2629	. . . . . . . . . . . 12 |- ( ( F ` b ) = ( F ` a ) <-> ( F ` a ) = ( F ` b ) )
50		lmghm 19031	. . . . . . . . . . . . . . 15 |- ( F e. ( S LMHom T ) -> F e. ( S GrpHom T ) )
51	1, 50	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F e. ( S GrpHom T ) )
52	51	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> F e. ( S GrpHom T ) )
53	13, 5, 4, 40	ghmeqker 17687	. . . . . . . . . . . . 13 |- ( ( F e. ( S GrpHom T ) /\ a e. B /\ b e. B ) -> ( ( F ` a ) = ( F ` b ) <-> ( a ( -g ` S ) b ) e. K ) )
54	52, 36, 38, 53	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> ( ( F ` a ) = ( F ` b ) <-> ( a ( -g ` S ) b ) e. K ) )
55	49, 54	syl5bb 272	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> ( ( F ` b ) = ( F ` a ) <-> ( a ( -g ` S ) b ) e. K ) )
56	55	biimpa 501	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> ( a ( -g ` S ) b ) e. K )
57		simplrr 801	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> b e. D )
58	13, 39, 10	lsmelvalix 18056	. . . . . . . . . 10 |- ( ( ( S e. LMod /\ K C_ B /\ D C_ B ) /\ ( ( a ( -g ` S ) b ) e. K /\ b e. D ) ) -> ( ( a ( -g ` S ) b ) ( +g ` S ) b ) e. ( K .(+) D ) )
59	44, 47, 48, 56, 57, 58	syl32anc 1334	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> ( ( a ( -g ` S ) b ) ( +g ` S ) b ) e. ( K .(+) D ) )
60	43, 59	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ph /\ ( a e. B /\ b e. D ) ) /\ ( F ` b ) = ( F ` a ) ) -> a e. ( K .(+) D ) )
61	60	ex 450	. . . . . . 7 |- ( ( ph /\ ( a e. B /\ b e. D ) ) -> ( ( F ` b ) = ( F ` a ) -> a e. ( K .(+) D ) ) )
62	61	anassrs 680	. . . . . 6 |- ( ( ( ph /\ a e. B ) /\ b e. D ) -> ( ( F ` b ) = ( F ` a ) -> a e. ( K .(+) D ) ) )
63	62	rexlimdva 3031	. . . . 5 |- ( ( ph /\ a e. B ) -> ( E. b e. D ( F ` b ) = ( F ` a ) -> a e. ( K .(+) D ) ) )
64	32, 63	mpd 15	. . . 4 |- ( ( ph /\ a e. B ) -> a e. ( K .(+) D ) )
65	64	ex 450	. . 3 |- ( ph -> ( a e. B -> a e. ( K .(+) D ) ) )
66	65	ssrdv 3609	. 2 |- ( ph -> B C_ ( K .(+) D ) )
67	15, 66	eqssd 3620	1 |- ( ph -> ( K .(+) D ) = B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   {csn 4177   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   GrpHom cghm 17657   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LMHom clmhm 19019

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lmhm 19022

This theorem is referenced by:  lmhmfgsplit  37656