Theorem lkreqN 34457

Description: Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lkreq.s	|- S = (Scalar ` W )
lkreq.r	|- R = ( Base ` S )
lkreq.o	|- .0. = ( 0g ` S )
lkreq.f	|- F = (LFnl ` W )
lkreq.k	|- K = (LKer ` W )
lkreq.d	|- D = (LDual ` W )
lkreq.t	|- .x. = ( .s ` D )
lkreq.w	|- ( ph -> W e. LVec )
lkreq.a	|- ( ph -> A e. ( R \ { .0. } ) )
lkreq.h	|- ( ph -> H e. F )
lkreq.g	|- ( ph -> G = ( A .x. H ) )

Assertion

Ref	Expression
lkreqN	|- ( ph -> ( K ` G ) = ( K ` H ) )

Proof of Theorem lkreqN

Step	Hyp	Ref	Expression
1		lkreq.g	. . . . . . . . 9 |- ( ph -> G = ( A .x. H ) )
2	1	eqeq1d 2624	. . . . . . . 8 |- ( ph -> ( G = ( 0g ` D ) <-> ( A .x. H ) = ( 0g ` D ) ) )
3		eqid 2622	. . . . . . . . . 10 |- ( Base ` D ) = ( Base ` D )
4		lkreq.t	. . . . . . . . . 10 |- .x. = ( .s ` D )
5		eqid 2622	. . . . . . . . . 10 |- (Scalar ` D ) = (Scalar ` D )
6		eqid 2622	. . . . . . . . . 10 |- ( Base ` (Scalar ` D ) ) = ( Base ` (Scalar ` D ) )
7		eqid 2622	. . . . . . . . . 10 |- ( 0g ` (Scalar ` D ) ) = ( 0g ` (Scalar ` D ) )
8		eqid 2622	. . . . . . . . . 10 |- ( 0g ` D ) = ( 0g ` D )
9		lkreq.d	. . . . . . . . . . 11 |- D = (LDual ` W )
10		lkreq.w	. . . . . . . . . . 11 |- ( ph -> W e. LVec )
11	9, 10	lduallvec 34441	. . . . . . . . . 10 |- ( ph -> D e. LVec )
12		lkreq.a	. . . . . . . . . . . 12 |- ( ph -> A e. ( R \ { .0. } ) )
13	12	eldifad 3586	. . . . . . . . . . 11 |- ( ph -> A e. R )
14		lkreq.s	. . . . . . . . . . . 12 |- S = (Scalar ` W )
15		lkreq.r	. . . . . . . . . . . 12 |- R = ( Base ` S )
16	14, 15, 9, 5, 6, 10	ldualsbase 34420	. . . . . . . . . . 11 |- ( ph -> ( Base ` (Scalar ` D ) ) = R )
17	13, 16	eleqtrrd 2704	. . . . . . . . . 10 |- ( ph -> A e. ( Base ` (Scalar ` D ) ) )
18		lkreq.f	. . . . . . . . . . 11 |- F = (LFnl ` W )
19		lkreq.h	. . . . . . . . . . 11 |- ( ph -> H e. F )
20	18, 9, 3, 10, 19	ldualelvbase 34414	. . . . . . . . . 10 |- ( ph -> H e. ( Base ` D ) )
21	3, 4, 5, 6, 7, 8, 11, 17, 20	lvecvs0or 19108	. . . . . . . . 9 |- ( ph -> ( ( A .x. H ) = ( 0g ` D ) <-> ( A = ( 0g ` (Scalar ` D ) ) \/ H = ( 0g ` D ) ) ) )
22		lkreq.o	. . . . . . . . . . . . 13 |- .0. = ( 0g ` S )
23		lveclmod 19106	. . . . . . . . . . . . . 14 |- ( W e. LVec -> W e. LMod )
24	10, 23	syl 17	. . . . . . . . . . . . 13 |- ( ph -> W e. LMod )
25	14, 22, 9, 5, 7, 24	ldual0 34434	. . . . . . . . . . . 12 |- ( ph -> ( 0g ` (Scalar ` D ) ) = .0. )
26	25	eqeq2d 2632	. . . . . . . . . . 11 |- ( ph -> ( A = ( 0g ` (Scalar ` D ) ) <-> A = .0. ) )
27		eldifsni 4320	. . . . . . . . . . . . . 14 |- ( A e. ( R \ { .0. } ) -> A =/= .0. )
28	12, 27	syl 17	. . . . . . . . . . . . 13 |- ( ph -> A =/= .0. )
29	28	a1d 25	. . . . . . . . . . . 12 |- ( ph -> ( H =/= ( 0g ` D ) -> A =/= .0. ) )
30	29	necon4d 2818	. . . . . . . . . . 11 |- ( ph -> ( A = .0. -> H = ( 0g ` D ) ) )
31	26, 30	sylbid 230	. . . . . . . . . 10 |- ( ph -> ( A = ( 0g ` (Scalar ` D ) ) -> H = ( 0g ` D ) ) )
32		idd 24	. . . . . . . . . 10 |- ( ph -> ( H = ( 0g ` D ) -> H = ( 0g ` D ) ) )
33	31, 32	jaod 395	. . . . . . . . 9 |- ( ph -> ( ( A = ( 0g ` (Scalar ` D ) ) \/ H = ( 0g ` D ) ) -> H = ( 0g ` D ) ) )
34	21, 33	sylbid 230	. . . . . . . 8 |- ( ph -> ( ( A .x. H ) = ( 0g ` D ) -> H = ( 0g ` D ) ) )
35	2, 34	sylbid 230	. . . . . . 7 |- ( ph -> ( G = ( 0g ` D ) -> H = ( 0g ` D ) ) )
36		nne 2798	. . . . . . 7 |- ( -. H =/= ( 0g ` D ) <-> H = ( 0g ` D ) )
37	35, 36	syl6ibr 242	. . . . . 6 |- ( ph -> ( G = ( 0g ` D ) -> -. H =/= ( 0g ` D ) ) )
38	37	con3d 148	. . . . 5 |- ( ph -> ( -. -. H =/= ( 0g ` D ) -> -. G = ( 0g ` D ) ) )
39	38	orrd 393	. . . 4 |- ( ph -> ( -. H =/= ( 0g ` D ) \/ -. G = ( 0g ` D ) ) )
40		ianor 509	. . . 4 |- ( -. ( H =/= ( 0g ` D ) /\ G = ( 0g ` D ) ) <-> ( -. H =/= ( 0g ` D ) \/ -. G = ( 0g ` D ) ) )
41	39, 40	sylibr 224	. . 3 |- ( ph -> -. ( H =/= ( 0g ` D ) /\ G = ( 0g ` D ) ) )
42		df-pss 3590	. . . . . 6 |- ( ( K ` H ) C. ( K ` G ) <-> ( ( K ` H ) C_ ( K ` G ) /\ ( K ` H ) =/= ( K ` G ) ) )
43		lkreq.k	. . . . . . 7 |- K = (LKer ` W )
44	18, 14, 15, 9, 4, 24, 13, 19	ldualvscl 34426	. . . . . . . 8 |- ( ph -> ( A .x. H ) e. F )
45	1, 44	eqeltrd 2701	. . . . . . 7 |- ( ph -> G e. F )
46	18, 43, 9, 8, 10, 19, 45	lkrpssN 34450	. . . . . 6 |- ( ph -> ( ( K ` H ) C. ( K ` G ) <-> ( H =/= ( 0g ` D ) /\ G = ( 0g ` D ) ) ) )
47	42, 46	syl5rbbr 275	. . . . 5 |- ( ph -> ( ( H =/= ( 0g ` D ) /\ G = ( 0g ` D ) ) <-> ( ( K ` H ) C_ ( K ` G ) /\ ( K ` H ) =/= ( K ` G ) ) ) )
48	14, 15, 18, 43, 9, 4, 10, 19, 13	lkrss 34455	. . . . . . 7 |- ( ph -> ( K ` H ) C_ ( K ` ( A .x. H ) ) )
49	1	fveq2d 6195	. . . . . . 7 |- ( ph -> ( K ` G ) = ( K ` ( A .x. H ) ) )
50	48, 49	sseqtr4d 3642	. . . . . 6 |- ( ph -> ( K ` H ) C_ ( K ` G ) )
51	50	biantrurd 529	. . . . 5 |- ( ph -> ( ( K ` H ) =/= ( K ` G ) <-> ( ( K ` H ) C_ ( K ` G ) /\ ( K ` H ) =/= ( K ` G ) ) ) )
52	47, 51	bitr4d 271	. . . 4 |- ( ph -> ( ( H =/= ( 0g ` D ) /\ G = ( 0g ` D ) ) <-> ( K ` H ) =/= ( K ` G ) ) )
53	52	necon2bbid 2837	. . 3 |- ( ph -> ( ( K ` H ) = ( K ` G ) <-> -. ( H =/= ( 0g ` D ) /\ G = ( 0g ` D ) ) ) )
54	41, 53	mpbird 247	. 2 |- ( ph -> ( K ` H ) = ( K ` G ) )
55	54	eqcomd 2628	1 |- ( ph -> ( K ` G ) = ( K ` H ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   C. wpss 3575   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863   LVecclvec 19102  LFnlclfn 34344  LKerclk 34372  LDualcld 34410

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-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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  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-dom 7957  df-sdom 7958  df-fin 7959  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  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-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lshyp 34264  df-lfl 34345  df-lkr 34373  df-ldual 34411

This theorem is referenced by:  lkrlspeqN  34458  lcdlkreq2N  36912