Theorem lkrpssN 34450

Description: Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lkrpss.f	|- F = (LFnl ` W )
lkrpss.k	|- K = (LKer ` W )
lkrpss.d	|- D = (LDual ` W )
lkrpss.o	|- .0. = ( 0g ` D )
lkrpss.w	|- ( ph -> W e. LVec )
lkrpss.g	|- ( ph -> G e. F )
lkrpss.h	|- ( ph -> H e. F )

Assertion

Ref	Expression
lkrpssN	|- ( ph -> ( ( K ` G ) C. ( K ` H ) <-> ( G =/= .0. /\ H = .0. ) ) )

Proof of Theorem lkrpssN

Step	Hyp	Ref	Expression
1		df-pss 3590	. . 3 |- ( ( K ` G ) C. ( K ` H ) <-> ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) )
2		simpr 477	. . . . . . . 8 |- ( ( ph /\ ( K ` G ) C. ( K ` H ) ) -> ( K ` G ) C. ( K ` H ) )
3		eqid 2622	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
4		lkrpss.f	. . . . . . . . . 10 |- F = (LFnl ` W )
5		lkrpss.k	. . . . . . . . . 10 |- K = (LKer ` W )
6		lkrpss.w	. . . . . . . . . . 11 |- ( ph -> W e. LVec )
7		lveclmod 19106	. . . . . . . . . . 11 |- ( W e. LVec -> W e. LMod )
8	6, 7	syl 17	. . . . . . . . . 10 |- ( ph -> W e. LMod )
9		lkrpss.h	. . . . . . . . . 10 |- ( ph -> H e. F )
10	3, 4, 5, 8, 9	lkrssv 34383	. . . . . . . . 9 |- ( ph -> ( K ` H ) C_ ( Base ` W ) )
11	10	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( K ` G ) C. ( K ` H ) ) -> ( K ` H ) C_ ( Base ` W ) )
12	2, 11	psssstrd 3716	. . . . . . 7 |- ( ( ph /\ ( K ` G ) C. ( K ` H ) ) -> ( K ` G ) C. ( Base ` W ) )
13	12	pssned 3705	. . . . . 6 |- ( ( ph /\ ( K ` G ) C. ( K ` H ) ) -> ( K ` G ) =/= ( Base ` W ) )
14	1, 13	sylan2br 493	. . . . 5 |- ( ( ph /\ ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) ) -> ( K ` G ) =/= ( Base ` W ) )
15		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> ( K ` G ) C_ ( K ` H ) )
16		eqid 2622	. . . . . . . . . . 11 |- (LSHyp ` W ) = (LSHyp ` W )
17	6	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> W e. LVec )
18		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) e. (LSHyp ` W ) ) -> ( K ` G ) e. (LSHyp ` W ) )
19		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( K ` H ) e. (LSHyp ` W ) )
20	10	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( K ` H ) C_ ( Base ` W ) )
21		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( K ` G ) = ( Base ` W ) )
22		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( K ` G ) C_ ( K ` H ) )
23	21, 22	eqsstr3d 3640	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( Base ` W ) C_ ( K ` H ) )
24	20, 23	eqssd 3620	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( K ` H ) = ( Base ` W ) )
25	3, 16, 4, 5, 6, 9	lkrshp4 34395	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( K ` H ) =/= ( Base ` W ) <-> ( K ` H ) e. (LSHyp ` W ) ) )
26	25	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( ( K ` H ) =/= ( Base ` W ) <-> ( K ` H ) e. (LSHyp ` W ) ) )
27	26	necon1bbid 2833	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( -. ( K ` H ) e. (LSHyp ` W ) <-> ( K ` H ) = ( Base ` W ) ) )
28	24, 27	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> -. ( K ` H ) e. (LSHyp ` W ) )
29	19, 28	pm2.21dd 186	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) /\ ( K ` G ) = ( Base ` W ) ) -> ( K ` G ) e. (LSHyp ` W ) )
30		lkrpss.g	. . . . . . . . . . . . . 14 |- ( ph -> G e. F )
31	3, 16, 4, 5, 6, 30	lkrshpor 34394	. . . . . . . . . . . . 13 |- ( ph -> ( ( K ` G ) e. (LSHyp ` W ) \/ ( K ` G ) = ( Base ` W ) ) )
32	31	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> ( ( K ` G ) e. (LSHyp ` W ) \/ ( K ` G ) = ( Base ` W ) ) )
33	18, 29, 32	mpjaodan 827	. . . . . . . . . . 11 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> ( K ` G ) e. (LSHyp ` W ) )
34		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> ( K ` H ) e. (LSHyp ` W ) )
35	16, 17, 33, 34	lshpcmp 34275	. . . . . . . . . 10 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> ( ( K ` G ) C_ ( K ` H ) <-> ( K ` G ) = ( K ` H ) ) )
36	15, 35	mpbid 222	. . . . . . . . 9 |- ( ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) /\ ( K ` H ) e. (LSHyp ` W ) ) -> ( K ` G ) = ( K ` H ) )
37	36	ex 450	. . . . . . . 8 |- ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) -> ( ( K ` H ) e. (LSHyp ` W ) -> ( K ` G ) = ( K ` H ) ) )
38	37	necon3ad 2807	. . . . . . 7 |- ( ( ph /\ ( K ` G ) C_ ( K ` H ) ) -> ( ( K ` G ) =/= ( K ` H ) -> -. ( K ` H ) e. (LSHyp ` W ) ) )
39	38	impr 649	. . . . . 6 |- ( ( ph /\ ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) ) -> -. ( K ` H ) e. (LSHyp ` W ) )
40	25	necon1bbid 2833	. . . . . . 7 |- ( ph -> ( -. ( K ` H ) e. (LSHyp ` W ) <-> ( K ` H ) = ( Base ` W ) ) )
41	40	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) ) -> ( -. ( K ` H ) e. (LSHyp ` W ) <-> ( K ` H ) = ( Base ` W ) ) )
42	39, 41	mpbid 222	. . . . 5 |- ( ( ph /\ ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) ) -> ( K ` H ) = ( Base ` W ) )
43	14, 42	jca 554	. . . 4 |- ( ( ph /\ ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) ) -> ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) )
44	3, 4, 5, 8, 30	lkrssv 34383	. . . . . . 7 |- ( ph -> ( K ` G ) C_ ( Base ` W ) )
45	44	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( K ` G ) C_ ( Base ` W ) )
46		simprr 796	. . . . . . 7 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( K ` H ) = ( Base ` W ) )
47	46	eqcomd 2628	. . . . . 6 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( Base ` W ) = ( K ` H ) )
48	45, 47	sseqtrd 3641	. . . . 5 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( K ` G ) C_ ( K ` H ) )
49		simprl 794	. . . . . 6 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( K ` G ) =/= ( Base ` W ) )
50	49, 47	neeqtrd 2863	. . . . 5 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( K ` G ) =/= ( K ` H ) )
51	48, 50	jca 554	. . . 4 |- ( ( ph /\ ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) -> ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) )
52	43, 51	impbida 877	. . 3 |- ( ph -> ( ( ( K ` G ) C_ ( K ` H ) /\ ( K ` G ) =/= ( K ` H ) ) <-> ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) )
53	1, 52	syl5bb 272	. 2 |- ( ph -> ( ( K ` G ) C. ( K ` H ) <-> ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) ) )
54		lkrpss.d	. . . . 5 |- D = (LDual ` W )
55		lkrpss.o	. . . . 5 |- .0. = ( 0g ` D )
56	3, 4, 5, 54, 55, 8, 30	lkr0f2 34448	. . . 4 |- ( ph -> ( ( K ` G ) = ( Base ` W ) <-> G = .0. ) )
57	56	necon3bid 2838	. . 3 |- ( ph -> ( ( K ` G ) =/= ( Base ` W ) <-> G =/= .0. ) )
58	3, 4, 5, 54, 55, 8, 9	lkr0f2 34448	. . 3 |- ( ph -> ( ( K ` H ) = ( Base ` W ) <-> H = .0. ) )
59	57, 58	anbi12d 747	. 2 |- ( ph -> ( ( ( K ` G ) =/= ( Base ` W ) /\ ( K ` H ) = ( Base ` W ) ) <-> ( G =/= .0. /\ H = .0. ) ) )
60	53, 59	bitrd 268	1 |- ( ph -> ( ( K ` G ) C. ( K ` H ) <-> ( G =/= .0. /\ H = .0. ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   C. wpss 3575   ` cfv 5888   Basecbs 15857   0gc0g 16100   LModclmod 18863   LVecclvec 19102  LSHypclsh 34262  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:  lkrss2N  34456  lkreqN  34457