Theorem lshpnelb 34271

Description: The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)

Hypotheses

Ref	Expression
lshpnelb.v	|- V = ( Base ` W )
lshpnelb.n	|- N = ( LSpan ` W )
lshpnelb.p	|- .(+) = ( LSSum ` W )
lshpnelb.h	|- H = (LSHyp ` W )
lshpnelb.w	|- ( ph -> W e. LVec )
lshpnelb.u	|- ( ph -> U e. H )
lshpnelb.x	|- ( ph -> X e. V )

Assertion

Ref	Expression
lshpnelb	|- ( ph -> ( -. X e. U <-> ( U .(+) ( N ` { X } ) ) = V ) )

Proof of Theorem lshpnelb

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpnelb.u	. . . . . 6 |- ( ph -> U e. H )
2		lshpnelb.v	. . . . . . 7 |- V = ( Base ` W )
3		lshpnelb.n	. . . . . . 7 |- N = ( LSpan ` W )
4		eqid 2622	. . . . . . 7 |- ( LSubSp ` W ) = ( LSubSp ` W )
5		lshpnelb.p	. . . . . . 7 |- .(+) = ( LSSum ` W )
6		lshpnelb.h	. . . . . . 7 |- H = (LSHyp ` W )
7		lshpnelb.w	. . . . . . . 8 |- ( ph -> W e. LVec )
8		lveclmod 19106	. . . . . . . 8 |- ( W e. LVec -> W e. LMod )
9	7, 8	syl 17	. . . . . . 7 |- ( ph -> W e. LMod )
10	2, 3, 4, 5, 6, 9	islshpsm 34267	. . . . . 6 |- ( ph -> ( U e. H <-> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( U .(+) ( N ` { v } ) ) = V ) ) )
11	1, 10	mpbid 222	. . . . 5 |- ( ph -> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( U .(+) ( N ` { v } ) ) = V ) )
12	11	simp3d 1075	. . . 4 |- ( ph -> E. v e. V ( U .(+) ( N ` { v } ) ) = V )
13	12	adantr 481	. . 3 |- ( ( ph /\ -. X e. U ) -> E. v e. V ( U .(+) ( N ` { v } ) ) = V )
14		simp1l 1085	. . . . . 6 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ph )
15		simp2 1062	. . . . . 6 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> v e. V )
16	4	lsssssubg 18958	. . . . . . . . . . . 12 |- ( W e. LMod -> ( LSubSp ` W ) C_ (SubGrp ` W ) )
17	9, 16	syl 17	. . . . . . . . . . 11 |- ( ph -> ( LSubSp ` W ) C_ (SubGrp ` W ) )
18	4, 6, 9, 1	lshplss 34268	. . . . . . . . . . 11 |- ( ph -> U e. ( LSubSp ` W ) )
19	17, 18	sseldd 3604	. . . . . . . . . 10 |- ( ph -> U e. (SubGrp ` W ) )
20		lshpnelb.x	. . . . . . . . . . . 12 |- ( ph -> X e. V )
21	2, 4, 3	lspsncl 18977	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` W ) )
22	9, 20, 21	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( N ` { X } ) e. ( LSubSp ` W ) )
23	17, 22	sseldd 3604	. . . . . . . . . 10 |- ( ph -> ( N ` { X } ) e. (SubGrp ` W ) )
24	5	lsmub1 18071	. . . . . . . . . 10 |- ( ( U e. (SubGrp ` W ) /\ ( N ` { X } ) e. (SubGrp ` W ) ) -> U C_ ( U .(+) ( N ` { X } ) ) )
25	19, 23, 24	syl2anc 693	. . . . . . . . 9 |- ( ph -> U C_ ( U .(+) ( N ` { X } ) ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. X e. U ) -> U C_ ( U .(+) ( N ` { X } ) ) )
27	5	lsmub2 18072	. . . . . . . . . . . 12 |- ( ( U e. (SubGrp ` W ) /\ ( N ` { X } ) e. (SubGrp ` W ) ) -> ( N ` { X } ) C_ ( U .(+) ( N ` { X } ) ) )
28	19, 23, 27	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( N ` { X } ) C_ ( U .(+) ( N ` { X } ) ) )
29	2, 3	lspsnid 18993	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) )
30	9, 20, 29	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> X e. ( N ` { X } ) )
31	28, 30	sseldd 3604	. . . . . . . . . 10 |- ( ph -> X e. ( U .(+) ( N ` { X } ) ) )
32		nelne1 2890	. . . . . . . . . 10 |- ( ( X e. ( U .(+) ( N ` { X } ) ) /\ -. X e. U ) -> ( U .(+) ( N ` { X } ) ) =/= U )
33	31, 32	sylan 488	. . . . . . . . 9 |- ( ( ph /\ -. X e. U ) -> ( U .(+) ( N ` { X } ) ) =/= U )
34	33	necomd 2849	. . . . . . . 8 |- ( ( ph /\ -. X e. U ) -> U =/= ( U .(+) ( N ` { X } ) ) )
35		df-pss 3590	. . . . . . . 8 |- ( U C. ( U .(+) ( N ` { X } ) ) <-> ( U C_ ( U .(+) ( N ` { X } ) ) /\ U =/= ( U .(+) ( N ` { X } ) ) ) )
36	26, 34, 35	sylanbrc 698	. . . . . . 7 |- ( ( ph /\ -. X e. U ) -> U C. ( U .(+) ( N ` { X } ) ) )
37	36	3ad2ant1 1082	. . . . . 6 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> U C. ( U .(+) ( N ` { X } ) ) )
38	4, 5	lsmcl 19083	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ U e. ( LSubSp ` W ) /\ ( N ` { X } ) e. ( LSubSp ` W ) ) -> ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) )
39	9, 18, 22, 38	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) )
40	2, 4	lssss 18937	. . . . . . . . . . 11 |- ( ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) -> ( U .(+) ( N ` { X } ) ) C_ V )
41	39, 40	syl 17	. . . . . . . . . 10 |- ( ph -> ( U .(+) ( N ` { X } ) ) C_ V )
42	41	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ V )
43		simpr 477	. . . . . . . . 9 |- ( ( ph /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { v } ) ) = V )
44	42, 43	sseqtr4d 3642	. . . . . . . 8 |- ( ( ph /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) )
45	44	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ -. X e. U ) /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) )
46	45	3adant2 1080	. . . . . 6 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) )
47	7	adantr 481	. . . . . . 7 |- ( ( ph /\ v e. V ) -> W e. LVec )
48	18	adantr 481	. . . . . . 7 |- ( ( ph /\ v e. V ) -> U e. ( LSubSp ` W ) )
49	39	adantr 481	. . . . . . 7 |- ( ( ph /\ v e. V ) -> ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) )
50		simpr 477	. . . . . . 7 |- ( ( ph /\ v e. V ) -> v e. V )
51	2, 4, 3, 5, 47, 48, 49, 50	lsmcv 19141	. . . . . 6 |- ( ( ( ph /\ v e. V ) /\ U C. ( U .(+) ( N ` { X } ) ) /\ ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) ) -> ( U .(+) ( N ` { X } ) ) = ( U .(+) ( N ` { v } ) ) )
52	14, 15, 37, 46, 51	syl211anc 1332	. . . . 5 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) = ( U .(+) ( N ` { v } ) ) )
53		simp3 1063	. . . . 5 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { v } ) ) = V )
54	52, 53	eqtrd 2656	. . . 4 |- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) = V )
55	54	rexlimdv3a 3033	. . 3 |- ( ( ph /\ -. X e. U ) -> ( E. v e. V ( U .(+) ( N ` { v } ) ) = V -> ( U .(+) ( N ` { X } ) ) = V ) )
56	13, 55	mpd 15	. 2 |- ( ( ph /\ -. X e. U ) -> ( U .(+) ( N ` { X } ) ) = V )
57	9	adantr 481	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> W e. LMod )
58	1	adantr 481	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> U e. H )
59	20	adantr 481	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> X e. V )
60		simpr 477	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) = V )
61	2, 3, 5, 6, 57, 58, 59, 60	lshpnel 34270	. 2 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> -. X e. U )
62	56, 61	impbida 877	1 |- ( ph -> ( -. X e. U <-> ( U .(+) ( N ` { X } ) ) = V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   C. wpss 3575   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857  SubGrpcsubg 17588   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102  LSHypclsh 34262

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-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  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-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  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

This theorem is referenced by:  lshpnel2N  34272  l1cvpat  34341  dochexmidat  36748