Theorem lshpnel2N 34272

Description: Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lshpnel2.v	|- V = ( Base ` W )
lshpnel2.s	|- S = ( LSubSp ` W )
lshpnel2.n	|- N = ( LSpan ` W )
lshpnel2.p	|- .(+) = ( LSSum ` W )
lshpnel2.h	|- H = (LSHyp ` W )
lshpnel2.w	|- ( ph -> W e. LVec )
lshpnel2.u	|- ( ph -> U e. S )
lshpnel2.t	|- ( ph -> U =/= V )
lshpnel2.x	|- ( ph -> X e. V )
lshpnel2.e	|- ( ph -> -. X e. U )

Assertion

Ref	Expression
lshpnel2N	|- ( ph -> ( U e. H <-> ( U .(+) ( N ` { X } ) ) = V ) )

Proof of Theorem lshpnel2N

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpnel2.e	. . . 4 |- ( ph -> -. X e. U )
2	1	adantr 481	. . 3 |- ( ( ph /\ U e. H ) -> -. X e. U )
3		lshpnel2.v	. . . 4 |- V = ( Base ` W )
4		lshpnel2.n	. . . 4 |- N = ( LSpan ` W )
5		lshpnel2.p	. . . 4 |- .(+) = ( LSSum ` W )
6		lshpnel2.h	. . . 4 |- H = (LSHyp ` W )
7		lshpnel2.w	. . . . 5 |- ( ph -> W e. LVec )
8	7	adantr 481	. . . 4 |- ( ( ph /\ U e. H ) -> W e. LVec )
9		simpr 477	. . . 4 |- ( ( ph /\ U e. H ) -> U e. H )
10		lshpnel2.x	. . . . 5 |- ( ph -> X e. V )
11	10	adantr 481	. . . 4 |- ( ( ph /\ U e. H ) -> X e. V )
12	3, 4, 5, 6, 8, 9, 11	lshpnelb 34271	. . 3 |- ( ( ph /\ U e. H ) -> ( -. X e. U <-> ( U .(+) ( N ` { X } ) ) = V ) )
13	2, 12	mpbid 222	. 2 |- ( ( ph /\ U e. H ) -> ( U .(+) ( N ` { X } ) ) = V )
14		lshpnel2.u	. . . 4 |- ( ph -> U e. S )
15	14	adantr 481	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> U e. S )
16		lshpnel2.t	. . . 4 |- ( ph -> U =/= V )
17	16	adantr 481	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> U =/= V )
18	10	adantr 481	. . . 4 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> X e. V )
19		lveclmod 19106	. . . . . . . . . . 11 |- ( W e. LVec -> W e. LMod )
20	7, 19	syl 17	. . . . . . . . . 10 |- ( ph -> W e. LMod )
21		lshpnel2.s	. . . . . . . . . . 11 |- S = ( LSubSp ` W )
22	21, 4	lspid 18982	. . . . . . . . . 10 |- ( ( W e. LMod /\ U e. S ) -> ( N ` U ) = U )
23	20, 14, 22	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( N ` U ) = U )
24	23	uneq1d 3766	. . . . . . . 8 |- ( ph -> ( ( N ` U ) u. ( N ` { X } ) ) = ( U u. ( N ` { X } ) ) )
25	24	fveq2d 6195	. . . . . . 7 |- ( ph -> ( N ` ( ( N ` U ) u. ( N ` { X } ) ) ) = ( N ` ( U u. ( N ` { X } ) ) ) )
26	3, 21	lssss 18937	. . . . . . . . 9 |- ( U e. S -> U C_ V )
27	14, 26	syl 17	. . . . . . . 8 |- ( ph -> U C_ V )
28	10	snssd 4340	. . . . . . . 8 |- ( ph -> { X } C_ V )
29	3, 4	lspun 18987	. . . . . . . 8 |- ( ( W e. LMod /\ U C_ V /\ { X } C_ V ) -> ( N ` ( U u. { X } ) ) = ( N ` ( ( N ` U ) u. ( N ` { X } ) ) ) )
30	20, 27, 28, 29	syl3anc 1326	. . . . . . 7 |- ( ph -> ( N ` ( U u. { X } ) ) = ( N ` ( ( N ` U ) u. ( N ` { X } ) ) ) )
31	3, 21, 4	lspsncl 18977	. . . . . . . . 9 |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )
32	20, 10, 31	syl2anc 693	. . . . . . . 8 |- ( ph -> ( N ` { X } ) e. S )
33	21, 4, 5	lsmsp 19086	. . . . . . . 8 |- ( ( W e. LMod /\ U e. S /\ ( N ` { X } ) e. S ) -> ( U .(+) ( N ` { X } ) ) = ( N ` ( U u. ( N ` { X } ) ) ) )
34	20, 14, 32, 33	syl3anc 1326	. . . . . . 7 |- ( ph -> ( U .(+) ( N ` { X } ) ) = ( N ` ( U u. ( N ` { X } ) ) ) )
35	25, 30, 34	3eqtr4rd 2667	. . . . . 6 |- ( ph -> ( U .(+) ( N ` { X } ) ) = ( N ` ( U u. { X } ) ) )
36	35	eqeq1d 2624	. . . . 5 |- ( ph -> ( ( U .(+) ( N ` { X } ) ) = V <-> ( N ` ( U u. { X } ) ) = V ) )
37	36	biimpa 501	. . . 4 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> ( N ` ( U u. { X } ) ) = V )
38		sneq 4187	. . . . . . . 8 |- ( v = X -> { v } = { X } )
39	38	uneq2d 3767	. . . . . . 7 |- ( v = X -> ( U u. { v } ) = ( U u. { X } ) )
40	39	fveq2d 6195	. . . . . 6 |- ( v = X -> ( N ` ( U u. { v } ) ) = ( N ` ( U u. { X } ) ) )
41	40	eqeq1d 2624	. . . . 5 |- ( v = X -> ( ( N ` ( U u. { v } ) ) = V <-> ( N ` ( U u. { X } ) ) = V ) )
42	41	rspcev 3309	. . . 4 |- ( ( X e. V /\ ( N ` ( U u. { X } ) ) = V ) -> E. v e. V ( N ` ( U u. { v } ) ) = V )
43	18, 37, 42	syl2anc 693	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> E. v e. V ( N ` ( U u. { v } ) ) = V )
44	7	adantr 481	. . . 4 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> W e. LVec )
45	3, 4, 21, 6	islshp 34266	. . . 4 |- ( W e. LVec -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )
46	44, 45	syl 17	. . 3 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )
47	15, 17, 43, 46	mpbir3and 1245	. 2 |- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> U e. H )
48	13, 47	impbida 877	1 |- ( ph -> ( U e. H <-> ( 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   u. cun 3572   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   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: (None)