Theorem lshpcmp 34275

Description: If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)

Hypotheses

Ref	Expression
lshpcmp.h	|- H = (LSHyp ` W )
lshpcmp.w	|- ( ph -> W e. LVec )
lshpcmp.t	|- ( ph -> T e. H )
lshpcmp.u	|- ( ph -> U e. H )

Assertion

Ref	Expression
lshpcmp	|- ( ph -> ( T C_ U <-> T = U ) )

Proof of Theorem lshpcmp

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` W ) = ( Base ` W )
2		lshpcmp.h	. . . . 5 |- H = (LSHyp ` W )
3		lshpcmp.w	. . . . . 6 |- ( ph -> W e. LVec )
4		lveclmod 19106	. . . . . 6 |- ( W e. LVec -> W e. LMod )
5	3, 4	syl 17	. . . . 5 |- ( ph -> W e. LMod )
6		lshpcmp.u	. . . . 5 |- ( ph -> U e. H )
7	1, 2, 5, 6	lshpne 34269	. . . 4 |- ( ph -> U =/= ( Base ` W ) )
8		eqid 2622	. . . . . . . 8 |- ( LSubSp ` W ) = ( LSubSp ` W )
9	8, 2, 5, 6	lshplss 34268	. . . . . . 7 |- ( ph -> U e. ( LSubSp ` W ) )
10	1, 8	lssss 18937	. . . . . . 7 |- ( U e. ( LSubSp ` W ) -> U C_ ( Base ` W ) )
11	9, 10	syl 17	. . . . . 6 |- ( ph -> U C_ ( Base ` W ) )
12		lshpcmp.t	. . . . . . . . 9 |- ( ph -> T e. H )
13		eqid 2622	. . . . . . . . . 10 |- ( LSpan ` W ) = ( LSpan ` W )
14		eqid 2622	. . . . . . . . . 10 |- ( LSSum ` W ) = ( LSSum ` W )
15	1, 13, 8, 14, 2, 5	islshpsm 34267	. . . . . . . . 9 |- ( ph -> ( T e. H <-> ( T e. ( LSubSp ` W ) /\ T =/= ( Base ` W ) /\ E. v e. ( Base ` W ) ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) )
16	12, 15	mpbid 222	. . . . . . . 8 |- ( ph -> ( T e. ( LSubSp ` W ) /\ T =/= ( Base ` W ) /\ E. v e. ( Base ` W ) ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) )
17	16	simp3d 1075	. . . . . . 7 |- ( ph -> E. v e. ( Base ` W ) ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) )
18		id 22	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( Base ` W ) ) -> ( ph /\ v e. ( Base ` W ) ) )
19	18	adantrr 753	. . . . . . . . . . . 12 |- ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) -> ( ph /\ v e. ( Base ` W ) ) )
20	3	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( Base ` W ) ) -> W e. LVec )
21	8, 2, 5, 12	lshplss 34268	. . . . . . . . . . . . . 14 |- ( ph -> T e. ( LSubSp ` W ) )
22	21	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( Base ` W ) ) -> T e. ( LSubSp ` W ) )
23	9	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( Base ` W ) ) -> U e. ( LSubSp ` W ) )
24		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( Base ` W ) ) -> v e. ( Base ` W ) )
25	1, 8, 13, 14, 20, 22, 23, 24	lsmcv 19141	. . . . . . . . . . . 12 |- ( ( ( ph /\ v e. ( Base ` W ) ) /\ T C. U /\ U C_ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) ) -> U = ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) )
26	19, 25	syl3an1 1359	. . . . . . . . . . 11 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) /\ T C. U /\ U C_ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) ) -> U = ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) )
27	26	3expia 1267	. . . . . . . . . 10 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) /\ T C. U ) -> ( U C_ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) -> U = ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) ) )
28		simplrr 801	. . . . . . . . . . 11 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) /\ T C. U ) -> ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) )
29	28	sseq2d 3633	. . . . . . . . . 10 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) /\ T C. U ) -> ( U C_ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) <-> U C_ ( Base ` W ) ) )
30	28	eqeq2d 2632	. . . . . . . . . 10 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) /\ T C. U ) -> ( U = ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) <-> U = ( Base ` W ) ) )
31	27, 29, 30	3imtr3d 282	. . . . . . . . 9 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) ) ) /\ T C. U ) -> ( U C_ ( Base ` W ) -> U = ( Base ` W ) ) )
32	31	exp42 639	. . . . . . . 8 |- ( ph -> ( v e. ( Base ` W ) -> ( ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) -> ( T C. U -> ( U C_ ( Base ` W ) -> U = ( Base ` W ) ) ) ) ) )
33	32	rexlimdv 3030	. . . . . . 7 |- ( ph -> ( E. v e. ( Base ` W ) ( T ( LSSum ` W ) ( ( LSpan ` W ) ` { v } ) ) = ( Base ` W ) -> ( T C. U -> ( U C_ ( Base ` W ) -> U = ( Base ` W ) ) ) ) )
34	17, 33	mpd 15	. . . . . 6 |- ( ph -> ( T C. U -> ( U C_ ( Base ` W ) -> U = ( Base ` W ) ) ) )
35	11, 34	mpid 44	. . . . 5 |- ( ph -> ( T C. U -> U = ( Base ` W ) ) )
36	35	necon3ad 2807	. . . 4 |- ( ph -> ( U =/= ( Base ` W ) -> -. T C. U ) )
37	7, 36	mpd 15	. . 3 |- ( ph -> -. T C. U )
38		df-pss 3590	. . . . 5 |- ( T C. U <-> ( T C_ U /\ T =/= U ) )
39	38	simplbi2 655	. . . 4 |- ( T C_ U -> ( T =/= U -> T C. U ) )
40	39	necon1bd 2812	. . 3 |- ( T C_ U -> ( -. T C. U -> T = U ) )
41	37, 40	syl5com 31	. 2 |- ( ph -> ( T C_ U -> T = U ) )
42		eqimss 3657	. 2 |- ( T = U -> T C_ U )
43	41, 42	impbid1 215	1 |- ( ph -> ( T C_ U <-> T = U ) )

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   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:  lshpinN  34276  lfl1dim  34408  lfl1dim2N  34409  lkrpssN  34450  dochlkr  36674  dochsatshpb  36741  lcfl9a  36794  lclkrlem2e  36800  lclkrlem2g  36802  lclkrlem2s  36814  lcfrlem25  36856  lcfrlem35  36866  hdmaplkr  37205