Theorem lsatcmp 34290

Description: If two atoms are comparable, they are equal. (atsseq 29206 analog.) TODO: can lspsncmp 19116 shorten this? (Contributed by NM, 25-Aug-2014.)

Hypotheses

Ref	Expression
lsatcmp.a	|- A = (LSAtoms ` W )
lsatcmp.w	|- ( ph -> W e. LVec )
lsatcmp.t	|- ( ph -> T e. A )
lsatcmp.u	|- ( ph -> U e. A )

Assertion

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

Proof of Theorem lsatcmp

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsatcmp.u	. . 3 |- ( ph -> U e. A )
2		lsatcmp.w	. . . . 5 |- ( ph -> W e. LVec )
3		lveclmod 19106	. . . . 5 |- ( W e. LVec -> W e. LMod )
4	2, 3	syl 17	. . . 4 |- ( ph -> W e. LMod )
5		eqid 2622	. . . . 5 |- ( Base ` W ) = ( Base ` W )
6		eqid 2622	. . . . 5 |- ( LSpan ` W ) = ( LSpan ` W )
7		eqid 2622	. . . . 5 |- ( 0g ` W ) = ( 0g ` W )
8		lsatcmp.a	. . . . 5 |- A = (LSAtoms ` W )
9	5, 6, 7, 8	islsat 34278	. . . 4 |- ( W e. LMod -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) ) )
10	4, 9	syl 17	. . 3 |- ( ph -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) ) )
11	1, 10	mpbid 222	. 2 |- ( ph -> E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) )
12		eldifsn 4317	. . . . 5 |- ( v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) <-> ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) )
13		lsatcmp.t	. . . . . . . . . . 11 |- ( ph -> T e. A )
14	7, 8, 4, 13	lsatn0 34286	. . . . . . . . . 10 |- ( ph -> T =/= { ( 0g ` W ) } )
15	14	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T =/= { ( 0g ` W ) } )
16	2	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> W e. LVec )
17		eqid 2622	. . . . . . . . . . . . . 14 |- ( LSubSp ` W ) = ( LSubSp ` W )
18	17, 8, 4, 13	lsatlssel 34284	. . . . . . . . . . . . 13 |- ( ph -> T e. ( LSubSp ` W ) )
19	18	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T e. ( LSubSp ` W ) )
20		simplrl 800	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> v e. ( Base ` W ) )
21		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T C_ ( ( LSpan ` W ) ` { v } ) )
22	5, 7, 17, 6	lspsnat 19145	. . . . . . . . . . . 12 |- ( ( ( W e. LVec /\ T e. ( LSubSp ` W ) /\ v e. ( Base ` W ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( T = ( ( LSpan ` W ) ` { v } ) \/ T = { ( 0g ` W ) } ) )
23	16, 19, 20, 21, 22	syl31anc 1329	. . . . . . . . . . 11 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( T = ( ( LSpan ` W ) ` { v } ) \/ T = { ( 0g ` W ) } ) )
24	23	ord 392	. . . . . . . . . 10 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( -. T = ( ( LSpan ` W ) ` { v } ) -> T = { ( 0g ` W ) } ) )
25	24	necon1ad 2811	. . . . . . . . 9 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( T =/= { ( 0g ` W ) } -> T = ( ( LSpan ` W ) ` { v } ) ) )
26	15, 25	mpd 15	. . . . . . . 8 |- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T = ( ( LSpan ` W ) ` { v } ) )
27	26	ex 450	. . . . . . 7 |- ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) -> T = ( ( LSpan ` W ) ` { v } ) ) )
28		eqimss 3657	. . . . . . 7 |- ( T = ( ( LSpan ` W ) ` { v } ) -> T C_ ( ( LSpan ` W ) ` { v } ) )
29	27, 28	impbid1 215	. . . . . 6 |- ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) )
30	29	ex 450	. . . . 5 |- ( ph -> ( ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) )
31	12, 30	syl5bi 232	. . . 4 |- ( ph -> ( v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) )
32		sseq2 3627	. . . . . 6 |- ( U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T C_ ( ( LSpan ` W ) ` { v } ) ) )
33		eqeq2 2633	. . . . . 6 |- ( U = ( ( LSpan ` W ) ` { v } ) -> ( T = U <-> T = ( ( LSpan ` W ) ` { v } ) ) )
34	32, 33	bibi12d 335	. . . . 5 |- ( U = ( ( LSpan ` W ) ` { v } ) -> ( ( T C_ U <-> T = U ) <-> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) )
35	34	biimprcd 240	. . . 4 |- ( ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) -> ( U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T = U ) ) )
36	31, 35	syl6 35	. . 3 |- ( ph -> ( v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) -> ( U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T = U ) ) ) )
37	36	rexlimdv 3030	. 2 |- ( ph -> ( E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T = U ) ) )
38	11, 37	mpd 15	1 |- ( ph -> ( T C_ U <-> T = U ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   ` cfv 5888   Basecbs 15857   0gc0g 16100   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102  LSAtomsclsa 34261

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-grp 17425  df-minusg 17426  df-sbg 17427  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-lsatoms 34263

This theorem is referenced by:  lsatcmp2  34291  lsatel  34292  lsatnem0  34332  dvh2dimatN  36729