Theorem lspprabs 19095

Description: Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)

Hypotheses

Ref	Expression
lspprabs.v	|- V = ( Base ` W )
lspprabs.p	|- .+ = ( +g ` W )
lspprabs.n	|- N = ( LSpan ` W )
lspprabs.w	|- ( ph -> W e. LMod )
lspprabs.x	|- ( ph -> X e. V )
lspprabs.y	|- ( ph -> Y e. V )

Assertion

Ref	Expression
lspprabs	|- ( ph -> ( N ` { X , ( X .+ Y ) } ) = ( N ` { X , Y } ) )

Proof of Theorem lspprabs

Step	Hyp	Ref	Expression
1		lspprabs.w	. . . . . . 7 |- ( ph -> W e. LMod )
2		eqid 2622	. . . . . . . 8 |- ( LSubSp ` W ) = ( LSubSp ` W )
3	2	lsssssubg 18958	. . . . . . 7 |- ( W e. LMod -> ( LSubSp ` W ) C_ (SubGrp ` W ) )
4	1, 3	syl 17	. . . . . 6 |- ( ph -> ( LSubSp ` W ) C_ (SubGrp ` W ) )
5		lspprabs.x	. . . . . . 7 |- ( ph -> X e. V )
6		lspprabs.v	. . . . . . . 8 |- V = ( Base ` W )
7		lspprabs.n	. . . . . . . 8 |- N = ( LSpan ` W )
8	6, 2, 7	lspsncl 18977	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` W ) )
9	1, 5, 8	syl2anc 693	. . . . . 6 |- ( ph -> ( N ` { X } ) e. ( LSubSp ` W ) )
10	4, 9	sseldd 3604	. . . . 5 |- ( ph -> ( N ` { X } ) e. (SubGrp ` W ) )
11		lspprabs.y	. . . . . . 7 |- ( ph -> Y e. V )
12	6, 2, 7	lspsncl 18977	. . . . . . 7 |- ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
13	1, 11, 12	syl2anc 693	. . . . . 6 |- ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
14	4, 13	sseldd 3604	. . . . 5 |- ( ph -> ( N ` { Y } ) e. (SubGrp ` W ) )
15		eqid 2622	. . . . . 6 |- ( LSSum ` W ) = ( LSSum ` W )
16	15	lsmub1 18071	. . . . 5 |- ( ( ( N ` { X } ) e. (SubGrp ` W ) /\ ( N ` { Y } ) e. (SubGrp ` W ) ) -> ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
17	10, 14, 16	syl2anc 693	. . . 4 |- ( ph -> ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
18	2, 15	lsmcl 19083	. . . . . 6 |- ( ( W e. LMod /\ ( N ` { X } ) e. ( LSubSp ` W ) /\ ( N ` { Y } ) e. ( LSubSp ` W ) ) -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) e. ( LSubSp ` W ) )
19	1, 9, 13, 18	syl3anc 1326	. . . . 5 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) e. ( LSubSp ` W ) )
20	6, 7	lspsnid 18993	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) )
21	1, 5, 20	syl2anc 693	. . . . . 6 |- ( ph -> X e. ( N ` { X } ) )
22	6, 7	lspsnid 18993	. . . . . . 7 |- ( ( W e. LMod /\ Y e. V ) -> Y e. ( N ` { Y } ) )
23	1, 11, 22	syl2anc 693	. . . . . 6 |- ( ph -> Y e. ( N ` { Y } ) )
24		lspprabs.p	. . . . . . 7 |- .+ = ( +g ` W )
25	24, 15	lsmelvali 18065	. . . . . 6 |- ( ( ( ( N ` { X } ) e. (SubGrp ` W ) /\ ( N ` { Y } ) e. (SubGrp ` W ) ) /\ ( X e. ( N ` { X } ) /\ Y e. ( N ` { Y } ) ) ) -> ( X .+ Y ) e. ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
26	10, 14, 21, 23, 25	syl22anc 1327	. . . . 5 |- ( ph -> ( X .+ Y ) e. ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
27	2, 7, 1, 19, 26	lspsnel5a 18996	. . . 4 |- ( ph -> ( N ` { ( X .+ Y ) } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
28	6, 24	lmodvacl 18877	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
29	1, 5, 11, 28	syl3anc 1326	. . . . . . 7 |- ( ph -> ( X .+ Y ) e. V )
30	6, 2, 7	lspsncl 18977	. . . . . . 7 |- ( ( W e. LMod /\ ( X .+ Y ) e. V ) -> ( N ` { ( X .+ Y ) } ) e. ( LSubSp ` W ) )
31	1, 29, 30	syl2anc 693	. . . . . 6 |- ( ph -> ( N ` { ( X .+ Y ) } ) e. ( LSubSp ` W ) )
32	4, 31	sseldd 3604	. . . . 5 |- ( ph -> ( N ` { ( X .+ Y ) } ) e. (SubGrp ` W ) )
33	4, 19	sseldd 3604	. . . . 5 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) e. (SubGrp ` W ) )
34	15	lsmlub 18078	. . . . 5 |- ( ( ( N ` { X } ) e. (SubGrp ` W ) /\ ( N ` { ( X .+ Y ) } ) e. (SubGrp ` W ) /\ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) e. (SubGrp ` W ) ) -> ( ( ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) /\ ( N ` { ( X .+ Y ) } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) <-> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) )
35	10, 32, 33, 34	syl3anc 1326	. . . 4 |- ( ph -> ( ( ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) /\ ( N ` { ( X .+ Y ) } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) <-> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) ) )
36	17, 27, 35	mpbi2and 956	. . 3 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
37	15	lsmub1 18071	. . . . 5 |- ( ( ( N ` { X } ) e. (SubGrp ` W ) /\ ( N ` { ( X .+ Y ) } ) e. (SubGrp ` W ) ) -> ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
38	10, 32, 37	syl2anc 693	. . . 4 |- ( ph -> ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
39	2, 15	lsmcl 19083	. . . . . 6 |- ( ( W e. LMod /\ ( N ` { X } ) e. ( LSubSp ` W ) /\ ( N ` { ( X .+ Y ) } ) e. ( LSubSp ` W ) ) -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) e. ( LSubSp ` W ) )
40	1, 9, 31, 39	syl3anc 1326	. . . . 5 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) e. ( LSubSp ` W ) )
41		eqid 2622	. . . . . . 7 |- ( -g ` W ) = ( -g ` W )
42	6, 7	lspsnid 18993	. . . . . . . 8 |- ( ( W e. LMod /\ ( X .+ Y ) e. V ) -> ( X .+ Y ) e. ( N ` { ( X .+ Y ) } ) )
43	1, 29, 42	syl2anc 693	. . . . . . 7 |- ( ph -> ( X .+ Y ) e. ( N ` { ( X .+ Y ) } ) )
44	41, 15, 32, 10, 43, 21	lsmelvalmi 18067	. . . . . 6 |- ( ph -> ( ( X .+ Y ) ( -g ` W ) X ) e. ( ( N ` { ( X .+ Y ) } ) ( LSSum ` W ) ( N ` { X } ) ) )
45		lmodabl 18910	. . . . . . . 8 |- ( W e. LMod -> W e. Abel )
46	1, 45	syl 17	. . . . . . 7 |- ( ph -> W e. Abel )
47	6, 24, 41	ablpncan2 18221	. . . . . . 7 |- ( ( W e. Abel /\ X e. V /\ Y e. V ) -> ( ( X .+ Y ) ( -g ` W ) X ) = Y )
48	46, 5, 11, 47	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( X .+ Y ) ( -g ` W ) X ) = Y )
49	15	lsmcom 18261	. . . . . . 7 |- ( ( W e. Abel /\ ( N ` { ( X .+ Y ) } ) e. (SubGrp ` W ) /\ ( N ` { X } ) e. (SubGrp ` W ) ) -> ( ( N ` { ( X .+ Y ) } ) ( LSSum ` W ) ( N ` { X } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
50	46, 32, 10, 49	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( N ` { ( X .+ Y ) } ) ( LSSum ` W ) ( N ` { X } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
51	44, 48, 50	3eltr3d 2715	. . . . 5 |- ( ph -> Y e. ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
52	2, 7, 1, 40, 51	lspsnel5a 18996	. . . 4 |- ( ph -> ( N ` { Y } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
53	4, 40	sseldd 3604	. . . . 5 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) e. (SubGrp ` W ) )
54	15	lsmlub 18078	. . . . 5 |- ( ( ( N ` { X } ) e. (SubGrp ` W ) /\ ( N ` { Y } ) e. (SubGrp ` W ) /\ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) e. (SubGrp ` W ) ) -> ( ( ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) /\ ( N ` { Y } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) ) <-> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) ) )
55	10, 14, 53, 54	syl3anc 1326	. . . 4 |- ( ph -> ( ( ( N ` { X } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) /\ ( N ` { Y } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) ) <-> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) ) )
56	38, 52, 55	mpbi2and 956	. . 3 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
57	36, 56	eqssd 3620	. 2 |- ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
58	6, 7, 15, 1, 5, 29	lsmpr 19089	. 2 |- ( ph -> ( N ` { X , ( X .+ Y ) } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
59	6, 7, 15, 1, 5, 11	lsmpr 19089	. 2 |- ( ph -> ( N ` { X , Y } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
60	57, 58, 59	3eqtr4d 2666	1 |- ( ph -> ( N ` { X , ( X .+ Y ) } ) = ( N ` { X , Y } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   -gcsg 17424  SubGrpcsubg 17588   LSSumclsm 18049   Abelcabl 18194   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971

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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  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-lmod 18865  df-lss 18933  df-lsp 18972

This theorem is referenced by:  lspabs2  19120  lspindp4  19137  mapdindp4  37012