Theorem lsatfixedN 34296

Description: Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 19128. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lsatfixed.v	|- V = ( Base ` W )
lsatfixed.p	|- .+ = ( +g ` W )
lsatfixed.o	|- .0. = ( 0g ` W )
lsatfixed.n	|- N = ( LSpan ` W )
lsatfixed.a	|- A = (LSAtoms ` W )
lsatfixed.w	|- ( ph -> W e. LVec )
lsatfixed.q	|- ( ph -> Q e. A )
lsatfixed.x	|- ( ph -> X e. V )
lsatfixed.y	|- ( ph -> Y e. V )
lsatfixed.e	|- ( ph -> Q =/= ( N ` { X } ) )
lsatfixed.f	|- ( ph -> Q =/= ( N ` { Y } ) )
lsatfixed.g	|- ( ph -> Q C_ ( N ` { X , Y } ) )

Assertion

Ref	Expression
lsatfixedN	|- ( ph -> E. z e. ( ( N ` { Y } ) \ { .0. } ) Q = ( N ` { ( X .+ z ) } ) )

Distinct variable groups:   z, N   z, .0.   z, .+   ph, z   z, Q   z, V   z, W   z, X   z, Y

Allowed substitution hint:   A( z)

Proof of Theorem lsatfixedN

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsatfixed.q	. . 3 |- ( ph -> Q e. A )
2		lsatfixed.w	. . . 4 |- ( ph -> W e. LVec )
3		lsatfixed.v	. . . . 5 |- V = ( Base ` W )
4		lsatfixed.n	. . . . 5 |- N = ( LSpan ` W )
5		lsatfixed.o	. . . . 5 |- .0. = ( 0g ` W )
6		lsatfixed.a	. . . . 5 |- A = (LSAtoms ` W )
7	3, 4, 5, 6	islsat 34278	. . . 4 |- ( W e. LVec -> ( Q e. A <-> E. w e. ( V \ { .0. } ) Q = ( N ` { w } ) ) )
8	2, 7	syl 17	. . 3 |- ( ph -> ( Q e. A <-> E. w e. ( V \ { .0. } ) Q = ( N ` { w } ) ) )
9	1, 8	mpbid 222	. 2 |- ( ph -> E. w e. ( V \ { .0. } ) Q = ( N ` { w } ) )
10		lsatfixed.p	. . . . 5 |- .+ = ( +g ` W )
11	2	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> W e. LVec )
12		simp2 1062	. . . . . 6 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> w e. ( V \ { .0. } ) )
13	12	eldifad 3586	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> w e. V )
14		lsatfixed.x	. . . . . 6 |- ( ph -> X e. V )
15	14	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> X e. V )
16		lsatfixed.y	. . . . . 6 |- ( ph -> Y e. V )
17	16	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> Y e. V )
18		simp3 1063	. . . . . . . 8 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> Q = ( N ` { w } ) )
19	18	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( N ` { w } ) = Q )
20		lsatfixed.e	. . . . . . . 8 |- ( ph -> Q =/= ( N ` { X } ) )
21	20	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> Q =/= ( N ` { X } ) )
22	19, 21	eqnetrd 2861	. . . . . 6 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( N ` { w } ) =/= ( N ` { X } ) )
23	3, 5, 4, 11, 12, 15, 22	lspsnne1 19117	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> -. w e. ( N ` { X } ) )
24		lsatfixed.f	. . . . . . . 8 |- ( ph -> Q =/= ( N ` { Y } ) )
25	24	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> Q =/= ( N ` { Y } ) )
26	19, 25	eqnetrd 2861	. . . . . 6 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( N ` { w } ) =/= ( N ` { Y } ) )
27	3, 5, 4, 11, 12, 17, 26	lspsnne1 19117	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> -. w e. ( N ` { Y } ) )
28		lsatfixed.g	. . . . . . . 8 |- ( ph -> Q C_ ( N ` { X , Y } ) )
29	28	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> Q C_ ( N ` { X , Y } ) )
30	19, 29	eqsstrd 3639	. . . . . 6 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( N ` { w } ) C_ ( N ` { X , Y } ) )
31		eqid 2622	. . . . . . 7 |- ( LSubSp ` W ) = ( LSubSp ` W )
32		lveclmod 19106	. . . . . . . . 9 |- ( W e. LVec -> W e. LMod )
33	2, 32	syl 17	. . . . . . . 8 |- ( ph -> W e. LMod )
34	33	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> W e. LMod )
35	3, 31, 4, 33, 14, 16	lspprcl 18978	. . . . . . . 8 |- ( ph -> ( N ` { X , Y } ) e. ( LSubSp ` W ) )
36	35	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( N ` { X , Y } ) e. ( LSubSp ` W ) )
37	3, 31, 4, 34, 36, 13	lspsnel5 18995	. . . . . 6 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( w e. ( N ` { X , Y } ) <-> ( N ` { w } ) C_ ( N ` { X , Y } ) ) )
38	30, 37	mpbird 247	. . . . 5 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> w e. ( N ` { X , Y } ) )
39	3, 10, 5, 4, 11, 13, 15, 17, 23, 27, 38	lspfixed 19128	. . . 4 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> E. z e. ( ( N ` { Y } ) \ { .0. } ) w e. ( N ` { ( X .+ z ) } ) )
40		simpl1 1064	. . . . . . . 8 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ph )
41	40, 2	syl 17	. . . . . . 7 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> W e. LVec )
42		simpl2 1065	. . . . . . 7 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> w e. ( V \ { .0. } ) )
43	40, 33	syl 17	. . . . . . . 8 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> W e. LMod )
44	40, 14	syl 17	. . . . . . . 8 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> X e. V )
45	16	snssd 4340	. . . . . . . . . . . 12 |- ( ph -> { Y } C_ V )
46	3, 4	lspssv 18983	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ { Y } C_ V ) -> ( N ` { Y } ) C_ V )
47	33, 45, 46	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( N ` { Y } ) C_ V )
48	47	ssdifssd 3748	. . . . . . . . . 10 |- ( ph -> ( ( N ` { Y } ) \ { .0. } ) C_ V )
49	48	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( ( N ` { Y } ) \ { .0. } ) C_ V )
50	49	sselda 3603	. . . . . . . 8 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> z e. V )
51	3, 10	lmodvacl 18877	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ z e. V ) -> ( X .+ z ) e. V )
52	43, 44, 50, 51	syl3anc 1326	. . . . . . 7 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ( X .+ z ) e. V )
53	3, 5, 4, 41, 42, 52	lspsncmp 19116	. . . . . 6 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ( ( N ` { w } ) C_ ( N ` { ( X .+ z ) } ) <-> ( N ` { w } ) = ( N ` { ( X .+ z ) } ) ) )
54	3, 31, 4	lspsncl 18977	. . . . . . . 8 |- ( ( W e. LMod /\ ( X .+ z ) e. V ) -> ( N ` { ( X .+ z ) } ) e. ( LSubSp ` W ) )
55	43, 52, 54	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ( N ` { ( X .+ z ) } ) e. ( LSubSp ` W ) )
56	42	eldifad 3586	. . . . . . 7 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> w e. V )
57	3, 31, 4, 43, 55, 56	lspsnel5 18995	. . . . . 6 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ( w e. ( N ` { ( X .+ z ) } ) <-> ( N ` { w } ) C_ ( N ` { ( X .+ z ) } ) ) )
58		simpl3 1066	. . . . . . 7 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> Q = ( N ` { w } ) )
59	58	eqeq1d 2624	. . . . . 6 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ( Q = ( N ` { ( X .+ z ) } ) <-> ( N ` { w } ) = ( N ` { ( X .+ z ) } ) ) )
60	53, 57, 59	3bitr4rd 301	. . . . 5 |- ( ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) /\ z e. ( ( N ` { Y } ) \ { .0. } ) ) -> ( Q = ( N ` { ( X .+ z ) } ) <-> w e. ( N ` { ( X .+ z ) } ) ) )
61	60	rexbidva 3049	. . . 4 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> ( E. z e. ( ( N ` { Y } ) \ { .0. } ) Q = ( N ` { ( X .+ z ) } ) <-> E. z e. ( ( N ` { Y } ) \ { .0. } ) w e. ( N ` { ( X .+ z ) } ) ) )
62	39, 61	mpbird 247	. . 3 |- ( ( ph /\ w e. ( V \ { .0. } ) /\ Q = ( N ` { w } ) ) -> E. z e. ( ( N ` { Y } ) \ { .0. } ) Q = ( N ` { ( X .+ z ) } ) )
63	62	rexlimdv3a 3033	. 2 |- ( ph -> ( E. w e. ( V \ { .0. } ) Q = ( N ` { w } ) -> E. z e. ( ( N ` { Y } ) \ { .0. } ) Q = ( N ` { ( X .+ z ) } ) ) )
64	9, 63	mpd 15	1 |- ( ph -> E. z e. ( ( N ` { Y } ) \ { .0. } ) Q = ( N ` { ( X .+ z ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   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-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-lsatoms 34263

This theorem is referenced by:  hdmaprnlem3eN  37150