Theorem lsslsp 13554

Description: Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)

Hypotheses

Ref	Expression
lsslsp.x	|- X = ( W ↾s U )
lsslsp.m	|- M = ( LSpan ` W )
lsslsp.n	|- N = ( LSpan ` X )
lsslsp.l	|- L = ( LSubSp ` W )

Assertion

Ref	Expression
lsslsp	|- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )

Proof of Theorem lsslsp

Step	Hyp	Ref	Expression
1		lsslsp.x	. . . . 5 |- X = ( W ↾s U )
2		lsslsp.l	. . . . 5 |- L = ( LSubSp ` W )
3	1, 2	lsslmod 13505	. . . 4 |- ( ( W e. LMod /\ U e. L ) -> X e. LMod )
4	3	3adant3 1017	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X e. LMod )
5		simp1 997	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> W e. LMod )
6		simp3 999	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ U )
7		eqid 2177	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
8	7, 2	lssssg 13485	. . . . . . 7 |- ( ( W e. LMod /\ U e. L ) -> U C_ ( Base ` W ) )
9	8	3adant3 1017	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U C_ ( Base ` W ) )
10	6, 9	sstrd 3167	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` W ) )
11		lsslsp.m	. . . . . 6 |- M = ( LSpan ` W )
12	7, 2, 11	lspcl 13516	. . . . 5 |- ( ( W e. LMod /\ G C_ ( Base ` W ) ) -> ( M ` G ) e. L )
13	5, 10, 12	syl2anc 411	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. L )
14	2, 11	lspssp 13528	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ U )
15		eqid 2177	. . . . . 6 |- ( LSubSp ` X ) = ( LSubSp ` X )
16	1, 2, 15	lsslss 13506	. . . . 5 |- ( ( W e. LMod /\ U e. L ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) )
17	16	3adant3 1017	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) )
18	13, 14, 17	mpbir2and 944	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. ( LSubSp ` X ) )
19	7, 11	lspssid 13525	. . . 4 |- ( ( W e. LMod /\ G C_ ( Base ` W ) ) -> G C_ ( M ` G ) )
20	5, 10, 19	syl2anc 411	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( M ` G ) )
21		lsslsp.n	. . . 4 |- N = ( LSpan ` X )
22	15, 21	lspssp 13528	. . 3 |- ( ( X e. LMod /\ ( M ` G ) e. ( LSubSp ` X ) /\ G C_ ( M ` G ) ) -> ( N ` G ) C_ ( M ` G ) )
23	4, 18, 20, 22	syl3anc 1238	. 2 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) C_ ( M ` G ) )
24	1	a1i 9	. . . . . . . 8 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X = ( W ↾s U ) )
25		eqidd 2178	. . . . . . . 8 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( Base ` W ) = ( Base ` W ) )
26	24, 25, 5, 9	ressbas2d 12531	. . . . . . 7 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U = ( Base ` X ) )
27	6, 26	sseqtrd 3195	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` X ) )
28		eqid 2177	. . . . . . 7 |- ( Base ` X ) = ( Base ` X )
29	28, 15, 21	lspcl 13516	. . . . . 6 |- ( ( X e. LMod /\ G C_ ( Base ` X ) ) -> ( N ` G ) e. ( LSubSp ` X ) )
30	4, 27, 29	syl2anc 411	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) e. ( LSubSp ` X ) )
31	1, 2, 15	lsslss 13506	. . . . . 6 |- ( ( W e. LMod /\ U e. L ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) )
32	31	3adant3 1017	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) )
33	30, 32	mpbid 147	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) )
34	33	simpld 112	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) e. L )
35	28, 21	lspssid 13525	. . . 4 |- ( ( X e. LMod /\ G C_ ( Base ` X ) ) -> G C_ ( N ` G ) )
36	4, 27, 35	syl2anc 411	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( N ` G ) )
37	2, 11	lspssp 13528	. . 3 |- ( ( W e. LMod /\ ( N ` G ) e. L /\ G C_ ( N ` G ) ) -> ( M ` G ) C_ ( N ` G ) )
38	5, 34, 36, 37	syl3anc 1238	. 2 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ ( N ` G ) )
39	23, 38	eqssd 3174	1 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3131   ` cfv 5218  (class class class)co 5878   Basecbs 12465   ↾_s cress 12466   LModclmod 13415   LSubSpclss 13480   LSpanclspn 13511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473  df-plusg 12552  df-mulr 12553  df-sca 12555  df-vsca 12556  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12826  df-grp 12888  df-minusg 12889  df-sbg 12890  df-subg 13044  df-mgp 13147  df-ur 13181  df-ring 13219  df-lmod 13417  df-lssm 13481  df-lsp 13512

This theorem is referenced by:  lss0v  13555