Theorem lsslsp 13675

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 13626	. . . 4 |- ( ( W e. LMod /\ U e. L ) -> X e. LMod )
4	3	3adant3 1018	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X e. LMod )
5		simp1 998	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> W e. LMod )
6		simp3 1000	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ U )
7		eqid 2187	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
8	7, 2	lssssg 13606	. . . . . . 7 |- ( ( W e. LMod /\ U e. L ) -> U C_ ( Base ` W ) )
9	8	3adant3 1018	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U C_ ( Base ` W ) )
10	6, 9	sstrd 3177	. . . . 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 13637	. . . . 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 13649	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ U )
15		eqid 2187	. . . . . 6 |- ( LSubSp ` X ) = ( LSubSp ` X )
16	1, 2, 15	lsslss 13627	. . . . 5 |- ( ( W e. LMod /\ U e. L ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) )
17	16	3adant3 1018	. . . 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 945	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. ( LSubSp ` X ) )
19	7, 11	lspssid 13646	. . . 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 13649	. . 3 |- ( ( X e. LMod /\ ( M ` G ) e. ( LSubSp ` X ) /\ G C_ ( M ` G ) ) -> ( N ` G ) C_ ( M ` G ) )
23	4, 18, 20, 22	syl3anc 1248	. 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 2188	. . . . . . . 8 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( Base ` W ) = ( Base ` W ) )
26	24, 25, 5, 9	ressbas2d 12542	. . . . . . 7 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U = ( Base ` X ) )
27	6, 26	sseqtrd 3205	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` X ) )
28		eqid 2187	. . . . . . 7 |- ( Base ` X ) = ( Base ` X )
29	28, 15, 21	lspcl 13637	. . . . . 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 13627	. . . . . 6 |- ( ( W e. LMod /\ U e. L ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) )
32	31	3adant3 1018	. . . . 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 13646	. . . 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 13649	. . 3 |- ( ( W e. LMod /\ ( N ` G ) e. L /\ G C_ ( N ` G ) ) -> ( M ` G ) C_ ( N ` G ) )
38	5, 34, 36, 37	syl3anc 1248	. 2 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ ( N ` G ) )
39	23, 38	eqssd 3184	1 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   C_ wss 3141   ` cfv 5228  (class class class)co 5888   Basecbs 12476   ↾_s cress 12477   LModclmod 13533   LSubSpclss 13598   LSpanclspn 13632

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-5 8995  df-6 8996  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-plusg 12564  df-mulr 12565  df-sca 12567  df-vsca 12568  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12909  df-minusg 12910  df-sbg 12911  df-subg 13070  df-mgp 13230  df-ur 13269  df-ring 13307  df-lmod 13535  df-lssm 13599  df-lsp 13633

This theorem is referenced by:  lss0v  13676