Theorem lsslsp 13925

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 13876	. . . 4 |- ( ( W e. LMod /\ U e. L ) -> X e. LMod )
4	3	3adant3 1019	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X e. LMod )
5		simp1 999	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> W e. LMod )
6		simp3 1001	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ U )
7		eqid 2193	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
8	7, 2	lssssg 13856	. . . . . . 7 |- ( ( W e. LMod /\ U e. L ) -> U C_ ( Base ` W ) )
9	8	3adant3 1019	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U C_ ( Base ` W ) )
10	6, 9	sstrd 3189	. . . . 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 13887	. . . . 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 13899	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ U )
15		eqid 2193	. . . . . 6 |- ( LSubSp ` X ) = ( LSubSp ` X )
16	1, 2, 15	lsslss 13877	. . . . 5 |- ( ( W e. LMod /\ U e. L ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) )
17	16	3adant3 1019	. . . 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 946	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. ( LSubSp ` X ) )
19	7, 11	lspssid 13896	. . . 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 13899	. . 3 |- ( ( X e. LMod /\ ( M ` G ) e. ( LSubSp ` X ) /\ G C_ ( M ` G ) ) -> ( N ` G ) C_ ( M ` G ) )
23	4, 18, 20, 22	syl3anc 1249	. 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 2194	. . . . . . . 8 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( Base ` W ) = ( Base ` W ) )
26	24, 25, 5, 9	ressbas2d 12686	. . . . . . 7 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U = ( Base ` X ) )
27	6, 26	sseqtrd 3217	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` X ) )
28		eqid 2193	. . . . . . 7 |- ( Base ` X ) = ( Base ` X )
29	28, 15, 21	lspcl 13887	. . . . . 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 13877	. . . . . 6 |- ( ( W e. LMod /\ U e. L ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) )
32	31	3adant3 1019	. . . . 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 13896	. . . 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 13899	. . 3 |- ( ( W e. LMod /\ ( N ` G ) e. L /\ G C_ ( N ` G ) ) -> ( M ` G ) C_ ( N ` G ) )
38	5, 34, 36, 37	syl3anc 1249	. 2 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ ( N ` G ) )
39	23, 38	eqssd 3196	1 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   LModclmod 13783   LSubSpclss 13848   LSpanclspn 13882

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240  df-mgp 13417  df-ur 13456  df-ring 13494  df-lmod 13785  df-lssm 13849  df-lsp 13883

This theorem is referenced by:  lss0v  13926