Theorem lsslsp 14266

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 14217	. . . 4 |- ( ( W e. LMod /\ U e. L ) -> X e. LMod )
4	3	3adant3 1020	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X e. LMod )
5		simp1 1000	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> W e. LMod )
6		simp3 1002	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ U )
7		eqid 2206	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
8	7, 2	lssssg 14197	. . . . . . 7 |- ( ( W e. LMod /\ U e. L ) -> U C_ ( Base ` W ) )
9	8	3adant3 1020	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U C_ ( Base ` W ) )
10	6, 9	sstrd 3207	. . . . 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 14228	. . . . 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 14240	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ U )
15		eqid 2206	. . . . . 6 |- ( LSubSp ` X ) = ( LSubSp ` X )
16	1, 2, 15	lsslss 14218	. . . . 5 |- ( ( W e. LMod /\ U e. L ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) )
17	16	3adant3 1020	. . . 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 947	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. ( LSubSp ` X ) )
19	7, 11	lspssid 14237	. . . 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 14240	. . 3 |- ( ( X e. LMod /\ ( M ` G ) e. ( LSubSp ` X ) /\ G C_ ( M ` G ) ) -> ( N ` G ) C_ ( M ` G ) )
23	4, 18, 20, 22	syl3anc 1250	. 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 2207	. . . . . . . 8 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( Base ` W ) = ( Base ` W ) )
26	24, 25, 5, 9	ressbas2d 12975	. . . . . . 7 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U = ( Base ` X ) )
27	6, 26	sseqtrd 3235	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` X ) )
28		eqid 2206	. . . . . . 7 |- ( Base ` X ) = ( Base ` X )
29	28, 15, 21	lspcl 14228	. . . . . 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 14218	. . . . . 6 |- ( ( W e. LMod /\ U e. L ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) )
32	31	3adant3 1020	. . . . 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 14237	. . . 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 14240	. . 3 |- ( ( W e. LMod /\ ( N ` G ) e. L /\ G C_ ( N ` G ) ) -> ( M ` G ) C_ ( N ` G ) )
38	5, 34, 36, 37	syl3anc 1250	. 2 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ ( N ` G ) )
39	23, 38	eqssd 3214	1 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2177   C_ wss 3170   ` cfv 5280  (class class class)co 5957   Basecbs 12907   ↾_s cress 12908   LModclmod 14124   LSubSpclss 14189   LSpanclspn 14223

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-mulr 12998  df-sca 13000  df-vsca 13001  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-sbg 13412  df-subg 13581  df-mgp 13758  df-ur 13797  df-ring 13835  df-lmod 14126  df-lssm 14190  df-lsp 14224

This theorem is referenced by:  lss0v  14267