Theorem lsslsp 14703

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 14654	. . . 4 |- ( ( W e. LMod /\ U e. L ) -> X e. LMod )
4	3	3adant3 1044	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> X e. LMod )
5		simp1 1024	. . . . 5 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> W e. LMod )
6		simp3 1026	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ U )
7		eqid 2234	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
8	7, 2	lssssg 14634	. . . . . . 7 |- ( ( W e. LMod /\ U e. L ) -> U C_ ( Base ` W ) )
9	8	3adant3 1044	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U C_ ( Base ` W ) )
10	6, 9	sstrd 3252	. . . . 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 14665	. . . . 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 14677	. . . 4 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ U )
15		eqid 2234	. . . . . 6 |- ( LSubSp ` X ) = ( LSubSp ` X )
16	1, 2, 15	lsslss 14655	. . . . 5 |- ( ( W e. LMod /\ U e. L ) -> ( ( M ` G ) e. ( LSubSp ` X ) <-> ( ( M ` G ) e. L /\ ( M ` G ) C_ U ) ) )
17	16	3adant3 1044	. . . 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 953	. . 3 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) e. ( LSubSp ` X ) )
19	7, 11	lspssid 14674	. . . 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 14677	. . 3 |- ( ( X e. LMod /\ ( M ` G ) e. ( LSubSp ` X ) /\ G C_ ( M ` G ) ) -> ( N ` G ) C_ ( M ` G ) )
23	4, 18, 20, 22	syl3anc 1274	. 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 2235	. . . . . . . 8 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( Base ` W ) = ( Base ` W ) )
26	24, 25, 5, 9	ressbas2d 13365	. . . . . . 7 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> U = ( Base ` X ) )
27	6, 26	sseqtrd 3280	. . . . . 6 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> G C_ ( Base ` X ) )
28		eqid 2234	. . . . . . 7 |- ( Base ` X ) = ( Base ` X )
29	28, 15, 21	lspcl 14665	. . . . . 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 14655	. . . . . 6 |- ( ( W e. LMod /\ U e. L ) -> ( ( N ` G ) e. ( LSubSp ` X ) <-> ( ( N ` G ) e. L /\ ( N ` G ) C_ U ) ) )
32	31	3adant3 1044	. . . . 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 14674	. . . 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 14677	. . 3 |- ( ( W e. LMod /\ ( N ` G ) e. L /\ G C_ ( N ` G ) ) -> ( M ` G ) C_ ( N ` G ) )
38	5, 34, 36, 37	syl3anc 1274	. 2 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( M ` G ) C_ ( N ` G ) )
39	23, 38	eqssd 3259	1 |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3214   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   LModclmod 14561   LSubSpclss 14626   LSpanclspn 14660

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-subg 13923  df-mgp 14160  df-ur 14203  df-ring 14241  df-lmod 14563  df-lssm 14627  df-lsp 14661

This theorem is referenced by:  lss0v  14704