Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lsslsp | Structured version Visualization version Unicode version |
Description: Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap and since we are computing a property of ? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015. |
Ref | Expression |
---|---|
lsslsp.x | ↾s |
lsslsp.m | |
lsslsp.n | |
lsslsp.l |
Ref | Expression |
---|---|
lsslsp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . 3 | |
2 | lsslsp.x | . . . . . . . 8 ↾s | |
3 | lsslsp.l | . . . . . . . 8 | |
4 | 2, 3 | lsslmod 18960 | . . . . . . 7 |
5 | 4 | 3adant3 1081 | . . . . . 6 |
6 | simp3 1063 | . . . . . . 7 | |
7 | eqid 2622 | . . . . . . . . . 10 | |
8 | 7, 3 | lssss 18937 | . . . . . . . . 9 |
9 | 8 | 3ad2ant2 1083 | . . . . . . . 8 |
10 | 2, 7 | ressbas2 15931 | . . . . . . . 8 |
11 | 9, 10 | syl 17 | . . . . . . 7 |
12 | 6, 11 | sseqtrd 3641 | . . . . . 6 |
13 | eqid 2622 | . . . . . . 7 | |
14 | eqid 2622 | . . . . . . 7 | |
15 | lsslsp.n | . . . . . . 7 | |
16 | 13, 14, 15 | lspcl 18976 | . . . . . 6 |
17 | 5, 12, 16 | syl2anc 693 | . . . . 5 |
18 | 2, 3, 14 | lsslss 18961 | . . . . . 6 |
19 | 18 | 3adant3 1081 | . . . . 5 |
20 | 17, 19 | mpbid 222 | . . . 4 |
21 | 20 | simpld 475 | . . 3 |
22 | 13, 15 | lspssid 18985 | . . . 4 |
23 | 5, 12, 22 | syl2anc 693 | . . 3 |
24 | lsslsp.m | . . . 4 | |
25 | 3, 24 | lspssp 18988 | . . 3 |
26 | 1, 21, 23, 25 | syl3anc 1326 | . 2 |
27 | 6, 9 | sstrd 3613 | . . . . 5 |
28 | 7, 3, 24 | lspcl 18976 | . . . . 5 |
29 | 1, 27, 28 | syl2anc 693 | . . . 4 |
30 | 3, 24 | lspssp 18988 | . . . 4 |
31 | 2, 3, 14 | lsslss 18961 | . . . . 5 |
32 | 31 | 3adant3 1081 | . . . 4 |
33 | 29, 30, 32 | mpbir2and 957 | . . 3 |
34 | 7, 24 | lspssid 18985 | . . . 4 |
35 | 1, 27, 34 | syl2anc 693 | . . 3 |
36 | 14, 15 | lspssp 18988 | . . 3 |
37 | 5, 33, 35, 36 | syl3anc 1326 | . 2 |
38 | 26, 37 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 clmod 18863 clss 18932 clspn 18971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lsp 18972 |
This theorem is referenced by: lss0v 19016 lsslindf 20169 islinds3 20173 lcdlsp 36910 islssfg 37640 |
Copyright terms: Public domain | W3C validator |