Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmfgcl | Structured version Visualization version Unicode version |
Description: The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
lsmfgcl.u | |
lsmfgcl.p | |
lsmfgcl.d | ↾s |
lsmfgcl.e | ↾s |
lsmfgcl.f | ↾s |
lsmfgcl.w | |
lsmfgcl.a | |
lsmfgcl.b | |
lsmfgcl.df | LFinGen |
lsmfgcl.ef | LFinGen |
Ref | Expression |
---|---|
lsmfgcl | LFinGen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmfgcl.f | . 2 ↾s | |
2 | lsmfgcl.df | . . . 4 LFinGen | |
3 | lsmfgcl.w | . . . . 5 | |
4 | lsmfgcl.a | . . . . 5 | |
5 | lsmfgcl.d | . . . . . 6 ↾s | |
6 | lsmfgcl.u | . . . . . 6 | |
7 | eqid 2622 | . . . . . 6 | |
8 | eqid 2622 | . . . . . 6 | |
9 | 5, 6, 7, 8 | islssfg2 37641 | . . . . 5 LFinGen |
10 | 3, 4, 9 | syl2anc 693 | . . . 4 LFinGen |
11 | 2, 10 | mpbid 222 | . . 3 |
12 | lsmfgcl.ef | . . . . . . . 8 LFinGen | |
13 | lsmfgcl.b | . . . . . . . . 9 | |
14 | lsmfgcl.e | . . . . . . . . . 10 ↾s | |
15 | 14, 6, 7, 8 | islssfg2 37641 | . . . . . . . . 9 LFinGen |
16 | 3, 13, 15 | syl2anc 693 | . . . . . . . 8 LFinGen |
17 | 12, 16 | mpbid 222 | . . . . . . 7 |
18 | 17 | adantr 481 | . . . . . 6 |
19 | inss1 3833 | . . . . . . . . . . . . . . 15 | |
20 | 19 | sseli 3599 | . . . . . . . . . . . . . 14 |
21 | 20 | elpwid 4170 | . . . . . . . . . . . . 13 |
22 | 19 | sseli 3599 | . . . . . . . . . . . . . 14 |
23 | 22 | elpwid 4170 | . . . . . . . . . . . . 13 |
24 | lsmfgcl.p | . . . . . . . . . . . . . 14 | |
25 | 8, 7, 24 | lsmsp2 19087 | . . . . . . . . . . . . 13 |
26 | 3, 21, 23, 25 | syl3an 1368 | . . . . . . . . . . . 12 |
27 | 26 | 3expb 1266 | . . . . . . . . . . 11 |
28 | 27 | oveq2d 6666 | . . . . . . . . . 10 ↾s ↾s |
29 | 3 | adantr 481 | . . . . . . . . . . 11 |
30 | unss 3787 | . . . . . . . . . . . . . 14 | |
31 | 30 | biimpi 206 | . . . . . . . . . . . . 13 |
32 | 21, 23, 31 | syl2an 494 | . . . . . . . . . . . 12 |
33 | 32 | adantl 482 | . . . . . . . . . . 11 |
34 | inss2 3834 | . . . . . . . . . . . . . 14 | |
35 | 34 | sseli 3599 | . . . . . . . . . . . . 13 |
36 | 34 | sseli 3599 | . . . . . . . . . . . . 13 |
37 | unfi 8227 | . . . . . . . . . . . . 13 | |
38 | 35, 36, 37 | syl2an 494 | . . . . . . . . . . . 12 |
39 | 38 | adantl 482 | . . . . . . . . . . 11 |
40 | eqid 2622 | . . . . . . . . . . . 12 ↾s ↾s | |
41 | 7, 8, 40 | islssfgi 37642 | . . . . . . . . . . 11 ↾s LFinGen |
42 | 29, 33, 39, 41 | syl3anc 1326 | . . . . . . . . . 10 ↾s LFinGen |
43 | 28, 42 | eqeltrd 2701 | . . . . . . . . 9 ↾s LFinGen |
44 | 43 | anassrs 680 | . . . . . . . 8 ↾s LFinGen |
45 | oveq2 6658 | . . . . . . . . . 10 | |
46 | 45 | oveq2d 6666 | . . . . . . . . 9 ↾s ↾s |
47 | 46 | eleq1d 2686 | . . . . . . . 8 ↾s LFinGen ↾s LFinGen |
48 | 44, 47 | syl5ibcom 235 | . . . . . . 7 ↾s LFinGen |
49 | 48 | rexlimdva 3031 | . . . . . 6 ↾s LFinGen |
50 | 18, 49 | mpd 15 | . . . . 5 ↾s LFinGen |
51 | oveq1 6657 | . . . . . . 7 | |
52 | 51 | oveq2d 6666 | . . . . . 6 ↾s ↾s |
53 | 52 | eleq1d 2686 | . . . . 5 ↾s LFinGen ↾s LFinGen |
54 | 50, 53 | syl5ibcom 235 | . . . 4 ↾s LFinGen |
55 | 54 | rexlimdva 3031 | . . 3 ↾s LFinGen |
56 | 11, 55 | mpd 15 | . 2 ↾s LFinGen |
57 | 1, 56 | syl5eqel 2705 | 1 LFinGen |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 cun 3572 cin 3573 wss 3574 cpw 4158 cfv 5888 (class class class)co 6650 cfn 7955 cbs 15857 ↾s cress 15858 clsm 18049 clmod 18863 clss 18932 clspn 18971 LFinGenclfig 37637 |
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-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 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-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lfig 37638 |
This theorem is referenced by: lmhmfgsplit 37656 |
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