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Mirrors > Home > MPE Home > Th. List > mpllsslem | Structured version Visualization version Unicode version |
Description: If is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set of finite bags (the primary applications being and for some ), then the set of all power series whose coefficient functions are supported on an element of is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.) |
Ref | Expression |
---|---|
mplsubglem.s | mPwSer |
mplsubglem.b | |
mplsubglem.z | |
mplsubglem.d | |
mplsubglem.i | |
mplsubglem.0 | |
mplsubglem.a | |
mplsubglem.y | |
mplsubglem.u | supp |
mpllsslem.r |
Ref | Expression |
---|---|
mpllsslem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplsubglem.s | . . 3 mPwSer | |
2 | mplsubglem.i | . . 3 | |
3 | mpllsslem.r | . . 3 | |
4 | 1, 2, 3 | psrsca 19389 | . 2 Scalar |
5 | eqidd 2623 | . 2 | |
6 | mplsubglem.b | . . 3 | |
7 | 6 | a1i 11 | . 2 |
8 | eqidd 2623 | . 2 | |
9 | eqidd 2623 | . 2 | |
10 | eqidd 2623 | . 2 | |
11 | mplsubglem.z | . . . 4 | |
12 | mplsubglem.d | . . . 4 | |
13 | mplsubglem.0 | . . . 4 | |
14 | mplsubglem.a | . . . 4 | |
15 | mplsubglem.y | . . . 4 | |
16 | mplsubglem.u | . . . 4 supp | |
17 | ringgrp 18552 | . . . . 5 | |
18 | 3, 17 | syl 17 | . . . 4 |
19 | 1, 6, 11, 12, 2, 13, 14, 15, 16, 18 | mplsubglem 19434 | . . 3 SubGrp |
20 | 6 | subgss 17595 | . . 3 SubGrp |
21 | 19, 20 | syl 17 | . 2 |
22 | eqid 2622 | . . . 4 | |
23 | 22 | subg0cl 17602 | . . 3 SubGrp |
24 | ne0i 3921 | . . 3 | |
25 | 19, 23, 24 | 3syl 18 | . 2 |
26 | 19 | adantr 481 | . . 3 SubGrp |
27 | eqid 2622 | . . . . . 6 | |
28 | eqid 2622 | . . . . . 6 | |
29 | 3 | adantr 481 | . . . . . 6 |
30 | simprl 794 | . . . . . 6 | |
31 | simprr 796 | . . . . . . . 8 | |
32 | 16 | adantr 481 | . . . . . . . . . 10 supp |
33 | 32 | eleq2d 2687 | . . . . . . . . 9 supp |
34 | oveq1 6657 | . . . . . . . . . . 11 supp supp | |
35 | 34 | eleq1d 2686 | . . . . . . . . . 10 supp supp |
36 | 35 | elrab 3363 | . . . . . . . . 9 supp supp |
37 | 33, 36 | syl6bb 276 | . . . . . . . 8 supp |
38 | 31, 37 | mpbid 222 | . . . . . . 7 supp |
39 | 38 | simpld 475 | . . . . . 6 |
40 | 1, 27, 28, 6, 29, 30, 39 | psrvscacl 19393 | . . . . 5 |
41 | ovex 6678 | . . . . . . 7 supp | |
42 | 41 | a1i 11 | . . . . . 6 supp |
43 | 38 | simprd 479 | . . . . . . 7 supp |
44 | 15 | expr 643 | . . . . . . . . . 10 |
45 | 44 | alrimiv 1855 | . . . . . . . . 9 |
46 | 45 | ralrimiva 2966 | . . . . . . . 8 |
47 | 46 | adantr 481 | . . . . . . 7 |
48 | sseq2 3627 | . . . . . . . . . 10 supp supp | |
49 | 48 | imbi1d 331 | . . . . . . . . 9 supp supp |
50 | 49 | albidv 1849 | . . . . . . . 8 supp supp |
51 | 50 | rspcv 3305 | . . . . . . 7 supp supp |
52 | 43, 47, 51 | sylc 65 | . . . . . 6 supp |
53 | 1, 28, 12, 6, 40 | psrelbas 19379 | . . . . . . 7 |
54 | eqid 2622 | . . . . . . . . 9 | |
55 | 30 | adantr 481 | . . . . . . . . 9 supp |
56 | 39 | adantr 481 | . . . . . . . . 9 supp |
57 | eldifi 3732 | . . . . . . . . . 10 supp | |
58 | 57 | adantl 482 | . . . . . . . . 9 supp |
59 | 1, 27, 28, 6, 54, 12, 55, 56, 58 | psrvscaval 19392 | . . . . . . . 8 supp |
60 | 1, 28, 12, 6, 39 | psrelbas 19379 | . . . . . . . . . 10 |
61 | ssid 3624 | . . . . . . . . . . 11 supp supp | |
62 | 61 | a1i 11 | . . . . . . . . . 10 supp supp |
63 | ovex 6678 | . . . . . . . . . . . 12 | |
64 | 12, 63 | rabex2 4815 | . . . . . . . . . . 11 |
65 | 64 | a1i 11 | . . . . . . . . . 10 |
66 | fvex 6201 | . . . . . . . . . . . 12 | |
67 | 11, 66 | eqeltri 2697 | . . . . . . . . . . 11 |
68 | 67 | a1i 11 | . . . . . . . . . 10 |
69 | 60, 62, 65, 68 | suppssr 7326 | . . . . . . . . 9 supp |
70 | 69 | oveq2d 6666 | . . . . . . . 8 supp |
71 | 28, 54, 11 | ringrz 18588 | . . . . . . . . . 10 |
72 | 3, 30, 71 | syl2an2r 876 | . . . . . . . . 9 |
73 | 72 | adantr 481 | . . . . . . . 8 supp |
74 | 59, 70, 73 | 3eqtrd 2660 | . . . . . . 7 supp |
75 | 53, 74 | suppss 7325 | . . . . . 6 supp supp |
76 | sseq1 3626 | . . . . . . . 8 supp supp supp supp | |
77 | eleq1 2689 | . . . . . . . 8 supp supp | |
78 | 76, 77 | imbi12d 334 | . . . . . . 7 supp supp supp supp supp |
79 | 78 | spcgv 3293 | . . . . . 6 supp supp supp supp supp |
80 | 42, 52, 75, 79 | syl3c 66 | . . . . 5 supp |
81 | 32 | eleq2d 2687 | . . . . . 6 supp |
82 | oveq1 6657 | . . . . . . . 8 supp supp | |
83 | 82 | eleq1d 2686 | . . . . . . 7 supp supp |
84 | 83 | elrab 3363 | . . . . . 6 supp supp |
85 | 81, 84 | syl6bb 276 | . . . . 5 supp |
86 | 40, 80, 85 | mpbir2and 957 | . . . 4 |
87 | 86 | 3adantr3 1222 | . . 3 |
88 | simpr3 1069 | . . 3 | |
89 | eqid 2622 | . . . 4 | |
90 | 89 | subgcl 17604 | . . 3 SubGrp |
91 | 26, 87, 88, 90 | syl3anc 1326 | . 2 |
92 | 4, 5, 7, 8, 9, 10, 21, 25, 91 | islssd 18936 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wal 1481 wceq 1483 wcel 1990 wne 2794 wral 2912 crab 2916 cvv 3200 cdif 3571 cun 3572 wss 3574 c0 3915 ccnv 5113 cima 5117 cfv 5888 (class class class)co 6650 supp csupp 7295 cmap 7857 cfn 7955 cn 11020 cn0 11292 cbs 15857 cplusg 15941 cmulr 15942 cvsca 15945 c0g 16100 cgrp 17422 SubGrpcsubg 17588 crg 18547 clss 18932 mPwSer cmps 19351 |
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-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 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-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-mgp 18490 df-ring 18549 df-lss 18933 df-psr 19356 |
This theorem is referenced by: mpllss 19438 |
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