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Mirrors > Home > MPE Home > Th. List > pgpfac1 | Structured version Visualization version Unicode version |
Description: Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (Contributed by Mario Carneiro, 27-Apr-2016.) |
Ref | Expression |
---|---|
pgpfac1.k | mrClsSubGrp |
pgpfac1.s | |
pgpfac1.b | |
pgpfac1.o | |
pgpfac1.e | gEx |
pgpfac1.z | |
pgpfac1.l | |
pgpfac1.p | pGrp |
pgpfac1.g | |
pgpfac1.n | |
pgpfac1.oe | |
pgpfac1.ab |
Ref | Expression |
---|---|
pgpfac1 | SubGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pgpfac1.g | . . 3 | |
2 | ablgrp 18198 | . . 3 | |
3 | pgpfac1.b | . . . 4 | |
4 | 3 | subgid 17596 | . . 3 SubGrp |
5 | 1, 2, 4 | 3syl 18 | . 2 SubGrp |
6 | pgpfac1.ab | . 2 | |
7 | pgpfac1.n | . . 3 | |
8 | eleq1 2689 | . . . . . . 7 SubGrp SubGrp | |
9 | eleq2 2690 | . . . . . . 7 | |
10 | 8, 9 | anbi12d 747 | . . . . . 6 SubGrp SubGrp |
11 | eqeq2 2633 | . . . . . . . 8 | |
12 | 11 | anbi2d 740 | . . . . . . 7 |
13 | 12 | rexbidv 3052 | . . . . . 6 SubGrp SubGrp |
14 | 10, 13 | imbi12d 334 | . . . . 5 SubGrp SubGrp SubGrp SubGrp |
15 | 14 | imbi2d 330 | . . . 4 SubGrp SubGrp SubGrp SubGrp |
16 | eleq1 2689 | . . . . . . 7 SubGrp SubGrp | |
17 | eleq2 2690 | . . . . . . 7 | |
18 | 16, 17 | anbi12d 747 | . . . . . 6 SubGrp SubGrp |
19 | eqeq2 2633 | . . . . . . . 8 | |
20 | 19 | anbi2d 740 | . . . . . . 7 |
21 | 20 | rexbidv 3052 | . . . . . 6 SubGrp SubGrp |
22 | 18, 21 | imbi12d 334 | . . . . 5 SubGrp SubGrp SubGrp SubGrp |
23 | 22 | imbi2d 330 | . . . 4 SubGrp SubGrp SubGrp SubGrp |
24 | bi2.04 376 | . . . . . . . . . . 11 SubGrp SubGrp SubGrp SubGrp | |
25 | impexp 462 | . . . . . . . . . . . 12 SubGrp SubGrp SubGrp SubGrp | |
26 | 25 | imbi2i 326 | . . . . . . . . . . 11 SubGrp SubGrp SubGrp SubGrp |
27 | impexp 462 | . . . . . . . . . . . 12 SubGrp SubGrp | |
28 | 27 | imbi2i 326 | . . . . . . . . . . 11 SubGrp SubGrp SubGrp SubGrp |
29 | 24, 26, 28 | 3bitr4i 292 | . . . . . . . . . 10 SubGrp SubGrp SubGrp SubGrp |
30 | 29 | imbi2i 326 | . . . . . . . . 9 SubGrp SubGrp SubGrp SubGrp |
31 | bi2.04 376 | . . . . . . . . 9 SubGrp SubGrp SubGrp SubGrp | |
32 | bi2.04 376 | . . . . . . . . 9 SubGrp SubGrp SubGrp SubGrp | |
33 | 30, 31, 32 | 3bitr4i 292 | . . . . . . . 8 SubGrp SubGrp SubGrp SubGrp |
34 | 33 | albii 1747 | . . . . . . 7 SubGrp SubGrp SubGrp SubGrp |
35 | df-ral 2917 | . . . . . . 7 SubGrp SubGrp SubGrp SubGrp | |
36 | r19.21v 2960 | . . . . . . 7 SubGrp SubGrp SubGrp SubGrp | |
37 | 34, 35, 36 | 3bitr2i 288 | . . . . . 6 SubGrp SubGrp SubGrp SubGrp |
38 | psseq1 3694 | . . . . . . . . . . 11 | |
39 | eleq2 2690 | . . . . . . . . . . 11 | |
40 | 38, 39 | anbi12d 747 | . . . . . . . . . 10 |
41 | ineq2 3808 | . . . . . . . . . . . . . 14 | |
42 | 41 | eqeq1d 2624 | . . . . . . . . . . . . 13 |
43 | oveq2 6658 | . . . . . . . . . . . . . 14 | |
44 | 43 | eqeq1d 2624 | . . . . . . . . . . . . 13 |
45 | 42, 44 | anbi12d 747 | . . . . . . . . . . . 12 |
46 | 45 | cbvrexv 3172 | . . . . . . . . . . 11 SubGrp SubGrp |
47 | eqeq2 2633 | . . . . . . . . . . . . 13 | |
48 | 47 | anbi2d 740 | . . . . . . . . . . . 12 |
49 | 48 | rexbidv 3052 | . . . . . . . . . . 11 SubGrp SubGrp |
50 | 46, 49 | syl5bb 272 | . . . . . . . . . 10 SubGrp SubGrp |
51 | 40, 50 | imbi12d 334 | . . . . . . . . 9 SubGrp SubGrp |
52 | 51 | cbvralv 3171 | . . . . . . . 8 SubGrp SubGrp SubGrp SubGrp |
53 | pgpfac1.k | . . . . . . . . . 10 mrClsSubGrp | |
54 | pgpfac1.s | . . . . . . . . . 10 | |
55 | pgpfac1.o | . . . . . . . . . 10 | |
56 | pgpfac1.e | . . . . . . . . . 10 gEx | |
57 | pgpfac1.z | . . . . . . . . . 10 | |
58 | pgpfac1.l | . . . . . . . . . 10 | |
59 | pgpfac1.p | . . . . . . . . . . 11 pGrp | |
60 | 59 | adantr 481 | . . . . . . . . . 10 SubGrp SubGrp SubGrp pGrp |
61 | 1 | adantr 481 | . . . . . . . . . 10 SubGrp SubGrp SubGrp |
62 | 7 | adantr 481 | . . . . . . . . . 10 SubGrp SubGrp SubGrp |
63 | pgpfac1.oe | . . . . . . . . . . 11 | |
64 | 63 | adantr 481 | . . . . . . . . . 10 SubGrp SubGrp SubGrp |
65 | simprrl 804 | . . . . . . . . . 10 SubGrp SubGrp SubGrp SubGrp | |
66 | simprrr 805 | . . . . . . . . . 10 SubGrp SubGrp SubGrp | |
67 | simprl 794 | . . . . . . . . . . 11 SubGrp SubGrp SubGrp SubGrp SubGrp | |
68 | 67, 52 | sylib 208 | . . . . . . . . . 10 SubGrp SubGrp SubGrp SubGrp SubGrp |
69 | 53, 54, 3, 55, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68 | pgpfac1lem5 18478 | . . . . . . . . 9 SubGrp SubGrp SubGrp SubGrp |
70 | 69 | exp32 631 | . . . . . . . 8 SubGrp SubGrp SubGrp SubGrp |
71 | 52, 70 | syl5bir 233 | . . . . . . 7 SubGrp SubGrp SubGrp SubGrp |
72 | 71 | a2i 14 | . . . . . 6 SubGrp SubGrp SubGrp SubGrp |
73 | 37, 72 | sylbi 207 | . . . . 5 SubGrp SubGrp SubGrp SubGrp |
74 | 73 | a1i 11 | . . . 4 SubGrp SubGrp SubGrp SubGrp |
75 | 15, 23, 74 | findcard3 8203 | . . 3 SubGrp SubGrp |
76 | 7, 75 | mpcom 38 | . 2 SubGrp SubGrp |
77 | 5, 6, 76 | mp2and 715 | 1 SubGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wceq 1483 wcel 1990 wral 2912 wrex 2913 cin 3573 wpss 3575 csn 4177 class class class wbr 4653 cfv 5888 (class class class)co 6650 cfn 7955 cbs 15857 c0g 16100 mrClscmrc 16243 cgrp 17422 SubGrpcsubg 17588 cod 17944 gExcgex 17945 pGrp cpgp 17946 clsm 18049 cabl 18194 |
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-inf2 8538 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 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-iin 4523 df-disj 4621 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-rpss 6937 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-ga 17723 df-cntz 17750 df-od 17948 df-gex 17949 df-pgp 17950 df-lsm 18051 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: pgpfaclem3 18482 |
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