Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pwsmulg | Structured version Visualization version Unicode version |
Description: Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
pwsmulg.y | s |
pwsmulg.b | |
pwsmulg.s | .g |
pwsmulg.t | .g |
Ref | Expression |
---|---|
pwsmulg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 790 | . . . 4 | |
2 | simplr 792 | . . . 4 | |
3 | simpr3 1069 | . . . 4 | |
4 | pwsmulg.y | . . . . 5 s | |
5 | pwsmulg.b | . . . . 5 | |
6 | 4, 5 | pwspjmhm 17368 | . . . 4 MndHom |
7 | 1, 2, 3, 6 | syl3anc 1326 | . . 3 MndHom |
8 | simpr1 1067 | . . 3 | |
9 | simpr2 1068 | . . 3 | |
10 | pwsmulg.s | . . . 4 .g | |
11 | pwsmulg.t | . . . 4 .g | |
12 | 5, 10, 11 | mhmmulg 17583 | . . 3 MndHom |
13 | 7, 8, 9, 12 | syl3anc 1326 | . 2 |
14 | 4 | pwsmnd 17325 | . . . . 5 |
15 | 14 | adantr 481 | . . . 4 |
16 | 5, 10 | mulgnn0cl 17558 | . . . 4 |
17 | 15, 8, 9, 16 | syl3anc 1326 | . . 3 |
18 | fveq1 6190 | . . . 4 | |
19 | eqid 2622 | . . . 4 | |
20 | fvex 6201 | . . . 4 | |
21 | 18, 19, 20 | fvmpt 6282 | . . 3 |
22 | 17, 21 | syl 17 | . 2 |
23 | fveq1 6190 | . . . . 5 | |
24 | fvex 6201 | . . . . 5 | |
25 | 23, 19, 24 | fvmpt 6282 | . . . 4 |
26 | 9, 25 | syl 17 | . . 3 |
27 | 26 | oveq2d 6666 | . 2 |
28 | 13, 22, 27 | 3eqtr3d 2664 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cmpt 4729 cfv 5888 (class class class)co 6650 cn0 11292 cbs 15857 s cpws 16107 cmnd 17294 MndHom cmhm 17333 .gcmg 17540 |
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 |
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-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 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-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-pws 16110 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-mulg 17541 |
This theorem is referenced by: evl1expd 19709 |
Copyright terms: Public domain | W3C validator |