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Mirrors > Home > MPE Home > Th. List > pwsco2rhm | Structured version Visualization version Unicode version |
Description: Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
pwsco2rhm.y | s |
pwsco2rhm.z | s |
pwsco2rhm.b | |
pwsco2rhm.a | |
pwsco2rhm.f | RingHom |
Ref | Expression |
---|---|
pwsco2rhm | RingHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsco2rhm.f | . . . . 5 RingHom | |
2 | rhmrcl1 18719 | . . . . 5 RingHom | |
3 | 1, 2 | syl 17 | . . . 4 |
4 | pwsco2rhm.a | . . . 4 | |
5 | pwsco2rhm.y | . . . . 5 s | |
6 | 5 | pwsring 18615 | . . . 4 |
7 | 3, 4, 6 | syl2anc 693 | . . 3 |
8 | rhmrcl2 18720 | . . . . 5 RingHom | |
9 | 1, 8 | syl 17 | . . . 4 |
10 | pwsco2rhm.z | . . . . 5 s | |
11 | 10 | pwsring 18615 | . . . 4 |
12 | 9, 4, 11 | syl2anc 693 | . . 3 |
13 | 7, 12 | jca 554 | . 2 |
14 | pwsco2rhm.b | . . . . 5 | |
15 | rhmghm 18725 | . . . . . . 7 RingHom | |
16 | 1, 15 | syl 17 | . . . . . 6 |
17 | ghmmhm 17670 | . . . . . 6 MndHom | |
18 | 16, 17 | syl 17 | . . . . 5 MndHom |
19 | 5, 10, 14, 4, 18 | pwsco2mhm 17371 | . . . 4 MndHom |
20 | ringgrp 18552 | . . . . . 6 | |
21 | 7, 20 | syl 17 | . . . . 5 |
22 | ringgrp 18552 | . . . . . 6 | |
23 | 12, 22 | syl 17 | . . . . 5 |
24 | ghmmhmb 17671 | . . . . 5 MndHom | |
25 | 21, 23, 24 | syl2anc 693 | . . . 4 MndHom |
26 | 19, 25 | eleqtrrd 2704 | . . 3 |
27 | eqid 2622 | . . . . 5 mulGrp s mulGrp s | |
28 | eqid 2622 | . . . . 5 mulGrp s mulGrp s | |
29 | eqid 2622 | . . . . 5 mulGrp s mulGrp s | |
30 | eqid 2622 | . . . . . . 7 mulGrp mulGrp | |
31 | eqid 2622 | . . . . . . 7 mulGrp mulGrp | |
32 | 30, 31 | rhmmhm 18722 | . . . . . 6 RingHom mulGrp MndHom mulGrp |
33 | 1, 32 | syl 17 | . . . . 5 mulGrp MndHom mulGrp |
34 | 27, 28, 29, 4, 33 | pwsco2mhm 17371 | . . . 4 mulGrp s mulGrp s MndHom mulGrp s |
35 | eqid 2622 | . . . . . . . . 9 | |
36 | 5, 35 | pwsbas 16147 | . . . . . . . 8 |
37 | 3, 4, 36 | syl2anc 693 | . . . . . . 7 |
38 | 37, 14 | syl6eqr 2674 | . . . . . 6 |
39 | 30 | ringmgp 18553 | . . . . . . . 8 mulGrp |
40 | 3, 39 | syl 17 | . . . . . . 7 mulGrp |
41 | 30, 35 | mgpbas 18495 | . . . . . . . 8 mulGrp |
42 | 27, 41 | pwsbas 16147 | . . . . . . 7 mulGrp mulGrp s |
43 | 40, 4, 42 | syl2anc 693 | . . . . . 6 mulGrp s |
44 | 38, 43 | eqtr3d 2658 | . . . . 5 mulGrp s |
45 | 44 | mpteq1d 4738 | . . . 4 mulGrp s |
46 | eqidd 2623 | . . . . 5 mulGrp mulGrp | |
47 | eqidd 2623 | . . . . 5 mulGrp mulGrp | |
48 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
49 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
50 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
51 | eqid 2622 | . . . . . . . 8 mulGrp s mulGrp s | |
52 | 5, 30, 27, 48, 49, 29, 50, 51 | pwsmgp 18618 | . . . . . . 7 mulGrp mulGrp s mulGrp mulGrp s |
53 | 3, 4, 52 | syl2anc 693 | . . . . . 6 mulGrp mulGrp s mulGrp mulGrp s |
54 | 53 | simpld 475 | . . . . 5 mulGrp mulGrp s |
55 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
56 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
57 | eqid 2622 | . . . . . . . 8 mulGrp s mulGrp s | |
58 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
59 | eqid 2622 | . . . . . . . 8 mulGrp s mulGrp s | |
60 | 10, 31, 28, 55, 56, 57, 58, 59 | pwsmgp 18618 | . . . . . . 7 mulGrp mulGrp s mulGrp mulGrp s |
61 | 9, 4, 60 | syl2anc 693 | . . . . . 6 mulGrp mulGrp s mulGrp mulGrp s |
62 | 61 | simpld 475 | . . . . 5 mulGrp mulGrp s |
63 | 53 | simprd 479 | . . . . . 6 mulGrp mulGrp s |
64 | 63 | oveqdr 6674 | . . . . 5 mulGrp mulGrp mulGrp mulGrp s |
65 | 61 | simprd 479 | . . . . . 6 mulGrp mulGrp s |
66 | 65 | oveqdr 6674 | . . . . 5 mulGrp mulGrp mulGrp mulGrp s |
67 | 46, 47, 54, 62, 64, 66 | mhmpropd 17341 | . . . 4 mulGrp MndHom mulGrp mulGrp s MndHom mulGrp s |
68 | 34, 45, 67 | 3eltr4d 2716 | . . 3 mulGrp MndHom mulGrp |
69 | 26, 68 | jca 554 | . 2 mulGrp MndHom mulGrp |
70 | 48, 55 | isrhm 18721 | . 2 RingHom mulGrp MndHom mulGrp |
71 | 13, 69, 70 | sylanbrc 698 | 1 RingHom |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cmpt 4729 ccom 5118 cfv 5888 (class class class)co 6650 cmap 7857 cbs 15857 cplusg 15941 s cpws 16107 cmnd 17294 MndHom cmhm 17333 cgrp 17422 cghm 17657 mulGrpcmgp 18489 crg 18547 RingHom crh 18712 |
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-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-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 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-grp 17425 df-minusg 17426 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-rnghom 18715 |
This theorem is referenced by: (None) |
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