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Mirrors > Home > MPE Home > Th. List > resmhm2 | Structured version Visualization version Unicode version |
Description: One direction of resmhm2b 17361. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
resmhm2.u | ↾s |
Ref | Expression |
---|---|
resmhm2 | MndHom SubMnd MndHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhmrcl1 17338 | . . 3 MndHom | |
2 | submrcl 17346 | . . 3 SubMnd | |
3 | 1, 2 | anim12i 590 | . 2 MndHom SubMnd |
4 | eqid 2622 | . . . . 5 | |
5 | eqid 2622 | . . . . 5 | |
6 | 4, 5 | mhmf 17340 | . . . 4 MndHom |
7 | resmhm2.u | . . . . . 6 ↾s | |
8 | 7 | submbas 17355 | . . . . 5 SubMnd |
9 | eqid 2622 | . . . . . 6 | |
10 | 9 | submss 17350 | . . . . 5 SubMnd |
11 | 8, 10 | eqsstr3d 3640 | . . . 4 SubMnd |
12 | fss 6056 | . . . 4 | |
13 | 6, 11, 12 | syl2an 494 | . . 3 MndHom SubMnd |
14 | eqid 2622 | . . . . . . . 8 | |
15 | eqid 2622 | . . . . . . . 8 | |
16 | 4, 14, 15 | mhmlin 17342 | . . . . . . 7 MndHom |
17 | 16 | 3expb 1266 | . . . . . 6 MndHom |
18 | 17 | adantlr 751 | . . . . 5 MndHom SubMnd |
19 | eqid 2622 | . . . . . . . 8 | |
20 | 7, 19 | ressplusg 15993 | . . . . . . 7 SubMnd |
21 | 20 | ad2antlr 763 | . . . . . 6 MndHom SubMnd |
22 | 21 | oveqd 6667 | . . . . 5 MndHom SubMnd |
23 | 18, 22 | eqtr4d 2659 | . . . 4 MndHom SubMnd |
24 | 23 | ralrimivva 2971 | . . 3 MndHom SubMnd |
25 | eqid 2622 | . . . . . 6 | |
26 | eqid 2622 | . . . . . 6 | |
27 | 25, 26 | mhm0 17343 | . . . . 5 MndHom |
28 | 27 | adantr 481 | . . . 4 MndHom SubMnd |
29 | eqid 2622 | . . . . . 6 | |
30 | 7, 29 | subm0 17356 | . . . . 5 SubMnd |
31 | 30 | adantl 482 | . . . 4 MndHom SubMnd |
32 | 28, 31 | eqtr4d 2659 | . . 3 MndHom SubMnd |
33 | 13, 24, 32 | 3jca 1242 | . 2 MndHom SubMnd |
34 | 4, 9, 14, 19, 25, 29 | ismhm 17337 | . 2 MndHom |
35 | 3, 33, 34 | sylanbrc 698 | 1 MndHom SubMnd MndHom |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wss 3574 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cplusg 15941 c0g 16100 cmnd 17294 MndHom cmhm 17333 SubMndcsubmnd 17334 |
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-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-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-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 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-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 |
This theorem is referenced by: resmhm2b 17361 resghm2 17677 zrhpsgnmhm 19930 lgseisenlem4 25103 |
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