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Mirrors > Home > MPE Home > Th. List > nghmfval | Structured version Visualization version Unicode version |
Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 |
Ref | Expression |
---|---|
nghmfval | NGHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 6659 | . . . . . 6 | |
2 | nmofval.1 | . . . . . 6 | |
3 | 1, 2 | syl6eqr 2674 | . . . . 5 |
4 | 3 | cnveqd 5298 | . . . 4 |
5 | 4 | imaeq1d 5465 | . . 3 |
6 | df-nghm 22513 | . . 3 NGHom NrmGrp NrmGrp | |
7 | 2 | ovexi 6679 | . . . . 5 |
8 | 7 | cnvex 7113 | . . . 4 |
9 | 8 | imaex 7104 | . . 3 |
10 | 5, 6, 9 | ovmpt2a 6791 | . 2 NrmGrp NrmGrp NGHom |
11 | 6 | mpt2ndm0 6875 | . . 3 NrmGrp NrmGrp NGHom |
12 | nmoffn 22515 | . . . . . . . . . 10 NrmGrp NrmGrp | |
13 | fndm 5990 | . . . . . . . . . 10 NrmGrp NrmGrp NrmGrp NrmGrp | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 NrmGrp NrmGrp |
15 | 14 | ndmov 6818 | . . . . . . . 8 NrmGrp NrmGrp |
16 | 2, 15 | syl5eq 2668 | . . . . . . 7 NrmGrp NrmGrp |
17 | 16 | cnveqd 5298 | . . . . . 6 NrmGrp NrmGrp |
18 | cnv0 5535 | . . . . . 6 | |
19 | 17, 18 | syl6eq 2672 | . . . . 5 NrmGrp NrmGrp |
20 | 19 | imaeq1d 5465 | . . . 4 NrmGrp NrmGrp |
21 | 0ima 5482 | . . . 4 | |
22 | 20, 21 | syl6eq 2672 | . . 3 NrmGrp NrmGrp |
23 | 11, 22 | eqtr4d 2659 | . 2 NrmGrp NrmGrp NGHom |
24 | 10, 23 | pm2.61i 176 | 1 NGHom |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wa 384 wceq 1483 wcel 1990 c0 3915 cxp 5112 ccnv 5113 cdm 5114 cima 5117 wfn 5883 (class class class)co 6650 cr 9935 NrmGrpcngp 22382 cnmo 22509 NGHom cnghm 22510 |
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 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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-ico 12181 df-nmo 22512 df-nghm 22513 |
This theorem is referenced by: isnghm 22527 |
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