| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dchr1re.g |
|- G = ( DChr ` N ) |
| 2 |
|
dchr1re.z |
|- Z = ( Z/nZ ` N ) |
| 3 |
|
dchr1re.o |
|- .1. = ( 0g ` G ) |
| 4 |
|
dchr1re.b |
|- B = ( Base ` Z ) |
| 5 |
|
dchr1re.n |
|- ( ph -> N e. NN ) |
| 6 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
| 7 |
1
|
dchrabl |
|- ( N e. NN -> G e. Abel ) |
| 8 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
| 9 |
6 3
|
grpidcl |
|- ( G e. Grp -> .1. e. ( Base ` G ) ) |
| 10 |
5 7 8 9
|
4syl |
|- ( ph -> .1. e. ( Base ` G ) ) |
| 11 |
1 2 6 4 10
|
dchrf |
|- ( ph -> .1. : B --> CC ) |
| 12 |
11
|
ffnd |
|- ( ph -> .1. Fn B ) |
| 13 |
|
simpr |
|- ( ( ( ph /\ x e. B ) /\ ( .1. ` x ) = 0 ) -> ( .1. ` x ) = 0 ) |
| 14 |
|
0re |
|- 0 e. RR |
| 15 |
13 14
|
eqeltrdi |
|- ( ( ( ph /\ x e. B ) /\ ( .1. ` x ) = 0 ) -> ( .1. ` x ) e. RR ) |
| 16 |
|
eqid |
|- ( Unit ` Z ) = ( Unit ` Z ) |
| 17 |
5
|
ad2antrr |
|- ( ( ( ph /\ x e. B ) /\ ( .1. ` x ) =/= 0 ) -> N e. NN ) |
| 18 |
10
|
adantr |
|- ( ( ph /\ x e. B ) -> .1. e. ( Base ` G ) ) |
| 19 |
|
simpr |
|- ( ( ph /\ x e. B ) -> x e. B ) |
| 20 |
1 2 6 4 16 18 19
|
dchrn0 |
|- ( ( ph /\ x e. B ) -> ( ( .1. ` x ) =/= 0 <-> x e. ( Unit ` Z ) ) ) |
| 21 |
20
|
biimpa |
|- ( ( ( ph /\ x e. B ) /\ ( .1. ` x ) =/= 0 ) -> x e. ( Unit ` Z ) ) |
| 22 |
1 2 3 16 17 21
|
dchr1 |
|- ( ( ( ph /\ x e. B ) /\ ( .1. ` x ) =/= 0 ) -> ( .1. ` x ) = 1 ) |
| 23 |
|
1re |
|- 1 e. RR |
| 24 |
22 23
|
eqeltrdi |
|- ( ( ( ph /\ x e. B ) /\ ( .1. ` x ) =/= 0 ) -> ( .1. ` x ) e. RR ) |
| 25 |
15 24
|
pm2.61dane |
|- ( ( ph /\ x e. B ) -> ( .1. ` x ) e. RR ) |
| 26 |
25
|
ralrimiva |
|- ( ph -> A. x e. B ( .1. ` x ) e. RR ) |
| 27 |
|
ffnfv |
|- ( .1. : B --> RR <-> ( .1. Fn B /\ A. x e. B ( .1. ` x ) e. RR ) ) |
| 28 |
12 26 27
|
sylanbrc |
|- ( ph -> .1. : B --> RR ) |