Step |
Hyp |
Ref |
Expression |
1 |
|
1re |
|- 1 e. RR |
2 |
1
|
ltnri |
|- -. 1 < 1 |
3 |
|
breq2 |
|- ( ( # ` ( Base ` R ) ) = 1 -> ( 1 < ( # ` ( Base ` R ) ) <-> 1 < 1 ) ) |
4 |
2 3
|
mtbiri |
|- ( ( # ` ( Base ` R ) ) = 1 -> -. 1 < ( # ` ( Base ` R ) ) ) |
5 |
4
|
adantl |
|- ( ( R e. Ring /\ ( # ` ( Base ` R ) ) = 1 ) -> -. 1 < ( # ` ( Base ` R ) ) ) |
6 |
5
|
intnand |
|- ( ( R e. Ring /\ ( # ` ( Base ` R ) ) = 1 ) -> -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) ) |
7 |
6
|
ex |
|- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 -> -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) ) ) |
8 |
|
ianor |
|- ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) <-> ( -. R e. Ring \/ -. 1 < ( # ` ( Base ` R ) ) ) ) |
9 |
|
pm2.21 |
|- ( -. R e. Ring -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) ) |
10 |
|
fvex |
|- ( Base ` R ) e. _V |
11 |
|
hashxrcl |
|- ( ( Base ` R ) e. _V -> ( # ` ( Base ` R ) ) e. RR* ) |
12 |
10 11
|
ax-mp |
|- ( # ` ( Base ` R ) ) e. RR* |
13 |
|
1xr |
|- 1 e. RR* |
14 |
|
xrlenlt |
|- ( ( ( # ` ( Base ` R ) ) e. RR* /\ 1 e. RR* ) -> ( ( # ` ( Base ` R ) ) <_ 1 <-> -. 1 < ( # ` ( Base ` R ) ) ) ) |
15 |
12 13 14
|
mp2an |
|- ( ( # ` ( Base ` R ) ) <_ 1 <-> -. 1 < ( # ` ( Base ` R ) ) ) |
16 |
15
|
bicomi |
|- ( -. 1 < ( # ` ( Base ` R ) ) <-> ( # ` ( Base ` R ) ) <_ 1 ) |
17 |
|
simpr |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) <_ 1 ) |
18 |
|
1nn0 |
|- 1 e. NN0 |
19 |
|
hashbnd |
|- ( ( ( Base ` R ) e. _V /\ 1 e. NN0 /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( Base ` R ) e. Fin ) |
20 |
10 18 17 19
|
mp3an12i |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( Base ` R ) e. Fin ) |
21 |
|
hashcl |
|- ( ( Base ` R ) e. Fin -> ( # ` ( Base ` R ) ) e. NN0 ) |
22 |
|
simpr |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) e. NN0 ) -> ( # ` ( Base ` R ) ) e. NN0 ) |
23 |
|
hasheq0 |
|- ( ( Base ` R ) e. _V -> ( ( # ` ( Base ` R ) ) = 0 <-> ( Base ` R ) = (/) ) ) |
24 |
10 23
|
mp1i |
|- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( # ` ( Base ` R ) ) = 0 <-> ( Base ` R ) = (/) ) ) |
25 |
24
|
biimpd |
|- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( # ` ( Base ` R ) ) = 0 -> ( Base ` R ) = (/) ) ) |
26 |
25
|
necon3d |
|- ( ( # ` ( Base ` R ) ) e. NN0 -> ( ( Base ` R ) =/= (/) -> ( # ` ( Base ` R ) ) =/= 0 ) ) |
27 |
26
|
impcom |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) e. NN0 ) -> ( # ` ( Base ` R ) ) =/= 0 ) |
28 |
|
elnnne0 |
|- ( ( # ` ( Base ` R ) ) e. NN <-> ( ( # ` ( Base ` R ) ) e. NN0 /\ ( # ` ( Base ` R ) ) =/= 0 ) ) |
29 |
22 27 28
|
sylanbrc |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) e. NN0 ) -> ( # ` ( Base ` R ) ) e. NN ) |
30 |
29
|
ex |
|- ( ( Base ` R ) =/= (/) -> ( ( # ` ( Base ` R ) ) e. NN0 -> ( # ` ( Base ` R ) ) e. NN ) ) |
31 |
30
|
adantr |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( ( # ` ( Base ` R ) ) e. NN0 -> ( # ` ( Base ` R ) ) e. NN ) ) |
32 |
21 31
|
syl5com |
|- ( ( Base ` R ) e. Fin -> ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) e. NN ) ) |
33 |
20 32
|
mpcom |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) e. NN ) |
34 |
|
nnle1eq1 |
|- ( ( # ` ( Base ` R ) ) e. NN -> ( ( # ` ( Base ` R ) ) <_ 1 <-> ( # ` ( Base ` R ) ) = 1 ) ) |
35 |
33 34
|
syl |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( ( # ` ( Base ` R ) ) <_ 1 <-> ( # ` ( Base ` R ) ) = 1 ) ) |
36 |
17 35
|
mpbid |
|- ( ( ( Base ` R ) =/= (/) /\ ( # ` ( Base ` R ) ) <_ 1 ) -> ( # ` ( Base ` R ) ) = 1 ) |
37 |
36
|
ex |
|- ( ( Base ` R ) =/= (/) -> ( ( # ` ( Base ` R ) ) <_ 1 -> ( # ` ( Base ` R ) ) = 1 ) ) |
38 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
39 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
40 |
39
|
grpbn0 |
|- ( R e. Grp -> ( Base ` R ) =/= (/) ) |
41 |
38 40
|
syl |
|- ( R e. Ring -> ( Base ` R ) =/= (/) ) |
42 |
37 41
|
syl11 |
|- ( ( # ` ( Base ` R ) ) <_ 1 -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) ) |
43 |
16 42
|
sylbi |
|- ( -. 1 < ( # ` ( Base ` R ) ) -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) ) |
44 |
9 43
|
jaoi |
|- ( ( -. R e. Ring \/ -. 1 < ( # ` ( Base ` R ) ) ) -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) ) |
45 |
8 44
|
sylbi |
|- ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) -> ( R e. Ring -> ( # ` ( Base ` R ) ) = 1 ) ) |
46 |
45
|
com12 |
|- ( R e. Ring -> ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) -> ( # ` ( Base ` R ) ) = 1 ) ) |
47 |
7 46
|
impbid |
|- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 <-> -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) ) ) |
48 |
39
|
isnzr2hash |
|- ( R e. NzRing <-> ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) ) |
49 |
48
|
bicomi |
|- ( ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) <-> R e. NzRing ) |
50 |
49
|
notbii |
|- ( -. ( R e. Ring /\ 1 < ( # ` ( Base ` R ) ) ) <-> -. R e. NzRing ) |
51 |
47 50
|
bitrdi |
|- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 <-> -. R e. NzRing ) ) |