Step |
Hyp |
Ref |
Expression |
1 |
|
irinitoringc.u |
|- ( ph -> U e. V ) |
2 |
|
irinitoringc.z |
|- ( ph -> ZZring e. U ) |
3 |
|
irinitoringc.c |
|- C = ( RingCat ` U ) |
4 |
|
zex |
|- ZZ e. _V |
5 |
4
|
mptex |
|- ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) e. _V |
6 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
7 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
8 |
3 6 1 7
|
ringchomfval |
|- ( ph -> ( Hom ` C ) = ( RingHom |` ( ( Base ` C ) X. ( Base ` C ) ) ) ) |
9 |
8
|
adantr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( Hom ` C ) = ( RingHom |` ( ( Base ` C ) X. ( Base ` C ) ) ) ) |
10 |
9
|
oveqd |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( ZZring ( Hom ` C ) r ) = ( ZZring ( RingHom |` ( ( Base ` C ) X. ( Base ` C ) ) ) r ) ) |
11 |
|
id |
|- ( ZZring e. U -> ZZring e. U ) |
12 |
|
zringring |
|- ZZring e. Ring |
13 |
12
|
a1i |
|- ( ZZring e. U -> ZZring e. Ring ) |
14 |
11 13
|
elind |
|- ( ZZring e. U -> ZZring e. ( U i^i Ring ) ) |
15 |
2 14
|
syl |
|- ( ph -> ZZring e. ( U i^i Ring ) ) |
16 |
3 6 1
|
ringcbas |
|- ( ph -> ( Base ` C ) = ( U i^i Ring ) ) |
17 |
15 16
|
eleqtrrd |
|- ( ph -> ZZring e. ( Base ` C ) ) |
18 |
17
|
adantr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ZZring e. ( Base ` C ) ) |
19 |
|
simpr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> r e. ( Base ` C ) ) |
20 |
18 19
|
ovresd |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( ZZring ( RingHom |` ( ( Base ` C ) X. ( Base ` C ) ) ) r ) = ( ZZring RingHom r ) ) |
21 |
16
|
eleq2d |
|- ( ph -> ( r e. ( Base ` C ) <-> r e. ( U i^i Ring ) ) ) |
22 |
|
elin |
|- ( r e. ( U i^i Ring ) <-> ( r e. U /\ r e. Ring ) ) |
23 |
22
|
simprbi |
|- ( r e. ( U i^i Ring ) -> r e. Ring ) |
24 |
21 23
|
syl6bi |
|- ( ph -> ( r e. ( Base ` C ) -> r e. Ring ) ) |
25 |
24
|
imp |
|- ( ( ph /\ r e. ( Base ` C ) ) -> r e. Ring ) |
26 |
|
eqid |
|- ( .g ` r ) = ( .g ` r ) |
27 |
|
eqid |
|- ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) = ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) |
28 |
|
eqid |
|- ( 1r ` r ) = ( 1r ` r ) |
29 |
26 27 28
|
mulgrhm2 |
|- ( r e. Ring -> ( ZZring RingHom r ) = { ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) } ) |
30 |
25 29
|
syl |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( ZZring RingHom r ) = { ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) } ) |
31 |
10 20 30
|
3eqtrd |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( ZZring ( Hom ` C ) r ) = { ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) } ) |
32 |
|
sneq |
|- ( f = ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) -> { f } = { ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) } ) |
33 |
32
|
eqeq2d |
|- ( f = ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) -> ( ( ZZring ( Hom ` C ) r ) = { f } <-> ( ZZring ( Hom ` C ) r ) = { ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) } ) ) |
34 |
33
|
spcegv |
|- ( ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) e. _V -> ( ( ZZring ( Hom ` C ) r ) = { ( z e. ZZ |-> ( z ( .g ` r ) ( 1r ` r ) ) ) } -> E. f ( ZZring ( Hom ` C ) r ) = { f } ) ) |
35 |
5 31 34
|
mpsyl |
|- ( ( ph /\ r e. ( Base ` C ) ) -> E. f ( ZZring ( Hom ` C ) r ) = { f } ) |
36 |
|
eusn |
|- ( E! f f e. ( ZZring ( Hom ` C ) r ) <-> E. f ( ZZring ( Hom ` C ) r ) = { f } ) |
37 |
35 36
|
sylibr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> E! f f e. ( ZZring ( Hom ` C ) r ) ) |
38 |
37
|
ralrimiva |
|- ( ph -> A. r e. ( Base ` C ) E! f f e. ( ZZring ( Hom ` C ) r ) ) |
39 |
3
|
ringccat |
|- ( U e. V -> C e. Cat ) |
40 |
1 39
|
syl |
|- ( ph -> C e. Cat ) |
41 |
12
|
a1i |
|- ( ph -> ZZring e. Ring ) |
42 |
2 41
|
elind |
|- ( ph -> ZZring e. ( U i^i Ring ) ) |
43 |
42 16
|
eleqtrrd |
|- ( ph -> ZZring e. ( Base ` C ) ) |
44 |
6 7 40 43
|
isinito |
|- ( ph -> ( ZZring e. ( InitO ` C ) <-> A. r e. ( Base ` C ) E! f f e. ( ZZring ( Hom ` C ) r ) ) ) |
45 |
38 44
|
mpbird |
|- ( ph -> ZZring e. ( InitO ` C ) ) |