Step |
Hyp |
Ref |
Expression |
1 |
|
zrinitorngc.u |
|- ( ph -> U e. V ) |
2 |
|
zrinitorngc.c |
|- C = ( RngCat ` U ) |
3 |
|
zrinitorngc.z |
|- ( ph -> Z e. ( Ring \ NzRing ) ) |
4 |
|
zrinitorngc.e |
|- ( ph -> Z e. U ) |
5 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
6 |
2 5 1
|
rngcbas |
|- ( ph -> ( Base ` C ) = ( U i^i Rng ) ) |
7 |
6
|
eleq2d |
|- ( ph -> ( r e. ( Base ` C ) <-> r e. ( U i^i Rng ) ) ) |
8 |
|
elin |
|- ( r e. ( U i^i Rng ) <-> ( r e. U /\ r e. Rng ) ) |
9 |
8
|
simprbi |
|- ( r e. ( U i^i Rng ) -> r e. Rng ) |
10 |
7 9
|
syl6bi |
|- ( ph -> ( r e. ( Base ` C ) -> r e. Rng ) ) |
11 |
10
|
imp |
|- ( ( ph /\ r e. ( Base ` C ) ) -> r e. Rng ) |
12 |
3
|
adantr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> Z e. ( Ring \ NzRing ) ) |
13 |
|
eqid |
|- ( Base ` Z ) = ( Base ` Z ) |
14 |
|
eqid |
|- ( 0g ` r ) = ( 0g ` r ) |
15 |
|
eqid |
|- ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) |
16 |
13 14 15
|
zrrnghm |
|- ( ( r e. Rng /\ Z e. ( Ring \ NzRing ) ) -> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) |
17 |
11 12 16
|
syl2anc |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) |
18 |
|
simpr |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) -> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) |
19 |
1
|
adantr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> U e. V ) |
20 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
21 |
|
eldifi |
|- ( Z e. ( Ring \ NzRing ) -> Z e. Ring ) |
22 |
|
ringrng |
|- ( Z e. Ring -> Z e. Rng ) |
23 |
3 21 22
|
3syl |
|- ( ph -> Z e. Rng ) |
24 |
4 23
|
elind |
|- ( ph -> Z e. ( U i^i Rng ) ) |
25 |
24 6
|
eleqtrrd |
|- ( ph -> Z e. ( Base ` C ) ) |
26 |
25
|
adantr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> Z e. ( Base ` C ) ) |
27 |
|
simpr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> r e. ( Base ` C ) ) |
28 |
2 5 19 20 26 27
|
rngchom |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( Z ( Hom ` C ) r ) = ( Z RngHomo r ) ) |
29 |
28
|
eqcomd |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( Z RngHomo r ) = ( Z ( Hom ` C ) r ) ) |
30 |
29
|
eleq2d |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) <-> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z ( Hom ` C ) r ) ) ) |
31 |
30
|
biimpa |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) -> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z ( Hom ` C ) r ) ) |
32 |
28
|
eleq2d |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( h e. ( Z ( Hom ` C ) r ) <-> h e. ( Z RngHomo r ) ) ) |
33 |
|
eqid |
|- ( Base ` r ) = ( Base ` r ) |
34 |
13 33
|
rnghmf |
|- ( h e. ( Z RngHomo r ) -> h : ( Base ` Z ) --> ( Base ` r ) ) |
35 |
32 34
|
syl6bi |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( h e. ( Z ( Hom ` C ) r ) -> h : ( Base ` Z ) --> ( Base ` r ) ) ) |
36 |
35
|
imp |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) -> h : ( Base ` Z ) --> ( Base ` r ) ) |
37 |
|
ffn |
|- ( h : ( Base ` Z ) --> ( Base ` r ) -> h Fn ( Base ` Z ) ) |
38 |
37
|
adantl |
|- ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) -> h Fn ( Base ` Z ) ) |
39 |
|
fvex |
|- ( 0g ` r ) e. _V |
40 |
39 15
|
fnmpti |
|- ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) Fn ( Base ` Z ) |
41 |
40
|
a1i |
|- ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) -> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) Fn ( Base ` Z ) ) |
42 |
32
|
biimpa |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) -> h e. ( Z RngHomo r ) ) |
43 |
|
rnghmghm |
|- ( h e. ( Z RngHomo r ) -> h e. ( Z GrpHom r ) ) |
44 |
|
eqid |
|- ( 0g ` Z ) = ( 0g ` Z ) |
45 |
44 14
|
ghmid |
|- ( h e. ( Z GrpHom r ) -> ( h ` ( 0g ` Z ) ) = ( 0g ` r ) ) |
46 |
42 43 45
|
3syl |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) -> ( h ` ( 0g ` Z ) ) = ( 0g ` r ) ) |
47 |
46
|
ad2antrr |
|- ( ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) /\ a e. ( Base ` Z ) ) -> ( h ` ( 0g ` Z ) ) = ( 0g ` r ) ) |
48 |
13 44
|
0ringbas |
|- ( Z e. ( Ring \ NzRing ) -> ( Base ` Z ) = { ( 0g ` Z ) } ) |
49 |
3 48
|
syl |
|- ( ph -> ( Base ` Z ) = { ( 0g ` Z ) } ) |
50 |
49
|
eleq2d |
|- ( ph -> ( a e. ( Base ` Z ) <-> a e. { ( 0g ` Z ) } ) ) |
51 |
|
elsni |
|- ( a e. { ( 0g ` Z ) } -> a = ( 0g ` Z ) ) |
52 |
51
|
fveq2d |
|- ( a e. { ( 0g ` Z ) } -> ( h ` a ) = ( h ` ( 0g ` Z ) ) ) |
53 |
50 52
|
syl6bi |
|- ( ph -> ( a e. ( Base ` Z ) -> ( h ` a ) = ( h ` ( 0g ` Z ) ) ) ) |
54 |
53
|
adantr |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( a e. ( Base ` Z ) -> ( h ` a ) = ( h ` ( 0g ` Z ) ) ) ) |
55 |
54
|
ad2antrr |
|- ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) -> ( a e. ( Base ` Z ) -> ( h ` a ) = ( h ` ( 0g ` Z ) ) ) ) |
56 |
55
|
imp |
|- ( ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) /\ a e. ( Base ` Z ) ) -> ( h ` a ) = ( h ` ( 0g ` Z ) ) ) |
57 |
|
eqidd |
|- ( a e. ( Base ` Z ) -> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) |
58 |
|
eqidd |
|- ( ( a e. ( Base ` Z ) /\ x = a ) -> ( 0g ` r ) = ( 0g ` r ) ) |
59 |
|
id |
|- ( a e. ( Base ` Z ) -> a e. ( Base ` Z ) ) |
60 |
39
|
a1i |
|- ( a e. ( Base ` Z ) -> ( 0g ` r ) e. _V ) |
61 |
57 58 59 60
|
fvmptd |
|- ( a e. ( Base ` Z ) -> ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ` a ) = ( 0g ` r ) ) |
62 |
61
|
adantl |
|- ( ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) /\ a e. ( Base ` Z ) ) -> ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ` a ) = ( 0g ` r ) ) |
63 |
47 56 62
|
3eqtr4d |
|- ( ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) /\ a e. ( Base ` Z ) ) -> ( h ` a ) = ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ` a ) ) |
64 |
38 41 63
|
eqfnfvd |
|- ( ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) /\ h : ( Base ` Z ) --> ( Base ` r ) ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) |
65 |
36 64
|
mpdan |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ h e. ( Z ( Hom ` C ) r ) ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) |
66 |
65
|
ex |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( h e. ( Z ( Hom ` C ) r ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) ) |
67 |
66
|
adantr |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) -> ( h e. ( Z ( Hom ` C ) r ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) ) |
68 |
67
|
alrimiv |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) -> A. h ( h e. ( Z ( Hom ` C ) r ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) ) |
69 |
18 31 68
|
3jca |
|- ( ( ( ph /\ r e. ( Base ` C ) ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) ) -> ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z ( Hom ` C ) r ) /\ A. h ( h e. ( Z ( Hom ` C ) r ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) ) ) |
70 |
17 69
|
mpdan |
|- ( ( ph /\ r e. ( Base ` C ) ) -> ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z ( Hom ` C ) r ) /\ A. h ( h e. ( Z ( Hom ` C ) r ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) ) ) |
71 |
|
eleq1 |
|- ( h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) -> ( h e. ( Z ( Hom ` C ) r ) <-> ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z ( Hom ` C ) r ) ) ) |
72 |
71
|
eqeu |
|- ( ( ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z RngHomo r ) /\ ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) e. ( Z ( Hom ` C ) r ) /\ A. h ( h e. ( Z ( Hom ` C ) r ) -> h = ( x e. ( Base ` Z ) |-> ( 0g ` r ) ) ) ) -> E! h h e. ( Z ( Hom ` C ) r ) ) |
73 |
70 72
|
syl |
|- ( ( ph /\ r e. ( Base ` C ) ) -> E! h h e. ( Z ( Hom ` C ) r ) ) |
74 |
73
|
ralrimiva |
|- ( ph -> A. r e. ( Base ` C ) E! h h e. ( Z ( Hom ` C ) r ) ) |
75 |
2
|
rngccat |
|- ( U e. V -> C e. Cat ) |
76 |
1 75
|
syl |
|- ( ph -> C e. Cat ) |
77 |
5 20 76 25
|
isinito |
|- ( ph -> ( Z e. ( InitO ` C ) <-> A. r e. ( Base ` C ) E! h h e. ( Z ( Hom ` C ) r ) ) ) |
78 |
74 77
|
mpbird |
|- ( ph -> Z e. ( InitO ` C ) ) |