Step |
Hyp |
Ref |
Expression |
1 |
|
rngurd.b |
|- ( ph -> B = ( Base ` R ) ) |
2 |
|
rngurd.p |
|- ( ph -> .x. = ( .r ` R ) ) |
3 |
|
rngurd.z |
|- ( ph -> .1. e. B ) |
4 |
|
rngurd.i |
|- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x ) |
5 |
|
rngurd.j |
|- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x ) |
6 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
7 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
8 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
9 |
6 7 8
|
dfur2 |
|- ( 1r ` R ) = ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) |
10 |
3 1
|
eleqtrd |
|- ( ph -> .1. e. ( Base ` R ) ) |
11 |
4 5
|
jca |
|- ( ( ph /\ x e. B ) -> ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) ) |
12 |
11
|
ralrimiva |
|- ( ph -> A. x e. B ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) ) |
13 |
2
|
adantr |
|- ( ( ph /\ x e. B ) -> .x. = ( .r ` R ) ) |
14 |
13
|
oveqd |
|- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = ( .1. ( .r ` R ) x ) ) |
15 |
14
|
eqeq1d |
|- ( ( ph /\ x e. B ) -> ( ( .1. .x. x ) = x <-> ( .1. ( .r ` R ) x ) = x ) ) |
16 |
13
|
oveqd |
|- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = ( x ( .r ` R ) .1. ) ) |
17 |
16
|
eqeq1d |
|- ( ( ph /\ x e. B ) -> ( ( x .x. .1. ) = x <-> ( x ( .r ` R ) .1. ) = x ) ) |
18 |
15 17
|
anbi12d |
|- ( ( ph /\ x e. B ) -> ( ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) <-> ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) ) |
19 |
1 18
|
raleqbidva |
|- ( ph -> ( A. x e. B ( ( .1. .x. x ) = x /\ ( x .x. .1. ) = x ) <-> A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) ) |
20 |
12 19
|
mpbid |
|- ( ph -> A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) |
21 |
1
|
eleq2d |
|- ( ph -> ( e e. B <-> e e. ( Base ` R ) ) ) |
22 |
13
|
oveqd |
|- ( ( ph /\ x e. B ) -> ( e .x. x ) = ( e ( .r ` R ) x ) ) |
23 |
22
|
eqeq1d |
|- ( ( ph /\ x e. B ) -> ( ( e .x. x ) = x <-> ( e ( .r ` R ) x ) = x ) ) |
24 |
13
|
oveqd |
|- ( ( ph /\ x e. B ) -> ( x .x. e ) = ( x ( .r ` R ) e ) ) |
25 |
24
|
eqeq1d |
|- ( ( ph /\ x e. B ) -> ( ( x .x. e ) = x <-> ( x ( .r ` R ) e ) = x ) ) |
26 |
23 25
|
anbi12d |
|- ( ( ph /\ x e. B ) -> ( ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) |
27 |
1 26
|
raleqbidva |
|- ( ph -> ( A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) |
28 |
21 27
|
anbi12d |
|- ( ph -> ( ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) <-> ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) ) |
29 |
4
|
ralrimiva |
|- ( ph -> A. x e. B ( .1. .x. x ) = x ) |
30 |
29
|
adantr |
|- ( ( ph /\ e e. B ) -> A. x e. B ( .1. .x. x ) = x ) |
31 |
|
simpr |
|- ( ( ph /\ e e. B ) -> e e. B ) |
32 |
|
simpr |
|- ( ( ( ph /\ e e. B ) /\ x = e ) -> x = e ) |
33 |
32
|
oveq2d |
|- ( ( ( ph /\ e e. B ) /\ x = e ) -> ( .1. .x. x ) = ( .1. .x. e ) ) |
34 |
33 32
|
eqeq12d |
|- ( ( ( ph /\ e e. B ) /\ x = e ) -> ( ( .1. .x. x ) = x <-> ( .1. .x. e ) = e ) ) |
35 |
31 34
|
rspcdv |
|- ( ( ph /\ e e. B ) -> ( A. x e. B ( .1. .x. x ) = x -> ( .1. .x. e ) = e ) ) |
36 |
30 35
|
mpd |
|- ( ( ph /\ e e. B ) -> ( .1. .x. e ) = e ) |
37 |
36
|
adantrr |
|- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> ( .1. .x. e ) = e ) |
38 |
3
|
adantr |
|- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> .1. e. B ) |
39 |
|
simprr |
|- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) |
40 |
|
oveq2 |
|- ( x = .1. -> ( e .x. x ) = ( e .x. .1. ) ) |
41 |
|
id |
|- ( x = .1. -> x = .1. ) |
42 |
40 41
|
eqeq12d |
|- ( x = .1. -> ( ( e .x. x ) = x <-> ( e .x. .1. ) = .1. ) ) |
43 |
|
oveq1 |
|- ( x = .1. -> ( x .x. e ) = ( .1. .x. e ) ) |
44 |
43 41
|
eqeq12d |
|- ( x = .1. -> ( ( x .x. e ) = x <-> ( .1. .x. e ) = .1. ) ) |
45 |
42 44
|
anbi12d |
|- ( x = .1. -> ( ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> ( ( e .x. .1. ) = .1. /\ ( .1. .x. e ) = .1. ) ) ) |
46 |
45
|
rspcva |
|- ( ( .1. e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) -> ( ( e .x. .1. ) = .1. /\ ( .1. .x. e ) = .1. ) ) |
47 |
46
|
simprd |
|- ( ( .1. e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) -> ( .1. .x. e ) = .1. ) |
48 |
38 39 47
|
syl2anc |
|- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> ( .1. .x. e ) = .1. ) |
49 |
37 48
|
eqtr3d |
|- ( ( ph /\ ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) -> e = .1. ) |
50 |
49
|
ex |
|- ( ph -> ( ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) -> e = .1. ) ) |
51 |
28 50
|
sylbird |
|- ( ph -> ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) -> e = .1. ) ) |
52 |
51
|
alrimiv |
|- ( ph -> A. e ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) -> e = .1. ) ) |
53 |
|
eleq1 |
|- ( e = .1. -> ( e e. ( Base ` R ) <-> .1. e. ( Base ` R ) ) ) |
54 |
|
oveq1 |
|- ( e = .1. -> ( e ( .r ` R ) x ) = ( .1. ( .r ` R ) x ) ) |
55 |
54
|
eqeq1d |
|- ( e = .1. -> ( ( e ( .r ` R ) x ) = x <-> ( .1. ( .r ` R ) x ) = x ) ) |
56 |
55
|
ovanraleqv |
|- ( e = .1. -> ( A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) <-> A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) ) |
57 |
53 56
|
anbi12d |
|- ( e = .1. -> ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) <-> ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) ) ) |
58 |
57
|
eqeu |
|- ( ( .1. e. ( Base ` R ) /\ ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) /\ A. e ( ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) -> e = .1. ) ) -> E! e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) |
59 |
10 10 20 52 58
|
syl121anc |
|- ( ph -> E! e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) |
60 |
57
|
iota2 |
|- ( ( .1. e. B /\ E! e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) -> ( ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) <-> ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) = .1. ) ) |
61 |
3 59 60
|
syl2anc |
|- ( ph -> ( ( .1. e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( .1. ( .r ` R ) x ) = x /\ ( x ( .r ` R ) .1. ) = x ) ) <-> ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) = .1. ) ) |
62 |
10 20 61
|
mpbi2and |
|- ( ph -> ( iota e ( e e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( e ( .r ` R ) x ) = x /\ ( x ( .r ` R ) e ) = x ) ) ) = .1. ) |
63 |
9 62
|
eqtr2id |
|- ( ph -> .1. = ( 1r ` R ) ) |