Step |
Hyp |
Ref |
Expression |
1 |
|
goaln0 |
|- ( A.g i a e. ( Fmla ` N ) -> N =/= (/) ) |
2 |
1
|
adantl |
|- ( ( N e. _om /\ A.g i a e. ( Fmla ` N ) ) -> N =/= (/) ) |
3 |
|
nnsuc |
|- ( ( N e. _om /\ N =/= (/) ) -> E. n e. _om N = suc n ) |
4 |
|
suceq |
|- ( x = (/) -> suc x = suc (/) ) |
5 |
4
|
fveq2d |
|- ( x = (/) -> ( Fmla ` suc x ) = ( Fmla ` suc (/) ) ) |
6 |
5
|
eleq2d |
|- ( x = (/) -> ( A.g i a e. ( Fmla ` suc x ) <-> A.g i a e. ( Fmla ` suc (/) ) ) ) |
7 |
5
|
eleq2d |
|- ( x = (/) -> ( a e. ( Fmla ` suc x ) <-> a e. ( Fmla ` suc (/) ) ) ) |
8 |
6 7
|
imbi12d |
|- ( x = (/) -> ( ( A.g i a e. ( Fmla ` suc x ) -> a e. ( Fmla ` suc x ) ) <-> ( A.g i a e. ( Fmla ` suc (/) ) -> a e. ( Fmla ` suc (/) ) ) ) ) |
9 |
|
suceq |
|- ( x = y -> suc x = suc y ) |
10 |
9
|
fveq2d |
|- ( x = y -> ( Fmla ` suc x ) = ( Fmla ` suc y ) ) |
11 |
10
|
eleq2d |
|- ( x = y -> ( A.g i a e. ( Fmla ` suc x ) <-> A.g i a e. ( Fmla ` suc y ) ) ) |
12 |
10
|
eleq2d |
|- ( x = y -> ( a e. ( Fmla ` suc x ) <-> a e. ( Fmla ` suc y ) ) ) |
13 |
11 12
|
imbi12d |
|- ( x = y -> ( ( A.g i a e. ( Fmla ` suc x ) -> a e. ( Fmla ` suc x ) ) <-> ( A.g i a e. ( Fmla ` suc y ) -> a e. ( Fmla ` suc y ) ) ) ) |
14 |
|
suceq |
|- ( x = suc y -> suc x = suc suc y ) |
15 |
14
|
fveq2d |
|- ( x = suc y -> ( Fmla ` suc x ) = ( Fmla ` suc suc y ) ) |
16 |
15
|
eleq2d |
|- ( x = suc y -> ( A.g i a e. ( Fmla ` suc x ) <-> A.g i a e. ( Fmla ` suc suc y ) ) ) |
17 |
15
|
eleq2d |
|- ( x = suc y -> ( a e. ( Fmla ` suc x ) <-> a e. ( Fmla ` suc suc y ) ) ) |
18 |
16 17
|
imbi12d |
|- ( x = suc y -> ( ( A.g i a e. ( Fmla ` suc x ) -> a e. ( Fmla ` suc x ) ) <-> ( A.g i a e. ( Fmla ` suc suc y ) -> a e. ( Fmla ` suc suc y ) ) ) ) |
19 |
|
suceq |
|- ( x = n -> suc x = suc n ) |
20 |
19
|
fveq2d |
|- ( x = n -> ( Fmla ` suc x ) = ( Fmla ` suc n ) ) |
21 |
20
|
eleq2d |
|- ( x = n -> ( A.g i a e. ( Fmla ` suc x ) <-> A.g i a e. ( Fmla ` suc n ) ) ) |
22 |
20
|
eleq2d |
|- ( x = n -> ( a e. ( Fmla ` suc x ) <-> a e. ( Fmla ` suc n ) ) ) |
23 |
21 22
|
imbi12d |
|- ( x = n -> ( ( A.g i a e. ( Fmla ` suc x ) -> a e. ( Fmla ` suc x ) ) <-> ( A.g i a e. ( Fmla ` suc n ) -> a e. ( Fmla ` suc n ) ) ) ) |
24 |
|
peano1 |
|- (/) e. _om |
25 |
|
df-goal |
|- A.g i a = <. 2o , <. i , a >. >. |
26 |
|
opex |
|- <. 2o , <. i , a >. >. e. _V |
27 |
25 26
|
eqeltri |
|- A.g i a e. _V |
28 |
|
isfmlasuc |
|- ( ( (/) e. _om /\ A.g i a e. _V ) -> ( A.g i a e. ( Fmla ` suc (/) ) <-> ( A.g i a e. ( Fmla ` (/) ) \/ E. u e. ( Fmla ` (/) ) ( E. v e. ( Fmla ` (/) ) A.g i a = ( u |g v ) \/ E. k e. _om A.g i a = A.g k u ) ) ) ) |
29 |
24 27 28
|
mp2an |
|- ( A.g i a e. ( Fmla ` suc (/) ) <-> ( A.g i a e. ( Fmla ` (/) ) \/ E. u e. ( Fmla ` (/) ) ( E. v e. ( Fmla ` (/) ) A.g i a = ( u |g v ) \/ E. k e. _om A.g i a = A.g k u ) ) ) |
30 |
|
eqeq1 |
|- ( x = A.g i a -> ( x = ( k e.g j ) <-> A.g i a = ( k e.g j ) ) ) |
31 |
30
|
2rexbidv |
|- ( x = A.g i a -> ( E. k e. _om E. j e. _om x = ( k e.g j ) <-> E. k e. _om E. j e. _om A.g i a = ( k e.g j ) ) ) |
32 |
|
fmla0 |
|- ( Fmla ` (/) ) = { x e. _V | E. k e. _om E. j e. _om x = ( k e.g j ) } |
33 |
31 32
|
elrab2 |
|- ( A.g i a e. ( Fmla ` (/) ) <-> ( A.g i a e. _V /\ E. k e. _om E. j e. _om A.g i a = ( k e.g j ) ) ) |
34 |
25
|
a1i |
|- ( ( k e. _om /\ j e. _om ) -> A.g i a = <. 2o , <. i , a >. >. ) |
35 |
|
goel |
|- ( ( k e. _om /\ j e. _om ) -> ( k e.g j ) = <. (/) , <. k , j >. >. ) |
36 |
34 35
|
eqeq12d |
|- ( ( k e. _om /\ j e. _om ) -> ( A.g i a = ( k e.g j ) <-> <. 2o , <. i , a >. >. = <. (/) , <. k , j >. >. ) ) |
37 |
|
2oex |
|- 2o e. _V |
38 |
|
opex |
|- <. i , a >. e. _V |
39 |
37 38
|
opth |
|- ( <. 2o , <. i , a >. >. = <. (/) , <. k , j >. >. <-> ( 2o = (/) /\ <. i , a >. = <. k , j >. ) ) |
40 |
|
2on0 |
|- 2o =/= (/) |
41 |
|
eqneqall |
|- ( 2o = (/) -> ( 2o =/= (/) -> a e. ( Fmla ` suc (/) ) ) ) |
42 |
40 41
|
mpi |
|- ( 2o = (/) -> a e. ( Fmla ` suc (/) ) ) |
43 |
42
|
adantr |
|- ( ( 2o = (/) /\ <. i , a >. = <. k , j >. ) -> a e. ( Fmla ` suc (/) ) ) |
44 |
39 43
|
sylbi |
|- ( <. 2o , <. i , a >. >. = <. (/) , <. k , j >. >. -> a e. ( Fmla ` suc (/) ) ) |
45 |
36 44
|
syl6bi |
|- ( ( k e. _om /\ j e. _om ) -> ( A.g i a = ( k e.g j ) -> a e. ( Fmla ` suc (/) ) ) ) |
46 |
45
|
rexlimdva |
|- ( k e. _om -> ( E. j e. _om A.g i a = ( k e.g j ) -> a e. ( Fmla ` suc (/) ) ) ) |
47 |
46
|
rexlimiv |
|- ( E. k e. _om E. j e. _om A.g i a = ( k e.g j ) -> a e. ( Fmla ` suc (/) ) ) |
48 |
33 47
|
simplbiim |
|- ( A.g i a e. ( Fmla ` (/) ) -> a e. ( Fmla ` suc (/) ) ) |
49 |
|
gonanegoal |
|- ( u |g v ) =/= A.g i a |
50 |
|
eqneqall |
|- ( ( u |g v ) = A.g i a -> ( ( u |g v ) =/= A.g i a -> a e. ( Fmla ` suc (/) ) ) ) |
51 |
49 50
|
mpi |
|- ( ( u |g v ) = A.g i a -> a e. ( Fmla ` suc (/) ) ) |
52 |
51
|
eqcoms |
|- ( A.g i a = ( u |g v ) -> a e. ( Fmla ` suc (/) ) ) |
53 |
52
|
a1i |
|- ( ( u e. ( Fmla ` (/) ) /\ v e. ( Fmla ` (/) ) ) -> ( A.g i a = ( u |g v ) -> a e. ( Fmla ` suc (/) ) ) ) |
54 |
53
|
rexlimdva |
|- ( u e. ( Fmla ` (/) ) -> ( E. v e. ( Fmla ` (/) ) A.g i a = ( u |g v ) -> a e. ( Fmla ` suc (/) ) ) ) |
55 |
|
df-goal |
|- A.g k u = <. 2o , <. k , u >. >. |
56 |
25 55
|
eqeq12i |
|- ( A.g i a = A.g k u <-> <. 2o , <. i , a >. >. = <. 2o , <. k , u >. >. ) |
57 |
37 38
|
opth |
|- ( <. 2o , <. i , a >. >. = <. 2o , <. k , u >. >. <-> ( 2o = 2o /\ <. i , a >. = <. k , u >. ) ) |
58 |
|
vex |
|- i e. _V |
59 |
|
vex |
|- a e. _V |
60 |
58 59
|
opth |
|- ( <. i , a >. = <. k , u >. <-> ( i = k /\ a = u ) ) |
61 |
|
eleq1w |
|- ( u = a -> ( u e. ( Fmla ` (/) ) <-> a e. ( Fmla ` (/) ) ) ) |
62 |
|
fmlasssuc |
|- ( (/) e. _om -> ( Fmla ` (/) ) C_ ( Fmla ` suc (/) ) ) |
63 |
24 62
|
ax-mp |
|- ( Fmla ` (/) ) C_ ( Fmla ` suc (/) ) |
64 |
63
|
sseli |
|- ( a e. ( Fmla ` (/) ) -> a e. ( Fmla ` suc (/) ) ) |
65 |
61 64
|
syl6bi |
|- ( u = a -> ( u e. ( Fmla ` (/) ) -> a e. ( Fmla ` suc (/) ) ) ) |
66 |
65
|
eqcoms |
|- ( a = u -> ( u e. ( Fmla ` (/) ) -> a e. ( Fmla ` suc (/) ) ) ) |
67 |
60 66
|
simplbiim |
|- ( <. i , a >. = <. k , u >. -> ( u e. ( Fmla ` (/) ) -> a e. ( Fmla ` suc (/) ) ) ) |
68 |
57 67
|
simplbiim |
|- ( <. 2o , <. i , a >. >. = <. 2o , <. k , u >. >. -> ( u e. ( Fmla ` (/) ) -> a e. ( Fmla ` suc (/) ) ) ) |
69 |
68
|
com12 |
|- ( u e. ( Fmla ` (/) ) -> ( <. 2o , <. i , a >. >. = <. 2o , <. k , u >. >. -> a e. ( Fmla ` suc (/) ) ) ) |
70 |
69
|
adantr |
|- ( ( u e. ( Fmla ` (/) ) /\ k e. _om ) -> ( <. 2o , <. i , a >. >. = <. 2o , <. k , u >. >. -> a e. ( Fmla ` suc (/) ) ) ) |
71 |
56 70
|
syl5bi |
|- ( ( u e. ( Fmla ` (/) ) /\ k e. _om ) -> ( A.g i a = A.g k u -> a e. ( Fmla ` suc (/) ) ) ) |
72 |
71
|
rexlimdva |
|- ( u e. ( Fmla ` (/) ) -> ( E. k e. _om A.g i a = A.g k u -> a e. ( Fmla ` suc (/) ) ) ) |
73 |
54 72
|
jaod |
|- ( u e. ( Fmla ` (/) ) -> ( ( E. v e. ( Fmla ` (/) ) A.g i a = ( u |g v ) \/ E. k e. _om A.g i a = A.g k u ) -> a e. ( Fmla ` suc (/) ) ) ) |
74 |
73
|
rexlimiv |
|- ( E. u e. ( Fmla ` (/) ) ( E. v e. ( Fmla ` (/) ) A.g i a = ( u |g v ) \/ E. k e. _om A.g i a = A.g k u ) -> a e. ( Fmla ` suc (/) ) ) |
75 |
48 74
|
jaoi |
|- ( ( A.g i a e. ( Fmla ` (/) ) \/ E. u e. ( Fmla ` (/) ) ( E. v e. ( Fmla ` (/) ) A.g i a = ( u |g v ) \/ E. k e. _om A.g i a = A.g k u ) ) -> a e. ( Fmla ` suc (/) ) ) |
76 |
29 75
|
sylbi |
|- ( A.g i a e. ( Fmla ` suc (/) ) -> a e. ( Fmla ` suc (/) ) ) |
77 |
|
goalrlem |
|- ( y e. _om -> ( ( A.g i a e. ( Fmla ` suc y ) -> a e. ( Fmla ` suc y ) ) -> ( A.g i a e. ( Fmla ` suc suc y ) -> a e. ( Fmla ` suc suc y ) ) ) ) |
78 |
8 13 18 23 76 77
|
finds |
|- ( n e. _om -> ( A.g i a e. ( Fmla ` suc n ) -> a e. ( Fmla ` suc n ) ) ) |
79 |
78
|
adantr |
|- ( ( n e. _om /\ N = suc n ) -> ( A.g i a e. ( Fmla ` suc n ) -> a e. ( Fmla ` suc n ) ) ) |
80 |
|
fveq2 |
|- ( N = suc n -> ( Fmla ` N ) = ( Fmla ` suc n ) ) |
81 |
80
|
eleq2d |
|- ( N = suc n -> ( A.g i a e. ( Fmla ` N ) <-> A.g i a e. ( Fmla ` suc n ) ) ) |
82 |
80
|
eleq2d |
|- ( N = suc n -> ( a e. ( Fmla ` N ) <-> a e. ( Fmla ` suc n ) ) ) |
83 |
81 82
|
imbi12d |
|- ( N = suc n -> ( ( A.g i a e. ( Fmla ` N ) -> a e. ( Fmla ` N ) ) <-> ( A.g i a e. ( Fmla ` suc n ) -> a e. ( Fmla ` suc n ) ) ) ) |
84 |
83
|
adantl |
|- ( ( n e. _om /\ N = suc n ) -> ( ( A.g i a e. ( Fmla ` N ) -> a e. ( Fmla ` N ) ) <-> ( A.g i a e. ( Fmla ` suc n ) -> a e. ( Fmla ` suc n ) ) ) ) |
85 |
79 84
|
mpbird |
|- ( ( n e. _om /\ N = suc n ) -> ( A.g i a e. ( Fmla ` N ) -> a e. ( Fmla ` N ) ) ) |
86 |
85
|
rexlimiva |
|- ( E. n e. _om N = suc n -> ( A.g i a e. ( Fmla ` N ) -> a e. ( Fmla ` N ) ) ) |
87 |
3 86
|
syl |
|- ( ( N e. _om /\ N =/= (/) ) -> ( A.g i a e. ( Fmla ` N ) -> a e. ( Fmla ` N ) ) ) |
88 |
87
|
impancom |
|- ( ( N e. _om /\ A.g i a e. ( Fmla ` N ) ) -> ( N =/= (/) -> a e. ( Fmla ` N ) ) ) |
89 |
2 88
|
mpd |
|- ( ( N e. _om /\ A.g i a e. ( Fmla ` N ) ) -> a e. ( Fmla ` N ) ) |