Step |
Hyp |
Ref |
Expression |
1 |
|
cantnfs.s |
|- S = dom ( A CNF B ) |
2 |
|
cantnfs.a |
|- ( ph -> A e. On ) |
3 |
|
cantnfs.b |
|- ( ph -> B e. On ) |
4 |
|
oemapval.t |
|- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) } |
5 |
|
cantnf.c |
|- ( ph -> C e. ( A ^o B ) ) |
6 |
|
cantnf.s |
|- ( ph -> C C_ ran ( A CNF B ) ) |
7 |
|
cantnf.e |
|- ( ph -> (/) e. C ) |
8 |
|
cantnf.x |
|- X = U. |^| { c e. On | C e. ( A ^o c ) } |
9 |
|
cantnf.p |
|- P = ( iota d E. a e. On E. b e. ( A ^o X ) ( d = <. a , b >. /\ ( ( ( A ^o X ) .o a ) +o b ) = C ) ) |
10 |
|
cantnf.y |
|- Y = ( 1st ` P ) |
11 |
|
cantnf.z |
|- Z = ( 2nd ` P ) |
12 |
|
cantnf.g |
|- ( ph -> G e. S ) |
13 |
|
cantnf.v |
|- ( ph -> ( ( A CNF B ) ` G ) = Z ) |
14 |
|
cantnf.f |
|- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) ) |
15 |
1 2 3 4 5 6 7
|
cantnflem2 |
|- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) ) |
16 |
|
eqid |
|- X = X |
17 |
|
eqid |
|- Y = Y |
18 |
|
eqid |
|- Z = Z |
19 |
16 17 18
|
3pm3.2i |
|- ( X = X /\ Y = Y /\ Z = Z ) |
20 |
8 9 10 11
|
oeeui |
|- ( ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) -> ( ( ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) /\ ( ( ( A ^o X ) .o Y ) +o Z ) = C ) <-> ( X = X /\ Y = Y /\ Z = Z ) ) ) |
21 |
19 20
|
mpbiri |
|- ( ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) -> ( ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) /\ ( ( ( A ^o X ) .o Y ) +o Z ) = C ) ) |
22 |
15 21
|
syl |
|- ( ph -> ( ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) /\ ( ( ( A ^o X ) .o Y ) +o Z ) = C ) ) |
23 |
22
|
simpld |
|- ( ph -> ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) ) |
24 |
23
|
simp1d |
|- ( ph -> X e. On ) |
25 |
|
oecl |
|- ( ( A e. On /\ X e. On ) -> ( A ^o X ) e. On ) |
26 |
2 24 25
|
syl2anc |
|- ( ph -> ( A ^o X ) e. On ) |
27 |
23
|
simp2d |
|- ( ph -> Y e. ( A \ 1o ) ) |
28 |
27
|
eldifad |
|- ( ph -> Y e. A ) |
29 |
|
onelon |
|- ( ( A e. On /\ Y e. A ) -> Y e. On ) |
30 |
2 28 29
|
syl2anc |
|- ( ph -> Y e. On ) |
31 |
|
dif1o |
|- ( Y e. ( A \ 1o ) <-> ( Y e. A /\ Y =/= (/) ) ) |
32 |
31
|
simprbi |
|- ( Y e. ( A \ 1o ) -> Y =/= (/) ) |
33 |
27 32
|
syl |
|- ( ph -> Y =/= (/) ) |
34 |
|
on0eln0 |
|- ( Y e. On -> ( (/) e. Y <-> Y =/= (/) ) ) |
35 |
30 34
|
syl |
|- ( ph -> ( (/) e. Y <-> Y =/= (/) ) ) |
36 |
33 35
|
mpbird |
|- ( ph -> (/) e. Y ) |
37 |
|
omword1 |
|- ( ( ( ( A ^o X ) e. On /\ Y e. On ) /\ (/) e. Y ) -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) ) |
38 |
26 30 36 37
|
syl21anc |
|- ( ph -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) ) |
39 |
|
omcl |
|- ( ( ( A ^o X ) e. On /\ Y e. On ) -> ( ( A ^o X ) .o Y ) e. On ) |
40 |
26 30 39
|
syl2anc |
|- ( ph -> ( ( A ^o X ) .o Y ) e. On ) |
41 |
23
|
simp3d |
|- ( ph -> Z e. ( A ^o X ) ) |
42 |
|
onelon |
|- ( ( ( A ^o X ) e. On /\ Z e. ( A ^o X ) ) -> Z e. On ) |
43 |
26 41 42
|
syl2anc |
|- ( ph -> Z e. On ) |
44 |
|
oaword1 |
|- ( ( ( ( A ^o X ) .o Y ) e. On /\ Z e. On ) -> ( ( A ^o X ) .o Y ) C_ ( ( ( A ^o X ) .o Y ) +o Z ) ) |
45 |
40 43 44
|
syl2anc |
|- ( ph -> ( ( A ^o X ) .o Y ) C_ ( ( ( A ^o X ) .o Y ) +o Z ) ) |
46 |
22
|
simprd |
|- ( ph -> ( ( ( A ^o X ) .o Y ) +o Z ) = C ) |
47 |
45 46
|
sseqtrd |
|- ( ph -> ( ( A ^o X ) .o Y ) C_ C ) |
48 |
38 47
|
sstrd |
|- ( ph -> ( A ^o X ) C_ C ) |
49 |
|
oecl |
|- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On ) |
50 |
2 3 49
|
syl2anc |
|- ( ph -> ( A ^o B ) e. On ) |
51 |
|
ontr2 |
|- ( ( ( A ^o X ) e. On /\ ( A ^o B ) e. On ) -> ( ( ( A ^o X ) C_ C /\ C e. ( A ^o B ) ) -> ( A ^o X ) e. ( A ^o B ) ) ) |
52 |
26 50 51
|
syl2anc |
|- ( ph -> ( ( ( A ^o X ) C_ C /\ C e. ( A ^o B ) ) -> ( A ^o X ) e. ( A ^o B ) ) ) |
53 |
48 5 52
|
mp2and |
|- ( ph -> ( A ^o X ) e. ( A ^o B ) ) |
54 |
15
|
simpld |
|- ( ph -> A e. ( On \ 2o ) ) |
55 |
|
oeord |
|- ( ( X e. On /\ B e. On /\ A e. ( On \ 2o ) ) -> ( X e. B <-> ( A ^o X ) e. ( A ^o B ) ) ) |
56 |
24 3 54 55
|
syl3anc |
|- ( ph -> ( X e. B <-> ( A ^o X ) e. ( A ^o B ) ) ) |
57 |
53 56
|
mpbird |
|- ( ph -> X e. B ) |
58 |
2
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> A e. On ) |
59 |
3
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> B e. On ) |
60 |
|
suppssdm |
|- ( G supp (/) ) C_ dom G |
61 |
1 2 3
|
cantnfs |
|- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) ) |
62 |
12 61
|
mpbid |
|- ( ph -> ( G : B --> A /\ G finSupp (/) ) ) |
63 |
62
|
simpld |
|- ( ph -> G : B --> A ) |
64 |
60 63
|
fssdm |
|- ( ph -> ( G supp (/) ) C_ B ) |
65 |
64
|
sselda |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> x e. B ) |
66 |
|
onelon |
|- ( ( B e. On /\ x e. B ) -> x e. On ) |
67 |
59 65 66
|
syl2anc |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> x e. On ) |
68 |
|
oecl |
|- ( ( A e. On /\ x e. On ) -> ( A ^o x ) e. On ) |
69 |
58 67 68
|
syl2anc |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( A ^o x ) e. On ) |
70 |
63
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> G : B --> A ) |
71 |
70 65
|
ffvelrnd |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( G ` x ) e. A ) |
72 |
|
onelon |
|- ( ( A e. On /\ ( G ` x ) e. A ) -> ( G ` x ) e. On ) |
73 |
58 71 72
|
syl2anc |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( G ` x ) e. On ) |
74 |
63
|
ffnd |
|- ( ph -> G Fn B ) |
75 |
7
|
elexd |
|- ( ph -> (/) e. _V ) |
76 |
|
elsuppfn |
|- ( ( G Fn B /\ B e. On /\ (/) e. _V ) -> ( x e. ( G supp (/) ) <-> ( x e. B /\ ( G ` x ) =/= (/) ) ) ) |
77 |
74 3 75 76
|
syl3anc |
|- ( ph -> ( x e. ( G supp (/) ) <-> ( x e. B /\ ( G ` x ) =/= (/) ) ) ) |
78 |
77
|
simplbda |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( G ` x ) =/= (/) ) |
79 |
|
on0eln0 |
|- ( ( G ` x ) e. On -> ( (/) e. ( G ` x ) <-> ( G ` x ) =/= (/) ) ) |
80 |
73 79
|
syl |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( (/) e. ( G ` x ) <-> ( G ` x ) =/= (/) ) ) |
81 |
78 80
|
mpbird |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> (/) e. ( G ` x ) ) |
82 |
|
omword1 |
|- ( ( ( ( A ^o x ) e. On /\ ( G ` x ) e. On ) /\ (/) e. ( G ` x ) ) -> ( A ^o x ) C_ ( ( A ^o x ) .o ( G ` x ) ) ) |
83 |
69 73 81 82
|
syl21anc |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( A ^o x ) C_ ( ( A ^o x ) .o ( G ` x ) ) ) |
84 |
|
eqid |
|- OrdIso ( _E , ( G supp (/) ) ) = OrdIso ( _E , ( G supp (/) ) ) |
85 |
12
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> G e. S ) |
86 |
|
eqid |
|- seqom ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( OrdIso ( _E , ( G supp (/) ) ) ` k ) ) .o ( G ` ( OrdIso ( _E , ( G supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) = seqom ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( OrdIso ( _E , ( G supp (/) ) ) ` k ) ) .o ( G ` ( OrdIso ( _E , ( G supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) |
87 |
1 58 59 84 85 86 65
|
cantnfle |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( ( A ^o x ) .o ( G ` x ) ) C_ ( ( A CNF B ) ` G ) ) |
88 |
13
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( ( A CNF B ) ` G ) = Z ) |
89 |
87 88
|
sseqtrd |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( ( A ^o x ) .o ( G ` x ) ) C_ Z ) |
90 |
83 89
|
sstrd |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( A ^o x ) C_ Z ) |
91 |
41
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> Z e. ( A ^o X ) ) |
92 |
26
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( A ^o X ) e. On ) |
93 |
|
ontr2 |
|- ( ( ( A ^o x ) e. On /\ ( A ^o X ) e. On ) -> ( ( ( A ^o x ) C_ Z /\ Z e. ( A ^o X ) ) -> ( A ^o x ) e. ( A ^o X ) ) ) |
94 |
69 92 93
|
syl2anc |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( ( ( A ^o x ) C_ Z /\ Z e. ( A ^o X ) ) -> ( A ^o x ) e. ( A ^o X ) ) ) |
95 |
90 91 94
|
mp2and |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( A ^o x ) e. ( A ^o X ) ) |
96 |
24
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> X e. On ) |
97 |
54
|
adantr |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> A e. ( On \ 2o ) ) |
98 |
|
oeord |
|- ( ( x e. On /\ X e. On /\ A e. ( On \ 2o ) ) -> ( x e. X <-> ( A ^o x ) e. ( A ^o X ) ) ) |
99 |
67 96 97 98
|
syl3anc |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> ( x e. X <-> ( A ^o x ) e. ( A ^o X ) ) ) |
100 |
95 99
|
mpbird |
|- ( ( ph /\ x e. ( G supp (/) ) ) -> x e. X ) |
101 |
100
|
ex |
|- ( ph -> ( x e. ( G supp (/) ) -> x e. X ) ) |
102 |
101
|
ssrdv |
|- ( ph -> ( G supp (/) ) C_ X ) |
103 |
1 2 3 12 57 28 102 14
|
cantnfp1 |
|- ( ph -> ( F e. S /\ ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) ) |
104 |
103
|
simprd |
|- ( ph -> ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) |
105 |
13
|
oveq2d |
|- ( ph -> ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) = ( ( ( A ^o X ) .o Y ) +o Z ) ) |
106 |
104 105 46
|
3eqtrd |
|- ( ph -> ( ( A CNF B ) ` F ) = C ) |
107 |
1 2 3
|
cantnff |
|- ( ph -> ( A CNF B ) : S --> ( A ^o B ) ) |
108 |
107
|
ffnd |
|- ( ph -> ( A CNF B ) Fn S ) |
109 |
103
|
simpld |
|- ( ph -> F e. S ) |
110 |
|
fnfvelrn |
|- ( ( ( A CNF B ) Fn S /\ F e. S ) -> ( ( A CNF B ) ` F ) e. ran ( A CNF B ) ) |
111 |
108 109 110
|
syl2anc |
|- ( ph -> ( ( A CNF B ) ` F ) e. ran ( A CNF B ) ) |
112 |
106 111
|
eqeltrrd |
|- ( ph -> C e. ran ( A CNF B ) ) |