Step |
Hyp |
Ref |
Expression |
1 |
|
opnvonmbllem2.x |
|- ( ph -> X e. Fin ) |
2 |
|
opnvonmbllem2.n |
|- S = dom ( voln ` X ) |
3 |
|
opnvonmbllem2.g |
|- ( ph -> G e. ( TopOpen ` ( RR^ ` X ) ) ) |
4 |
|
opnvonmbl.k |
|- K = { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } |
5 |
|
eqid |
|- ( dist ` ( RR^ ` X ) ) = ( dist ` ( RR^ ` X ) ) |
6 |
5
|
rrxmetfi |
|- ( X e. Fin -> ( dist ` ( RR^ ` X ) ) e. ( Met ` ( RR ^m X ) ) ) |
7 |
1 6
|
syl |
|- ( ph -> ( dist ` ( RR^ ` X ) ) e. ( Met ` ( RR ^m X ) ) ) |
8 |
|
metxmet |
|- ( ( dist ` ( RR^ ` X ) ) e. ( Met ` ( RR ^m X ) ) -> ( dist ` ( RR^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) ) |
9 |
7 8
|
syl |
|- ( ph -> ( dist ` ( RR^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) ) |
10 |
9
|
adantr |
|- ( ( ph /\ x e. G ) -> ( dist ` ( RR^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) ) |
11 |
|
eqid |
|- ( RR^ ` X ) = ( RR^ ` X ) |
12 |
11
|
rrxval |
|- ( X e. Fin -> ( RR^ ` X ) = ( toCPreHil ` ( RRfld freeLMod X ) ) ) |
13 |
1 12
|
syl |
|- ( ph -> ( RR^ ` X ) = ( toCPreHil ` ( RRfld freeLMod X ) ) ) |
14 |
13
|
fveq2d |
|- ( ph -> ( TopOpen ` ( RR^ ` X ) ) = ( TopOpen ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) ) |
15 |
|
ovex |
|- ( RRfld freeLMod X ) e. _V |
16 |
|
eqid |
|- ( toCPreHil ` ( RRfld freeLMod X ) ) = ( toCPreHil ` ( RRfld freeLMod X ) ) |
17 |
|
eqid |
|- ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) = ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) |
18 |
|
eqid |
|- ( TopOpen ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) = ( TopOpen ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) |
19 |
16 17 18
|
tcphtopn |
|- ( ( RRfld freeLMod X ) e. _V -> ( TopOpen ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) = ( MetOpen ` ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) ) ) |
20 |
15 19
|
ax-mp |
|- ( TopOpen ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) = ( MetOpen ` ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) ) |
21 |
20
|
a1i |
|- ( ph -> ( TopOpen ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) = ( MetOpen ` ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) ) ) |
22 |
13
|
eqcomd |
|- ( ph -> ( toCPreHil ` ( RRfld freeLMod X ) ) = ( RR^ ` X ) ) |
23 |
22
|
fveq2d |
|- ( ph -> ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) = ( dist ` ( RR^ ` X ) ) ) |
24 |
23
|
fveq2d |
|- ( ph -> ( MetOpen ` ( dist ` ( toCPreHil ` ( RRfld freeLMod X ) ) ) ) = ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) ) |
25 |
14 21 24
|
3eqtrd |
|- ( ph -> ( TopOpen ` ( RR^ ` X ) ) = ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) ) |
26 |
3 25
|
eleqtrd |
|- ( ph -> G e. ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) ) |
27 |
26
|
adantr |
|- ( ( ph /\ x e. G ) -> G e. ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) ) |
28 |
|
simpr |
|- ( ( ph /\ x e. G ) -> x e. G ) |
29 |
|
eqid |
|- ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) = ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) |
30 |
29
|
mopni2 |
|- ( ( ( dist ` ( RR^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) /\ G e. ( MetOpen ` ( dist ` ( RR^ ` X ) ) ) /\ x e. G ) -> E. e e. RR+ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) |
31 |
10 27 28 30
|
syl3anc |
|- ( ( ph /\ x e. G ) -> E. e e. RR+ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) |
32 |
1
|
ad2antrr |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> X e. Fin ) |
33 |
|
eqid |
|- ( TopOpen ` ( RR^ ` X ) ) = ( TopOpen ` ( RR^ ` X ) ) |
34 |
33
|
rrxtoponfi |
|- ( X e. Fin -> ( TopOpen ` ( RR^ ` X ) ) e. ( TopOn ` ( RR ^m X ) ) ) |
35 |
1 34
|
syl |
|- ( ph -> ( TopOpen ` ( RR^ ` X ) ) e. ( TopOn ` ( RR ^m X ) ) ) |
36 |
|
toponss |
|- ( ( ( TopOpen ` ( RR^ ` X ) ) e. ( TopOn ` ( RR ^m X ) ) /\ G e. ( TopOpen ` ( RR^ ` X ) ) ) -> G C_ ( RR ^m X ) ) |
37 |
35 3 36
|
syl2anc |
|- ( ph -> G C_ ( RR ^m X ) ) |
38 |
37
|
adantr |
|- ( ( ph /\ x e. G ) -> G C_ ( RR ^m X ) ) |
39 |
38 28
|
sseldd |
|- ( ( ph /\ x e. G ) -> x e. ( RR ^m X ) ) |
40 |
39
|
adantr |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> x e. ( RR ^m X ) ) |
41 |
|
simpr |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> e e. RR+ ) |
42 |
32 40 41
|
hoiqssbl |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> E. c e. ( QQ ^m X ) E. d e. ( QQ ^m X ) ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) |
43 |
42
|
3adant3 |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> E. c e. ( QQ ^m X ) E. d e. ( QQ ^m X ) ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) |
44 |
|
nfv |
|- F/ i ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) |
45 |
|
nfv |
|- F/ i ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) |
46 |
|
nfcv |
|- F/_ i x |
47 |
|
nfixp1 |
|- F/_ i X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) |
48 |
46 47
|
nfel |
|- F/ i x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) |
49 |
|
nfcv |
|- F/_ i ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) |
50 |
47 49
|
nfss |
|- F/ i X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) |
51 |
48 50
|
nfan |
|- F/ i ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) |
52 |
44 45 51
|
nf3an |
|- F/ i ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) |
53 |
1
|
adantr |
|- ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> X e. Fin ) |
54 |
53
|
3ad2ant1 |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> X e. Fin ) |
55 |
|
elmapi |
|- ( c e. ( QQ ^m X ) -> c : X --> QQ ) |
56 |
55
|
adantr |
|- ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> c : X --> QQ ) |
57 |
56
|
3ad2ant2 |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> c : X --> QQ ) |
58 |
|
elmapi |
|- ( d e. ( QQ ^m X ) -> d : X --> QQ ) |
59 |
58
|
adantl |
|- ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> d : X --> QQ ) |
60 |
59
|
3ad2ant2 |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> d : X --> QQ ) |
61 |
|
simp3r |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) |
62 |
|
simp1r |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) |
63 |
|
simp3l |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) ) |
64 |
|
eqid |
|- ( i e. X |-> <. ( c ` i ) , ( d ` i ) >. ) = ( i e. X |-> <. ( c ` i ) , ( d ` i ) >. ) |
65 |
52 54 57 60 61 62 63 4 64
|
opnvonmbllem1 |
|- ( ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) |
66 |
65
|
3exp |
|- ( ( ph /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> ( ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) ) |
67 |
66
|
adantlr |
|- ( ( ( ph /\ x e. G ) /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> ( ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) ) |
68 |
67
|
3adant2 |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> ( ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) ) |
69 |
68
|
rexlimdvv |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> ( E. c e. ( QQ ^m X ) E. d e. ( QQ ^m X ) ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) |
70 |
43 69
|
mpd |
|- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) |
71 |
70
|
3exp |
|- ( ( ph /\ x e. G ) -> ( e e. RR+ -> ( ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) ) |
72 |
71
|
rexlimdv |
|- ( ( ph /\ x e. G ) -> ( E. e e. RR+ ( x ( ball ` ( dist ` ( RR^ ` X ) ) ) e ) C_ G -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) |
73 |
31 72
|
mpd |
|- ( ( ph /\ x e. G ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) |
74 |
|
eliun |
|- ( x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) <-> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) |
75 |
73 74
|
sylibr |
|- ( ( ph /\ x e. G ) -> x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) ) |
76 |
75
|
ralrimiva |
|- ( ph -> A. x e. G x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) ) |
77 |
|
dfss3 |
|- ( G C_ U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) <-> A. x e. G x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) ) |
78 |
76 77
|
sylibr |
|- ( ph -> G C_ U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) ) |
79 |
4
|
eleq2i |
|- ( h e. K <-> h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } ) |
80 |
79
|
biimpi |
|- ( h e. K -> h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } ) |
81 |
80
|
adantl |
|- ( ( ph /\ h e. K ) -> h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } ) |
82 |
|
rabid |
|- ( h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } <-> ( h e. ( ( QQ X. QQ ) ^m X ) /\ X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) ) |
83 |
81 82
|
sylib |
|- ( ( ph /\ h e. K ) -> ( h e. ( ( QQ X. QQ ) ^m X ) /\ X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) ) |
84 |
83
|
simprd |
|- ( ( ph /\ h e. K ) -> X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) |
85 |
84
|
ralrimiva |
|- ( ph -> A. h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) |
86 |
|
iunss |
|- ( U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G <-> A. h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) |
87 |
85 86
|
sylibr |
|- ( ph -> U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) |
88 |
78 87
|
eqssd |
|- ( ph -> G = U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) ) |
89 |
1 2
|
dmovnsal |
|- ( ph -> S e. SAlg ) |
90 |
|
ssrab2 |
|- { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } C_ ( ( QQ X. QQ ) ^m X ) |
91 |
4 90
|
eqsstri |
|- K C_ ( ( QQ X. QQ ) ^m X ) |
92 |
91
|
a1i |
|- ( ph -> K C_ ( ( QQ X. QQ ) ^m X ) ) |
93 |
|
qct |
|- QQ ~<_ _om |
94 |
93
|
a1i |
|- ( ph -> QQ ~<_ _om ) |
95 |
|
xpct |
|- ( ( QQ ~<_ _om /\ QQ ~<_ _om ) -> ( QQ X. QQ ) ~<_ _om ) |
96 |
94 94 95
|
syl2anc |
|- ( ph -> ( QQ X. QQ ) ~<_ _om ) |
97 |
96 1
|
mpct |
|- ( ph -> ( ( QQ X. QQ ) ^m X ) ~<_ _om ) |
98 |
|
ssct |
|- ( ( K C_ ( ( QQ X. QQ ) ^m X ) /\ ( ( QQ X. QQ ) ^m X ) ~<_ _om ) -> K ~<_ _om ) |
99 |
92 97 98
|
syl2anc |
|- ( ph -> K ~<_ _om ) |
100 |
|
reex |
|- RR e. _V |
101 |
100 100
|
xpex |
|- ( RR X. RR ) e. _V |
102 |
|
qssre |
|- QQ C_ RR |
103 |
|
xpss12 |
|- ( ( QQ C_ RR /\ QQ C_ RR ) -> ( QQ X. QQ ) C_ ( RR X. RR ) ) |
104 |
102 102 103
|
mp2an |
|- ( QQ X. QQ ) C_ ( RR X. RR ) |
105 |
|
mapss |
|- ( ( ( RR X. RR ) e. _V /\ ( QQ X. QQ ) C_ ( RR X. RR ) ) -> ( ( QQ X. QQ ) ^m X ) C_ ( ( RR X. RR ) ^m X ) ) |
106 |
101 104 105
|
mp2an |
|- ( ( QQ X. QQ ) ^m X ) C_ ( ( RR X. RR ) ^m X ) |
107 |
91
|
sseli |
|- ( h e. K -> h e. ( ( QQ X. QQ ) ^m X ) ) |
108 |
106 107
|
sselid |
|- ( h e. K -> h e. ( ( RR X. RR ) ^m X ) ) |
109 |
|
elmapi |
|- ( h e. ( ( RR X. RR ) ^m X ) -> h : X --> ( RR X. RR ) ) |
110 |
108 109
|
syl |
|- ( h e. K -> h : X --> ( RR X. RR ) ) |
111 |
110
|
adantl |
|- ( ( ph /\ h e. K ) -> h : X --> ( RR X. RR ) ) |
112 |
|
2fveq3 |
|- ( k = i -> ( 1st ` ( h ` k ) ) = ( 1st ` ( h ` i ) ) ) |
113 |
112
|
cbvmptv |
|- ( k e. X |-> ( 1st ` ( h ` k ) ) ) = ( i e. X |-> ( 1st ` ( h ` i ) ) ) |
114 |
|
2fveq3 |
|- ( k = i -> ( 2nd ` ( h ` k ) ) = ( 2nd ` ( h ` i ) ) ) |
115 |
114
|
cbvmptv |
|- ( k e. X |-> ( 2nd ` ( h ` k ) ) ) = ( i e. X |-> ( 2nd ` ( h ` i ) ) ) |
116 |
111 113 115
|
hoicoto2 |
|- ( ( ph /\ h e. K ) -> X_ i e. X ( ( [,) o. h ) ` i ) = X_ i e. X ( ( ( k e. X |-> ( 1st ` ( h ` k ) ) ) ` i ) [,) ( ( k e. X |-> ( 2nd ` ( h ` k ) ) ) ` i ) ) ) |
117 |
1
|
adantr |
|- ( ( ph /\ h e. K ) -> X e. Fin ) |
118 |
111
|
ffvelrnda |
|- ( ( ( ph /\ h e. K ) /\ k e. X ) -> ( h ` k ) e. ( RR X. RR ) ) |
119 |
|
xp1st |
|- ( ( h ` k ) e. ( RR X. RR ) -> ( 1st ` ( h ` k ) ) e. RR ) |
120 |
118 119
|
syl |
|- ( ( ( ph /\ h e. K ) /\ k e. X ) -> ( 1st ` ( h ` k ) ) e. RR ) |
121 |
120
|
fmpttd |
|- ( ( ph /\ h e. K ) -> ( k e. X |-> ( 1st ` ( h ` k ) ) ) : X --> RR ) |
122 |
|
xp2nd |
|- ( ( h ` k ) e. ( RR X. RR ) -> ( 2nd ` ( h ` k ) ) e. RR ) |
123 |
118 122
|
syl |
|- ( ( ( ph /\ h e. K ) /\ k e. X ) -> ( 2nd ` ( h ` k ) ) e. RR ) |
124 |
123
|
fmpttd |
|- ( ( ph /\ h e. K ) -> ( k e. X |-> ( 2nd ` ( h ` k ) ) ) : X --> RR ) |
125 |
117 2 121 124
|
hoimbl |
|- ( ( ph /\ h e. K ) -> X_ i e. X ( ( ( k e. X |-> ( 1st ` ( h ` k ) ) ) ` i ) [,) ( ( k e. X |-> ( 2nd ` ( h ` k ) ) ) ` i ) ) e. S ) |
126 |
116 125
|
eqeltrd |
|- ( ( ph /\ h e. K ) -> X_ i e. X ( ( [,) o. h ) ` i ) e. S ) |
127 |
89 99 126
|
saliuncl |
|- ( ph -> U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) e. S ) |
128 |
88 127
|
eqeltrd |
|- ( ph -> G e. S ) |