Step |
Hyp |
Ref |
Expression |
0 |
|
copws |
|- ordPwSer |
1 |
|
vi |
|- i |
2 |
|
cvv |
|- _V |
3 |
|
vs |
|- s |
4 |
|
vr |
|- r |
5 |
1
|
cv |
|- i |
6 |
5 5
|
cxp |
|- ( i X. i ) |
7 |
6
|
cpw |
|- ~P ( i X. i ) |
8 |
|
cmps |
|- mPwSer |
9 |
3
|
cv |
|- s |
10 |
5 9 8
|
co |
|- ( i mPwSer s ) |
11 |
|
vp |
|- p |
12 |
11
|
cv |
|- p |
13 |
|
csts |
|- sSet |
14 |
|
cple |
|- le |
15 |
|
cnx |
|- ndx |
16 |
15 14
|
cfv |
|- ( le ` ndx ) |
17 |
|
vx |
|- x |
18 |
|
vy |
|- y |
19 |
17
|
cv |
|- x |
20 |
18
|
cv |
|- y |
21 |
19 20
|
cpr |
|- { x , y } |
22 |
|
cbs |
|- Base |
23 |
12 22
|
cfv |
|- ( Base ` p ) |
24 |
21 23
|
wss |
|- { x , y } C_ ( Base ` p ) |
25 |
|
vh |
|- h |
26 |
|
cn0 |
|- NN0 |
27 |
|
cmap |
|- ^m |
28 |
26 5 27
|
co |
|- ( NN0 ^m i ) |
29 |
25
|
cv |
|- h |
30 |
29
|
ccnv |
|- `' h |
31 |
|
cn |
|- NN |
32 |
30 31
|
cima |
|- ( `' h " NN ) |
33 |
|
cfn |
|- Fin |
34 |
32 33
|
wcel |
|- ( `' h " NN ) e. Fin |
35 |
34 25 28
|
crab |
|- { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } |
36 |
|
vd |
|- d |
37 |
|
vz |
|- z |
38 |
36
|
cv |
|- d |
39 |
37
|
cv |
|- z |
40 |
39 19
|
cfv |
|- ( x ` z ) |
41 |
|
cplt |
|- lt |
42 |
9 41
|
cfv |
|- ( lt ` s ) |
43 |
39 20
|
cfv |
|- ( y ` z ) |
44 |
40 43 42
|
wbr |
|- ( x ` z ) ( lt ` s ) ( y ` z ) |
45 |
|
vw |
|- w |
46 |
45
|
cv |
|- w |
47 |
4
|
cv |
|- r |
48 |
|
cltb |
|- |
49 |
47 5 48
|
co |
|- ( r |
50 |
46 39 49
|
wbr |
|- w ( r |
51 |
46 19
|
cfv |
|- ( x ` w ) |
52 |
46 20
|
cfv |
|- ( y ` w ) |
53 |
51 52
|
wceq |
|- ( x ` w ) = ( y ` w ) |
54 |
50 53
|
wi |
|- ( w ( r ( x ` w ) = ( y ` w ) ) |
55 |
54 45 38
|
wral |
|- A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) |
56 |
44 55
|
wa |
|- ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) |
57 |
56 37 38
|
wrex |
|- E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) |
58 |
57 36 35
|
wsbc |
|- [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) |
59 |
19 20
|
wceq |
|- x = y |
60 |
58 59
|
wo |
|- ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) |
61 |
24 60
|
wa |
|- ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) |
62 |
61 17 18
|
copab |
|- { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } |
63 |
16 62
|
cop |
|- <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. |
64 |
12 63 13
|
co |
|- ( p sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) |
65 |
11 10 64
|
csb |
|- [_ ( i mPwSer s ) / p ]_ ( p sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) |
66 |
4 7 65
|
cmpt |
|- ( r e. ~P ( i X. i ) |-> [_ ( i mPwSer s ) / p ]_ ( p sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) ) |
67 |
1 3 2 2 66
|
cmpo |
|- ( i e. _V , s e. _V |-> ( r e. ~P ( i X. i ) |-> [_ ( i mPwSer s ) / p ]_ ( p sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) ) ) |
68 |
0 67
|
wceq |
|- ordPwSer = ( i e. _V , s e. _V |-> ( r e. ~P ( i X. i ) |-> [_ ( i mPwSer s ) / p ]_ ( p sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ ( Base ` p ) /\ ( [. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]. E. z e. d ( ( x ` z ) ( lt ` s ) ( y ` z ) /\ A. w e. d ( w ( r ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) ) ) |