Step |
Hyp |
Ref |
Expression |
0 |
|
cpfl |
|- polyFld |
1 |
|
vr |
|- r |
2 |
|
cvv |
|- _V |
3 |
|
vp |
|- p |
4 |
|
cpl1 |
|- Poly1 |
5 |
1
|
cv |
|- r |
6 |
5 4
|
cfv |
|- ( Poly1 ` r ) |
7 |
|
vs |
|- s |
8 |
|
crsp |
|- RSpan |
9 |
7
|
cv |
|- s |
10 |
9 8
|
cfv |
|- ( RSpan ` s ) |
11 |
3
|
cv |
|- p |
12 |
11
|
csn |
|- { p } |
13 |
12 10
|
cfv |
|- ( ( RSpan ` s ) ` { p } ) |
14 |
|
vi |
|- i |
15 |
|
vz |
|- z |
16 |
|
cbs |
|- Base |
17 |
5 16
|
cfv |
|- ( Base ` r ) |
18 |
15
|
cv |
|- z |
19 |
|
cvsca |
|- .s |
20 |
9 19
|
cfv |
|- ( .s ` s ) |
21 |
|
cur |
|- 1r |
22 |
9 21
|
cfv |
|- ( 1r ` s ) |
23 |
18 22 20
|
co |
|- ( z ( .s ` s ) ( 1r ` s ) ) |
24 |
|
cqg |
|- ~QG |
25 |
14
|
cv |
|- i |
26 |
9 25 24
|
co |
|- ( s ~QG i ) |
27 |
23 26
|
cec |
|- [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) |
28 |
15 17 27
|
cmpt |
|- ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) |
29 |
|
vf |
|- f |
30 |
|
cqus |
|- /s |
31 |
9 26 30
|
co |
|- ( s /s ( s ~QG i ) ) |
32 |
|
vt |
|- t |
33 |
32
|
cv |
|- t |
34 |
|
ctng |
|- toNrmGrp |
35 |
|
vn |
|- n |
36 |
|
cabv |
|- AbsVal |
37 |
33 36
|
cfv |
|- ( AbsVal ` t ) |
38 |
35
|
cv |
|- n |
39 |
29
|
cv |
|- f |
40 |
38 39
|
ccom |
|- ( n o. f ) |
41 |
|
cnm |
|- norm |
42 |
5 41
|
cfv |
|- ( norm ` r ) |
43 |
40 42
|
wceq |
|- ( n o. f ) = ( norm ` r ) |
44 |
43 35 37
|
crio |
|- ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) |
45 |
33 44 34
|
co |
|- ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) |
46 |
|
csts |
|- sSet |
47 |
|
cple |
|- le |
48 |
|
cnx |
|- ndx |
49 |
48 47
|
cfv |
|- ( le ` ndx ) |
50 |
33 16
|
cfv |
|- ( Base ` t ) |
51 |
|
vq |
|- q |
52 |
|
cdg1 |
|- deg1 |
53 |
51
|
cv |
|- q |
54 |
5 53 52
|
co |
|- ( r deg1 q ) |
55 |
|
clt |
|- < |
56 |
5 11 52
|
co |
|- ( r deg1 p ) |
57 |
54 56 55
|
wbr |
|- ( r deg1 q ) < ( r deg1 p ) |
58 |
57 51 18
|
crio |
|- ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) |
59 |
15 50 58
|
cmpt |
|- ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) |
60 |
|
vg |
|- g |
61 |
60
|
cv |
|- g |
62 |
61
|
ccnv |
|- `' g |
63 |
9 47
|
cfv |
|- ( le ` s ) |
64 |
63 61
|
ccom |
|- ( ( le ` s ) o. g ) |
65 |
62 64
|
ccom |
|- ( `' g o. ( ( le ` s ) o. g ) ) |
66 |
60 59 65
|
csb |
|- [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) |
67 |
49 66
|
cop |
|- <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. |
68 |
45 67 46
|
co |
|- ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) |
69 |
32 31 68
|
csb |
|- [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) |
70 |
69 39
|
cop |
|- <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. |
71 |
29 28 70
|
csb |
|- [_ ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. |
72 |
14 13 71
|
csb |
|- [_ ( ( RSpan ` s ) ` { p } ) / i ]_ [_ ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. |
73 |
7 6 72
|
csb |
|- [_ ( Poly1 ` r ) / s ]_ [_ ( ( RSpan ` s ) ` { p } ) / i ]_ [_ ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. |
74 |
1 3 2 2 73
|
cmpo |
|- ( r e. _V , p e. _V |-> [_ ( Poly1 ` r ) / s ]_ [_ ( ( RSpan ` s ) ` { p } ) / i ]_ [_ ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. ) |
75 |
0 74
|
wceq |
|- polyFld = ( r e. _V , p e. _V |-> [_ ( Poly1 ` r ) / s ]_ [_ ( ( RSpan ` s ) ` { p } ) / i ]_ [_ ( z e. ( Base ` r ) |-> [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , [_ ( z e. ( Base ` t ) |-> ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. ) |