Step |
Hyp |
Ref |
Expression |
1 |
|
hgt750lemg.f |
|- F = ( c e. R |-> ( c o. T ) ) |
2 |
|
hgt750lemg.t |
|- ( ph -> T : ( 0 ..^ 3 ) -1-1-onto-> ( 0 ..^ 3 ) ) |
3 |
|
hgt750lemg.n |
|- ( ph -> N : ( 0 ..^ 3 ) --> NN ) |
4 |
|
hgt750lemg.l |
|- ( ph -> L : NN --> RR ) |
5 |
|
hgt750lemg.1 |
|- ( ph -> N e. R ) |
6 |
|
2fveq3 |
|- ( a = ( T ` b ) -> ( L ` ( N ` a ) ) = ( L ` ( N ` ( T ` b ) ) ) ) |
7 |
|
tpfi |
|- { 0 , 1 , 2 } e. Fin |
8 |
7
|
a1i |
|- ( ph -> { 0 , 1 , 2 } e. Fin ) |
9 |
|
fzo0to3tp |
|- ( 0 ..^ 3 ) = { 0 , 1 , 2 } |
10 |
|
f1oeq23 |
|- ( ( ( 0 ..^ 3 ) = { 0 , 1 , 2 } /\ ( 0 ..^ 3 ) = { 0 , 1 , 2 } ) -> ( T : ( 0 ..^ 3 ) -1-1-onto-> ( 0 ..^ 3 ) <-> T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } ) ) |
11 |
9 9 10
|
mp2an |
|- ( T : ( 0 ..^ 3 ) -1-1-onto-> ( 0 ..^ 3 ) <-> T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } ) |
12 |
2 11
|
sylib |
|- ( ph -> T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } ) |
13 |
|
eqidd |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( T ` b ) = ( T ` b ) ) |
14 |
4
|
adantr |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> L : NN --> RR ) |
15 |
3
|
adantr |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> N : ( 0 ..^ 3 ) --> NN ) |
16 |
|
simpr |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> a e. { 0 , 1 , 2 } ) |
17 |
16 9
|
eleqtrrdi |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> a e. ( 0 ..^ 3 ) ) |
18 |
15 17
|
ffvelrnd |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> ( N ` a ) e. NN ) |
19 |
14 18
|
ffvelrnd |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> ( L ` ( N ` a ) ) e. RR ) |
20 |
19
|
recnd |
|- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> ( L ` ( N ` a ) ) e. CC ) |
21 |
6 8 12 13 20
|
fprodf1o |
|- ( ph -> prod_ a e. { 0 , 1 , 2 } ( L ` ( N ` a ) ) = prod_ b e. { 0 , 1 , 2 } ( L ` ( N ` ( T ` b ) ) ) ) |
22 |
1
|
a1i |
|- ( ph -> F = ( c e. R |-> ( c o. T ) ) ) |
23 |
|
simpr |
|- ( ( ph /\ c = N ) -> c = N ) |
24 |
23
|
coeq1d |
|- ( ( ph /\ c = N ) -> ( c o. T ) = ( N o. T ) ) |
25 |
|
f1of |
|- ( T : ( 0 ..^ 3 ) -1-1-onto-> ( 0 ..^ 3 ) -> T : ( 0 ..^ 3 ) --> ( 0 ..^ 3 ) ) |
26 |
2 25
|
syl |
|- ( ph -> T : ( 0 ..^ 3 ) --> ( 0 ..^ 3 ) ) |
27 |
|
ovexd |
|- ( ph -> ( 0 ..^ 3 ) e. _V ) |
28 |
26 27
|
fexd |
|- ( ph -> T e. _V ) |
29 |
|
coexg |
|- ( ( N e. R /\ T e. _V ) -> ( N o. T ) e. _V ) |
30 |
5 28 29
|
syl2anc |
|- ( ph -> ( N o. T ) e. _V ) |
31 |
22 24 5 30
|
fvmptd |
|- ( ph -> ( F ` N ) = ( N o. T ) ) |
32 |
31
|
adantr |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( F ` N ) = ( N o. T ) ) |
33 |
32
|
fveq1d |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( ( F ` N ) ` b ) = ( ( N o. T ) ` b ) ) |
34 |
|
f1ofun |
|- ( T : ( 0 ..^ 3 ) -1-1-onto-> ( 0 ..^ 3 ) -> Fun T ) |
35 |
2 34
|
syl |
|- ( ph -> Fun T ) |
36 |
35
|
adantr |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> Fun T ) |
37 |
|
f1odm |
|- ( T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } -> dom T = { 0 , 1 , 2 } ) |
38 |
12 37
|
syl |
|- ( ph -> dom T = { 0 , 1 , 2 } ) |
39 |
38
|
eleq2d |
|- ( ph -> ( b e. dom T <-> b e. { 0 , 1 , 2 } ) ) |
40 |
39
|
biimpar |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> b e. dom T ) |
41 |
|
fvco |
|- ( ( Fun T /\ b e. dom T ) -> ( ( N o. T ) ` b ) = ( N ` ( T ` b ) ) ) |
42 |
36 40 41
|
syl2anc |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( ( N o. T ) ` b ) = ( N ` ( T ` b ) ) ) |
43 |
33 42
|
eqtr2d |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( N ` ( T ` b ) ) = ( ( F ` N ) ` b ) ) |
44 |
43
|
fveq2d |
|- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( L ` ( N ` ( T ` b ) ) ) = ( L ` ( ( F ` N ) ` b ) ) ) |
45 |
44
|
prodeq2dv |
|- ( ph -> prod_ b e. { 0 , 1 , 2 } ( L ` ( N ` ( T ` b ) ) ) = prod_ b e. { 0 , 1 , 2 } ( L ` ( ( F ` N ) ` b ) ) ) |
46 |
21 45
|
eqtr2d |
|- ( ph -> prod_ b e. { 0 , 1 , 2 } ( L ` ( ( F ` N ) ` b ) ) = prod_ a e. { 0 , 1 , 2 } ( L ` ( N ` a ) ) ) |
47 |
|
2fveq3 |
|- ( b = 0 -> ( L ` ( ( F ` N ) ` b ) ) = ( L ` ( ( F ` N ) ` 0 ) ) ) |
48 |
|
2fveq3 |
|- ( b = 1 -> ( L ` ( ( F ` N ) ` b ) ) = ( L ` ( ( F ` N ) ` 1 ) ) ) |
49 |
|
c0ex |
|- 0 e. _V |
50 |
49
|
a1i |
|- ( ph -> 0 e. _V ) |
51 |
|
1ex |
|- 1 e. _V |
52 |
51
|
a1i |
|- ( ph -> 1 e. _V ) |
53 |
31
|
fveq1d |
|- ( ph -> ( ( F ` N ) ` 0 ) = ( ( N o. T ) ` 0 ) ) |
54 |
49
|
tpid1 |
|- 0 e. { 0 , 1 , 2 } |
55 |
54 38
|
eleqtrrid |
|- ( ph -> 0 e. dom T ) |
56 |
|
fvco |
|- ( ( Fun T /\ 0 e. dom T ) -> ( ( N o. T ) ` 0 ) = ( N ` ( T ` 0 ) ) ) |
57 |
35 55 56
|
syl2anc |
|- ( ph -> ( ( N o. T ) ` 0 ) = ( N ` ( T ` 0 ) ) ) |
58 |
53 57
|
eqtrd |
|- ( ph -> ( ( F ` N ) ` 0 ) = ( N ` ( T ` 0 ) ) ) |
59 |
54 9
|
eleqtrri |
|- 0 e. ( 0 ..^ 3 ) |
60 |
59
|
a1i |
|- ( ph -> 0 e. ( 0 ..^ 3 ) ) |
61 |
26 60
|
ffvelrnd |
|- ( ph -> ( T ` 0 ) e. ( 0 ..^ 3 ) ) |
62 |
3 61
|
ffvelrnd |
|- ( ph -> ( N ` ( T ` 0 ) ) e. NN ) |
63 |
58 62
|
eqeltrd |
|- ( ph -> ( ( F ` N ) ` 0 ) e. NN ) |
64 |
4 63
|
ffvelrnd |
|- ( ph -> ( L ` ( ( F ` N ) ` 0 ) ) e. RR ) |
65 |
64
|
recnd |
|- ( ph -> ( L ` ( ( F ` N ) ` 0 ) ) e. CC ) |
66 |
31
|
fveq1d |
|- ( ph -> ( ( F ` N ) ` 1 ) = ( ( N o. T ) ` 1 ) ) |
67 |
51
|
tpid2 |
|- 1 e. { 0 , 1 , 2 } |
68 |
67 38
|
eleqtrrid |
|- ( ph -> 1 e. dom T ) |
69 |
|
fvco |
|- ( ( Fun T /\ 1 e. dom T ) -> ( ( N o. T ) ` 1 ) = ( N ` ( T ` 1 ) ) ) |
70 |
35 68 69
|
syl2anc |
|- ( ph -> ( ( N o. T ) ` 1 ) = ( N ` ( T ` 1 ) ) ) |
71 |
66 70
|
eqtrd |
|- ( ph -> ( ( F ` N ) ` 1 ) = ( N ` ( T ` 1 ) ) ) |
72 |
67 9
|
eleqtrri |
|- 1 e. ( 0 ..^ 3 ) |
73 |
72
|
a1i |
|- ( ph -> 1 e. ( 0 ..^ 3 ) ) |
74 |
26 73
|
ffvelrnd |
|- ( ph -> ( T ` 1 ) e. ( 0 ..^ 3 ) ) |
75 |
3 74
|
ffvelrnd |
|- ( ph -> ( N ` ( T ` 1 ) ) e. NN ) |
76 |
71 75
|
eqeltrd |
|- ( ph -> ( ( F ` N ) ` 1 ) e. NN ) |
77 |
4 76
|
ffvelrnd |
|- ( ph -> ( L ` ( ( F ` N ) ` 1 ) ) e. RR ) |
78 |
77
|
recnd |
|- ( ph -> ( L ` ( ( F ` N ) ` 1 ) ) e. CC ) |
79 |
|
0ne1 |
|- 0 =/= 1 |
80 |
79
|
a1i |
|- ( ph -> 0 =/= 1 ) |
81 |
|
2fveq3 |
|- ( b = 2 -> ( L ` ( ( F ` N ) ` b ) ) = ( L ` ( ( F ` N ) ` 2 ) ) ) |
82 |
|
2ex |
|- 2 e. _V |
83 |
82
|
a1i |
|- ( ph -> 2 e. _V ) |
84 |
31
|
fveq1d |
|- ( ph -> ( ( F ` N ) ` 2 ) = ( ( N o. T ) ` 2 ) ) |
85 |
82
|
tpid3 |
|- 2 e. { 0 , 1 , 2 } |
86 |
85 38
|
eleqtrrid |
|- ( ph -> 2 e. dom T ) |
87 |
|
fvco |
|- ( ( Fun T /\ 2 e. dom T ) -> ( ( N o. T ) ` 2 ) = ( N ` ( T ` 2 ) ) ) |
88 |
35 86 87
|
syl2anc |
|- ( ph -> ( ( N o. T ) ` 2 ) = ( N ` ( T ` 2 ) ) ) |
89 |
84 88
|
eqtrd |
|- ( ph -> ( ( F ` N ) ` 2 ) = ( N ` ( T ` 2 ) ) ) |
90 |
85 9
|
eleqtrri |
|- 2 e. ( 0 ..^ 3 ) |
91 |
90
|
a1i |
|- ( ph -> 2 e. ( 0 ..^ 3 ) ) |
92 |
26 91
|
ffvelrnd |
|- ( ph -> ( T ` 2 ) e. ( 0 ..^ 3 ) ) |
93 |
3 92
|
ffvelrnd |
|- ( ph -> ( N ` ( T ` 2 ) ) e. NN ) |
94 |
89 93
|
eqeltrd |
|- ( ph -> ( ( F ` N ) ` 2 ) e. NN ) |
95 |
4 94
|
ffvelrnd |
|- ( ph -> ( L ` ( ( F ` N ) ` 2 ) ) e. RR ) |
96 |
95
|
recnd |
|- ( ph -> ( L ` ( ( F ` N ) ` 2 ) ) e. CC ) |
97 |
|
0ne2 |
|- 0 =/= 2 |
98 |
97
|
a1i |
|- ( ph -> 0 =/= 2 ) |
99 |
|
1ne2 |
|- 1 =/= 2 |
100 |
99
|
a1i |
|- ( ph -> 1 =/= 2 ) |
101 |
47 48 50 52 65 78 80 81 83 96 98 100
|
prodtp |
|- ( ph -> prod_ b e. { 0 , 1 , 2 } ( L ` ( ( F ` N ) ` b ) ) = ( ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( L ` ( ( F ` N ) ` 1 ) ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) ) |
102 |
|
2fveq3 |
|- ( a = 0 -> ( L ` ( N ` a ) ) = ( L ` ( N ` 0 ) ) ) |
103 |
|
2fveq3 |
|- ( a = 1 -> ( L ` ( N ` a ) ) = ( L ` ( N ` 1 ) ) ) |
104 |
3 60
|
ffvelrnd |
|- ( ph -> ( N ` 0 ) e. NN ) |
105 |
4 104
|
ffvelrnd |
|- ( ph -> ( L ` ( N ` 0 ) ) e. RR ) |
106 |
105
|
recnd |
|- ( ph -> ( L ` ( N ` 0 ) ) e. CC ) |
107 |
3 73
|
ffvelrnd |
|- ( ph -> ( N ` 1 ) e. NN ) |
108 |
4 107
|
ffvelrnd |
|- ( ph -> ( L ` ( N ` 1 ) ) e. RR ) |
109 |
108
|
recnd |
|- ( ph -> ( L ` ( N ` 1 ) ) e. CC ) |
110 |
|
2fveq3 |
|- ( a = 2 -> ( L ` ( N ` a ) ) = ( L ` ( N ` 2 ) ) ) |
111 |
3 91
|
ffvelrnd |
|- ( ph -> ( N ` 2 ) e. NN ) |
112 |
4 111
|
ffvelrnd |
|- ( ph -> ( L ` ( N ` 2 ) ) e. RR ) |
113 |
112
|
recnd |
|- ( ph -> ( L ` ( N ` 2 ) ) e. CC ) |
114 |
102 103 50 52 106 109 80 110 83 113 98 100
|
prodtp |
|- ( ph -> prod_ a e. { 0 , 1 , 2 } ( L ` ( N ` a ) ) = ( ( ( L ` ( N ` 0 ) ) x. ( L ` ( N ` 1 ) ) ) x. ( L ` ( N ` 2 ) ) ) ) |
115 |
46 101 114
|
3eqtr3d |
|- ( ph -> ( ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( L ` ( ( F ` N ) ` 1 ) ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) = ( ( ( L ` ( N ` 0 ) ) x. ( L ` ( N ` 1 ) ) ) x. ( L ` ( N ` 2 ) ) ) ) |
116 |
65 78 96
|
mulassd |
|- ( ph -> ( ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( L ` ( ( F ` N ) ` 1 ) ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) = ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( ( L ` ( ( F ` N ) ` 1 ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) ) ) |
117 |
106 109 113
|
mulassd |
|- ( ph -> ( ( ( L ` ( N ` 0 ) ) x. ( L ` ( N ` 1 ) ) ) x. ( L ` ( N ` 2 ) ) ) = ( ( L ` ( N ` 0 ) ) x. ( ( L ` ( N ` 1 ) ) x. ( L ` ( N ` 2 ) ) ) ) ) |
118 |
115 116 117
|
3eqtr3d |
|- ( ph -> ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( ( L ` ( ( F ` N ) ` 1 ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) ) = ( ( L ` ( N ` 0 ) ) x. ( ( L ` ( N ` 1 ) ) x. ( L ` ( N ` 2 ) ) ) ) ) |