Step |
Hyp |
Ref |
Expression |
1 |
|
fmuldfeqlem1.1 |
|- F/ f ph |
2 |
|
fmuldfeqlem1.2 |
|- F/ g ph |
3 |
|
fmuldfeqlem1.3 |
|- F/_ t Y |
4 |
|
fmuldfeqlem1.5 |
|- P = ( f e. Y , g e. Y |-> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) ) |
5 |
|
fmuldfeqlem1.6 |
|- F = ( t e. T |-> ( i e. ( 1 ... M ) |-> ( ( U ` i ) ` t ) ) ) |
6 |
|
fmuldfeqlem1.7 |
|- ( ph -> T e. _V ) |
7 |
|
fmuldfeqlem1.8 |
|- ( ph -> U : ( 1 ... M ) --> Y ) |
8 |
|
fmuldfeqlem1.9 |
|- ( ( ph /\ f e. Y /\ g e. Y ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. Y ) |
9 |
|
fmuldfeqlem1.10 |
|- ( ph -> N e. ( 1 ... M ) ) |
10 |
|
fmuldfeqlem1.11 |
|- ( ph -> ( N + 1 ) e. ( 1 ... M ) ) |
11 |
|
fmuldfeqlem1.12 |
|- ( ph -> ( ( seq 1 ( P , U ) ` N ) ` t ) = ( seq 1 ( x. , ( F ` t ) ) ` N ) ) |
12 |
|
fmuldfeqlem1.13 |
|- ( ( ph /\ f e. Y ) -> f : T --> RR ) |
13 |
|
ovex |
|- ( 1 ... M ) e. _V |
14 |
13
|
mptex |
|- ( i e. ( 1 ... M ) |-> ( ( U ` i ) ` t ) ) e. _V |
15 |
5
|
fvmpt2 |
|- ( ( t e. T /\ ( i e. ( 1 ... M ) |-> ( ( U ` i ) ` t ) ) e. _V ) -> ( F ` t ) = ( i e. ( 1 ... M ) |-> ( ( U ` i ) ` t ) ) ) |
16 |
14 15
|
mpan2 |
|- ( t e. T -> ( F ` t ) = ( i e. ( 1 ... M ) |-> ( ( U ` i ) ` t ) ) ) |
17 |
|
fveq2 |
|- ( i = j -> ( U ` i ) = ( U ` j ) ) |
18 |
17
|
fveq1d |
|- ( i = j -> ( ( U ` i ) ` t ) = ( ( U ` j ) ` t ) ) |
19 |
18
|
cbvmptv |
|- ( i e. ( 1 ... M ) |-> ( ( U ` i ) ` t ) ) = ( j e. ( 1 ... M ) |-> ( ( U ` j ) ` t ) ) |
20 |
16 19
|
eqtrdi |
|- ( t e. T -> ( F ` t ) = ( j e. ( 1 ... M ) |-> ( ( U ` j ) ` t ) ) ) |
21 |
20
|
adantl |
|- ( ( ph /\ t e. T ) -> ( F ` t ) = ( j e. ( 1 ... M ) |-> ( ( U ` j ) ` t ) ) ) |
22 |
|
fveq2 |
|- ( j = ( N + 1 ) -> ( U ` j ) = ( U ` ( N + 1 ) ) ) |
23 |
22
|
fveq1d |
|- ( j = ( N + 1 ) -> ( ( U ` j ) ` t ) = ( ( U ` ( N + 1 ) ) ` t ) ) |
24 |
23
|
adantl |
|- ( ( ( ph /\ t e. T ) /\ j = ( N + 1 ) ) -> ( ( U ` j ) ` t ) = ( ( U ` ( N + 1 ) ) ` t ) ) |
25 |
10
|
adantr |
|- ( ( ph /\ t e. T ) -> ( N + 1 ) e. ( 1 ... M ) ) |
26 |
7 10
|
ffvelrnd |
|- ( ph -> ( U ` ( N + 1 ) ) e. Y ) |
27 |
26
|
ancli |
|- ( ph -> ( ph /\ ( U ` ( N + 1 ) ) e. Y ) ) |
28 |
|
nfcv |
|- F/_ f ( U ` ( N + 1 ) ) |
29 |
|
nfv |
|- F/ f ( U ` ( N + 1 ) ) e. Y |
30 |
1 29
|
nfan |
|- F/ f ( ph /\ ( U ` ( N + 1 ) ) e. Y ) |
31 |
|
nfv |
|- F/ f ( U ` ( N + 1 ) ) : T --> RR |
32 |
30 31
|
nfim |
|- F/ f ( ( ph /\ ( U ` ( N + 1 ) ) e. Y ) -> ( U ` ( N + 1 ) ) : T --> RR ) |
33 |
|
eleq1 |
|- ( f = ( U ` ( N + 1 ) ) -> ( f e. Y <-> ( U ` ( N + 1 ) ) e. Y ) ) |
34 |
33
|
anbi2d |
|- ( f = ( U ` ( N + 1 ) ) -> ( ( ph /\ f e. Y ) <-> ( ph /\ ( U ` ( N + 1 ) ) e. Y ) ) ) |
35 |
|
feq1 |
|- ( f = ( U ` ( N + 1 ) ) -> ( f : T --> RR <-> ( U ` ( N + 1 ) ) : T --> RR ) ) |
36 |
34 35
|
imbi12d |
|- ( f = ( U ` ( N + 1 ) ) -> ( ( ( ph /\ f e. Y ) -> f : T --> RR ) <-> ( ( ph /\ ( U ` ( N + 1 ) ) e. Y ) -> ( U ` ( N + 1 ) ) : T --> RR ) ) ) |
37 |
28 32 36 12
|
vtoclgf |
|- ( ( U ` ( N + 1 ) ) e. Y -> ( ( ph /\ ( U ` ( N + 1 ) ) e. Y ) -> ( U ` ( N + 1 ) ) : T --> RR ) ) |
38 |
26 27 37
|
sylc |
|- ( ph -> ( U ` ( N + 1 ) ) : T --> RR ) |
39 |
38
|
ffvelrnda |
|- ( ( ph /\ t e. T ) -> ( ( U ` ( N + 1 ) ) ` t ) e. RR ) |
40 |
21 24 25 39
|
fvmptd |
|- ( ( ph /\ t e. T ) -> ( ( F ` t ) ` ( N + 1 ) ) = ( ( U ` ( N + 1 ) ) ` t ) ) |
41 |
40
|
oveq2d |
|- ( ( ph /\ t e. T ) -> ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( F ` t ) ` ( N + 1 ) ) ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) |
42 |
|
elfzuz |
|- ( N e. ( 1 ... M ) -> N e. ( ZZ>= ` 1 ) ) |
43 |
9 42
|
syl |
|- ( ph -> N e. ( ZZ>= ` 1 ) ) |
44 |
|
seqp1 |
|- ( N e. ( ZZ>= ` 1 ) -> ( seq 1 ( x. , ( F ` t ) ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( F ` t ) ` ( N + 1 ) ) ) ) |
45 |
43 44
|
syl |
|- ( ph -> ( seq 1 ( x. , ( F ` t ) ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( F ` t ) ` ( N + 1 ) ) ) ) |
46 |
45
|
adantr |
|- ( ( ph /\ t e. T ) -> ( seq 1 ( x. , ( F ` t ) ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( F ` t ) ` ( N + 1 ) ) ) ) |
47 |
|
seqp1 |
|- ( N e. ( ZZ>= ` 1 ) -> ( seq 1 ( P , U ) ` ( N + 1 ) ) = ( ( seq 1 ( P , U ) ` N ) P ( U ` ( N + 1 ) ) ) ) |
48 |
43 47
|
syl |
|- ( ph -> ( seq 1 ( P , U ) ` ( N + 1 ) ) = ( ( seq 1 ( P , U ) ` N ) P ( U ` ( N + 1 ) ) ) ) |
49 |
|
nfcv |
|- F/_ h ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) |
50 |
|
nfcv |
|- F/_ l ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) |
51 |
|
nfcv |
|- F/_ f ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) |
52 |
|
nfcv |
|- F/_ g ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) |
53 |
|
fveq1 |
|- ( f = h -> ( f ` t ) = ( h ` t ) ) |
54 |
|
fveq1 |
|- ( g = l -> ( g ` t ) = ( l ` t ) ) |
55 |
53 54
|
oveqan12d |
|- ( ( f = h /\ g = l ) -> ( ( f ` t ) x. ( g ` t ) ) = ( ( h ` t ) x. ( l ` t ) ) ) |
56 |
55
|
mpteq2dv |
|- ( ( f = h /\ g = l ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) ) |
57 |
49 50 51 52 56
|
cbvmpo |
|- ( f e. Y , g e. Y |-> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) ) = ( h e. Y , l e. Y |-> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) ) |
58 |
4 57
|
eqtri |
|- P = ( h e. Y , l e. Y |-> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) ) |
59 |
58
|
a1i |
|- ( ph -> P = ( h e. Y , l e. Y |-> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) ) ) |
60 |
|
nfcv |
|- F/_ t 1 |
61 |
|
nfmpt1 |
|- F/_ t ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) |
62 |
3 3 61
|
nfmpo |
|- F/_ t ( f e. Y , g e. Y |-> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) ) |
63 |
4 62
|
nfcxfr |
|- F/_ t P |
64 |
|
nfcv |
|- F/_ t U |
65 |
60 63 64
|
nfseq |
|- F/_ t seq 1 ( P , U ) |
66 |
|
nfcv |
|- F/_ t N |
67 |
65 66
|
nffv |
|- F/_ t ( seq 1 ( P , U ) ` N ) |
68 |
67
|
nfeq2 |
|- F/ t h = ( seq 1 ( P , U ) ` N ) |
69 |
|
nfv |
|- F/ t l = ( U ` ( N + 1 ) ) |
70 |
68 69
|
nfan |
|- F/ t ( h = ( seq 1 ( P , U ) ` N ) /\ l = ( U ` ( N + 1 ) ) ) |
71 |
|
fveq1 |
|- ( h = ( seq 1 ( P , U ) ` N ) -> ( h ` t ) = ( ( seq 1 ( P , U ) ` N ) ` t ) ) |
72 |
71
|
ad2antrr |
|- ( ( ( h = ( seq 1 ( P , U ) ` N ) /\ l = ( U ` ( N + 1 ) ) ) /\ t e. T ) -> ( h ` t ) = ( ( seq 1 ( P , U ) ` N ) ` t ) ) |
73 |
|
fveq1 |
|- ( l = ( U ` ( N + 1 ) ) -> ( l ` t ) = ( ( U ` ( N + 1 ) ) ` t ) ) |
74 |
73
|
ad2antlr |
|- ( ( ( h = ( seq 1 ( P , U ) ` N ) /\ l = ( U ` ( N + 1 ) ) ) /\ t e. T ) -> ( l ` t ) = ( ( U ` ( N + 1 ) ) ` t ) ) |
75 |
72 74
|
oveq12d |
|- ( ( ( h = ( seq 1 ( P , U ) ` N ) /\ l = ( U ` ( N + 1 ) ) ) /\ t e. T ) -> ( ( h ` t ) x. ( l ` t ) ) = ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) |
76 |
70 75
|
mpteq2da |
|- ( ( h = ( seq 1 ( P , U ) ` N ) /\ l = ( U ` ( N + 1 ) ) ) -> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) = ( t e. T |-> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) ) |
77 |
76
|
adantl |
|- ( ( ph /\ ( h = ( seq 1 ( P , U ) ` N ) /\ l = ( U ` ( N + 1 ) ) ) ) -> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) = ( t e. T |-> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) ) |
78 |
|
eqid |
|- ( seq 1 ( P , U ) ` N ) = ( seq 1 ( P , U ) ` N ) |
79 |
|
3simpc |
|- ( ( ph /\ h e. Y /\ l e. Y ) -> ( h e. Y /\ l e. Y ) ) |
80 |
|
nfcv |
|- F/_ f h |
81 |
|
nfcv |
|- F/_ g h |
82 |
|
nfcv |
|- F/_ g l |
83 |
|
nfv |
|- F/ f h e. Y |
84 |
|
nfv |
|- F/ f g e. Y |
85 |
1 83 84
|
nf3an |
|- F/ f ( ph /\ h e. Y /\ g e. Y ) |
86 |
|
nfv |
|- F/ f ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) e. Y |
87 |
85 86
|
nfim |
|- F/ f ( ( ph /\ h e. Y /\ g e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) e. Y ) |
88 |
|
nfv |
|- F/ g h e. Y |
89 |
|
nfv |
|- F/ g l e. Y |
90 |
2 88 89
|
nf3an |
|- F/ g ( ph /\ h e. Y /\ l e. Y ) |
91 |
|
nfv |
|- F/ g ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) e. Y |
92 |
90 91
|
nfim |
|- F/ g ( ( ph /\ h e. Y /\ l e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) e. Y ) |
93 |
|
eleq1 |
|- ( f = h -> ( f e. Y <-> h e. Y ) ) |
94 |
93
|
3anbi2d |
|- ( f = h -> ( ( ph /\ f e. Y /\ g e. Y ) <-> ( ph /\ h e. Y /\ g e. Y ) ) ) |
95 |
53
|
oveq1d |
|- ( f = h -> ( ( f ` t ) x. ( g ` t ) ) = ( ( h ` t ) x. ( g ` t ) ) ) |
96 |
95
|
mpteq2dv |
|- ( f = h -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) ) |
97 |
96
|
eleq1d |
|- ( f = h -> ( ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. Y <-> ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) e. Y ) ) |
98 |
94 97
|
imbi12d |
|- ( f = h -> ( ( ( ph /\ f e. Y /\ g e. Y ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. Y ) <-> ( ( ph /\ h e. Y /\ g e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) e. Y ) ) ) |
99 |
|
eleq1 |
|- ( g = l -> ( g e. Y <-> l e. Y ) ) |
100 |
99
|
3anbi3d |
|- ( g = l -> ( ( ph /\ h e. Y /\ g e. Y ) <-> ( ph /\ h e. Y /\ l e. Y ) ) ) |
101 |
54
|
oveq2d |
|- ( g = l -> ( ( h ` t ) x. ( g ` t ) ) = ( ( h ` t ) x. ( l ` t ) ) ) |
102 |
101
|
mpteq2dv |
|- ( g = l -> ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) ) |
103 |
102
|
eleq1d |
|- ( g = l -> ( ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) e. Y <-> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) e. Y ) ) |
104 |
100 103
|
imbi12d |
|- ( g = l -> ( ( ( ph /\ h e. Y /\ g e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( g ` t ) ) ) e. Y ) <-> ( ( ph /\ h e. Y /\ l e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) e. Y ) ) ) |
105 |
80 81 82 87 92 98 104 8
|
vtocl2gf |
|- ( ( h e. Y /\ l e. Y ) -> ( ( ph /\ h e. Y /\ l e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) e. Y ) ) |
106 |
79 105
|
mpcom |
|- ( ( ph /\ h e. Y /\ l e. Y ) -> ( t e. T |-> ( ( h ` t ) x. ( l ` t ) ) ) e. Y ) |
107 |
58 78 9 7 106 6
|
fmulcl |
|- ( ph -> ( seq 1 ( P , U ) ` N ) e. Y ) |
108 |
|
mptexg |
|- ( T e. _V -> ( t e. T |-> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) e. _V ) |
109 |
6 108
|
syl |
|- ( ph -> ( t e. T |-> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) e. _V ) |
110 |
59 77 107 26 109
|
ovmpod |
|- ( ph -> ( ( seq 1 ( P , U ) ` N ) P ( U ` ( N + 1 ) ) ) = ( t e. T |-> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) ) |
111 |
48 110
|
eqtrd |
|- ( ph -> ( seq 1 ( P , U ) ` ( N + 1 ) ) = ( t e. T |-> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) ) |
112 |
107
|
ancli |
|- ( ph -> ( ph /\ ( seq 1 ( P , U ) ` N ) e. Y ) ) |
113 |
|
nfcv |
|- F/_ f 1 |
114 |
|
nfmpo1 |
|- F/_ f ( f e. Y , g e. Y |-> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) ) |
115 |
4 114
|
nfcxfr |
|- F/_ f P |
116 |
|
nfcv |
|- F/_ f U |
117 |
113 115 116
|
nfseq |
|- F/_ f seq 1 ( P , U ) |
118 |
|
nfcv |
|- F/_ f N |
119 |
117 118
|
nffv |
|- F/_ f ( seq 1 ( P , U ) ` N ) |
120 |
119
|
nfel1 |
|- F/ f ( seq 1 ( P , U ) ` N ) e. Y |
121 |
1 120
|
nfan |
|- F/ f ( ph /\ ( seq 1 ( P , U ) ` N ) e. Y ) |
122 |
|
nfcv |
|- F/_ f T |
123 |
|
nfcv |
|- F/_ f RR |
124 |
119 122 123
|
nff |
|- F/ f ( seq 1 ( P , U ) ` N ) : T --> RR |
125 |
121 124
|
nfim |
|- F/ f ( ( ph /\ ( seq 1 ( P , U ) ` N ) e. Y ) -> ( seq 1 ( P , U ) ` N ) : T --> RR ) |
126 |
|
eleq1 |
|- ( f = ( seq 1 ( P , U ) ` N ) -> ( f e. Y <-> ( seq 1 ( P , U ) ` N ) e. Y ) ) |
127 |
126
|
anbi2d |
|- ( f = ( seq 1 ( P , U ) ` N ) -> ( ( ph /\ f e. Y ) <-> ( ph /\ ( seq 1 ( P , U ) ` N ) e. Y ) ) ) |
128 |
|
feq1 |
|- ( f = ( seq 1 ( P , U ) ` N ) -> ( f : T --> RR <-> ( seq 1 ( P , U ) ` N ) : T --> RR ) ) |
129 |
127 128
|
imbi12d |
|- ( f = ( seq 1 ( P , U ) ` N ) -> ( ( ( ph /\ f e. Y ) -> f : T --> RR ) <-> ( ( ph /\ ( seq 1 ( P , U ) ` N ) e. Y ) -> ( seq 1 ( P , U ) ` N ) : T --> RR ) ) ) |
130 |
119 125 129 12
|
vtoclgf |
|- ( ( seq 1 ( P , U ) ` N ) e. Y -> ( ( ph /\ ( seq 1 ( P , U ) ` N ) e. Y ) -> ( seq 1 ( P , U ) ` N ) : T --> RR ) ) |
131 |
107 112 130
|
sylc |
|- ( ph -> ( seq 1 ( P , U ) ` N ) : T --> RR ) |
132 |
131
|
ffvelrnda |
|- ( ( ph /\ t e. T ) -> ( ( seq 1 ( P , U ) ` N ) ` t ) e. RR ) |
133 |
132 39
|
remulcld |
|- ( ( ph /\ t e. T ) -> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) e. RR ) |
134 |
111 133
|
fvmpt2d |
|- ( ( ph /\ t e. T ) -> ( ( seq 1 ( P , U ) ` ( N + 1 ) ) ` t ) = ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) |
135 |
11
|
oveq1d |
|- ( ph -> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) |
136 |
135
|
adantr |
|- ( ( ph /\ t e. T ) -> ( ( ( seq 1 ( P , U ) ` N ) ` t ) x. ( ( U ` ( N + 1 ) ) ` t ) ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) |
137 |
134 136
|
eqtrd |
|- ( ( ph /\ t e. T ) -> ( ( seq 1 ( P , U ) ` ( N + 1 ) ) ` t ) = ( ( seq 1 ( x. , ( F ` t ) ) ` N ) x. ( ( U ` ( N + 1 ) ) ` t ) ) ) |
138 |
41 46 137
|
3eqtr4rd |
|- ( ( ph /\ t e. T ) -> ( ( seq 1 ( P , U ) ` ( N + 1 ) ) ` t ) = ( seq 1 ( x. , ( F ` t ) ) ` ( N + 1 ) ) ) |