Step |
Hyp |
Ref |
Expression |
1 |
|
amgm.1 |
⊢ 𝑀 = ( mulGrp ‘ ℂfld ) |
2 |
|
amgm.2 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
3 |
|
amgm.3 |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
4 |
|
amgm.4 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ+ ) |
5 |
|
cnfld0 |
⊢ 0 = ( 0g ‘ ℂfld ) |
6 |
|
cnring |
⊢ ℂfld ∈ Ring |
7 |
|
ringabl |
⊢ ( ℂfld ∈ Ring → ℂfld ∈ Abel ) |
8 |
6 7
|
mp1i |
⊢ ( 𝜑 → ℂfld ∈ Abel ) |
9 |
|
resubdrg |
⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ℝfld ∈ DivRing ) |
10 |
9
|
simpli |
⊢ ℝ ∈ ( SubRing ‘ ℂfld ) |
11 |
|
subrgsubg |
⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) → ℝ ∈ ( SubGrp ‘ ℂfld ) ) |
12 |
10 11
|
mp1i |
⊢ ( 𝜑 → ℝ ∈ ( SubGrp ‘ ℂfld ) ) |
13 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ+ ) |
14 |
13
|
relogcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
15 |
14
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
16 |
15
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) : 𝐴 ⟶ ℝ ) |
17 |
|
c0ex |
⊢ 0 ∈ V |
18 |
17
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
19 |
16 2 18
|
fdmfifsupp |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) finSupp 0 ) |
20 |
5 8 2 12 16 19
|
gsumsubgcl |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ∈ ℝ ) |
21 |
20
|
recnd |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ∈ ℂ ) |
22 |
|
hashnncl |
⊢ ( 𝐴 ∈ Fin → ( ( ♯ ‘ 𝐴 ) ∈ ℕ ↔ 𝐴 ≠ ∅ ) ) |
23 |
2 22
|
syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) ∈ ℕ ↔ 𝐴 ≠ ∅ ) ) |
24 |
3 23
|
mpbird |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
25 |
24
|
nncnd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℂ ) |
26 |
24
|
nnne0d |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ≠ 0 ) |
27 |
21 25 26
|
divnegd |
⊢ ( 𝜑 → - ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) = ( - ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ) |
28 |
14
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
29 |
2 28
|
gsumfsum |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = Σ 𝑘 ∈ 𝐴 ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
30 |
28
|
negnegd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → - - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) = ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
31 |
30
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 - - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) = Σ 𝑘 ∈ 𝐴 ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
32 |
15
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
33 |
2 32
|
fsumneg |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 - - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) = - Σ 𝑘 ∈ 𝐴 - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
34 |
29 31 33
|
3eqtr2rd |
⊢ ( 𝜑 → - Σ 𝑘 ∈ 𝐴 - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
35 |
2 32
|
gsumfsum |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = Σ 𝑘 ∈ 𝐴 - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
36 |
35
|
negeqd |
⊢ ( 𝜑 → - ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = - Σ 𝑘 ∈ 𝐴 - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
37 |
4
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) ) |
38 |
|
relogf1o |
⊢ ( log ↾ ℝ+ ) : ℝ+ –1-1-onto→ ℝ |
39 |
|
f1of |
⊢ ( ( log ↾ ℝ+ ) : ℝ+ –1-1-onto→ ℝ → ( log ↾ ℝ+ ) : ℝ+ ⟶ ℝ ) |
40 |
38 39
|
mp1i |
⊢ ( 𝜑 → ( log ↾ ℝ+ ) : ℝ+ ⟶ ℝ ) |
41 |
40
|
feqmptd |
⊢ ( 𝜑 → ( log ↾ ℝ+ ) = ( 𝑥 ∈ ℝ+ ↦ ( ( log ↾ ℝ+ ) ‘ 𝑥 ) ) ) |
42 |
|
fvres |
⊢ ( 𝑥 ∈ ℝ+ → ( ( log ↾ ℝ+ ) ‘ 𝑥 ) = ( log ‘ 𝑥 ) ) |
43 |
42
|
mpteq2ia |
⊢ ( 𝑥 ∈ ℝ+ ↦ ( ( log ↾ ℝ+ ) ‘ 𝑥 ) ) = ( 𝑥 ∈ ℝ+ ↦ ( log ‘ 𝑥 ) ) |
44 |
41 43
|
eqtrdi |
⊢ ( 𝜑 → ( log ↾ ℝ+ ) = ( 𝑥 ∈ ℝ+ ↦ ( log ‘ 𝑥 ) ) ) |
45 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑘 ) → ( log ‘ 𝑥 ) = ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
46 |
13 37 44 45
|
fmptco |
⊢ ( 𝜑 → ( ( log ↾ ℝ+ ) ∘ 𝐹 ) = ( 𝑘 ∈ 𝐴 ↦ ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
47 |
46
|
oveq2d |
⊢ ( 𝜑 → ( ℂfld Σg ( ( log ↾ ℝ+ ) ∘ 𝐹 ) ) = ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
48 |
34 36 47
|
3eqtr4d |
⊢ ( 𝜑 → - ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ℂfld Σg ( ( log ↾ ℝ+ ) ∘ 𝐹 ) ) ) |
49 |
1
|
oveq1i |
⊢ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) = ( ( mulGrp ‘ ℂfld ) ↾s ( ℂ ∖ { 0 } ) ) |
50 |
49
|
rpmsubg |
⊢ ℝ+ ∈ ( SubGrp ‘ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) ) |
51 |
|
subgsubm |
⊢ ( ℝ+ ∈ ( SubGrp ‘ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) ) → ℝ+ ∈ ( SubMnd ‘ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) ) ) |
52 |
50 51
|
ax-mp |
⊢ ℝ+ ∈ ( SubMnd ‘ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) ) |
53 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
54 |
|
cndrng |
⊢ ℂfld ∈ DivRing |
55 |
53 5 54
|
drngui |
⊢ ( ℂ ∖ { 0 } ) = ( Unit ‘ ℂfld ) |
56 |
55 1
|
unitsubm |
⊢ ( ℂfld ∈ Ring → ( ℂ ∖ { 0 } ) ∈ ( SubMnd ‘ 𝑀 ) ) |
57 |
|
eqid |
⊢ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) = ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) |
58 |
57
|
subsubm |
⊢ ( ( ℂ ∖ { 0 } ) ∈ ( SubMnd ‘ 𝑀 ) → ( ℝ+ ∈ ( SubMnd ‘ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) ) ↔ ( ℝ+ ∈ ( SubMnd ‘ 𝑀 ) ∧ ℝ+ ⊆ ( ℂ ∖ { 0 } ) ) ) ) |
59 |
6 56 58
|
mp2b |
⊢ ( ℝ+ ∈ ( SubMnd ‘ ( 𝑀 ↾s ( ℂ ∖ { 0 } ) ) ) ↔ ( ℝ+ ∈ ( SubMnd ‘ 𝑀 ) ∧ ℝ+ ⊆ ( ℂ ∖ { 0 } ) ) ) |
60 |
52 59
|
mpbi |
⊢ ( ℝ+ ∈ ( SubMnd ‘ 𝑀 ) ∧ ℝ+ ⊆ ( ℂ ∖ { 0 } ) ) |
61 |
60
|
simpli |
⊢ ℝ+ ∈ ( SubMnd ‘ 𝑀 ) |
62 |
|
eqid |
⊢ ( 𝑀 ↾s ℝ+ ) = ( 𝑀 ↾s ℝ+ ) |
63 |
62
|
submbas |
⊢ ( ℝ+ ∈ ( SubMnd ‘ 𝑀 ) → ℝ+ = ( Base ‘ ( 𝑀 ↾s ℝ+ ) ) ) |
64 |
61 63
|
ax-mp |
⊢ ℝ+ = ( Base ‘ ( 𝑀 ↾s ℝ+ ) ) |
65 |
|
cnfld1 |
⊢ 1 = ( 1r ‘ ℂfld ) |
66 |
1 65
|
ringidval |
⊢ 1 = ( 0g ‘ 𝑀 ) |
67 |
62 66
|
subm0 |
⊢ ( ℝ+ ∈ ( SubMnd ‘ 𝑀 ) → 1 = ( 0g ‘ ( 𝑀 ↾s ℝ+ ) ) ) |
68 |
61 67
|
ax-mp |
⊢ 1 = ( 0g ‘ ( 𝑀 ↾s ℝ+ ) ) |
69 |
|
cncrng |
⊢ ℂfld ∈ CRing |
70 |
1
|
crngmgp |
⊢ ( ℂfld ∈ CRing → 𝑀 ∈ CMnd ) |
71 |
69 70
|
mp1i |
⊢ ( 𝜑 → 𝑀 ∈ CMnd ) |
72 |
62
|
submmnd |
⊢ ( ℝ+ ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑀 ↾s ℝ+ ) ∈ Mnd ) |
73 |
61 72
|
mp1i |
⊢ ( 𝜑 → ( 𝑀 ↾s ℝ+ ) ∈ Mnd ) |
74 |
62
|
subcmn |
⊢ ( ( 𝑀 ∈ CMnd ∧ ( 𝑀 ↾s ℝ+ ) ∈ Mnd ) → ( 𝑀 ↾s ℝ+ ) ∈ CMnd ) |
75 |
71 73 74
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 ↾s ℝ+ ) ∈ CMnd ) |
76 |
|
df-refld |
⊢ ℝfld = ( ℂfld ↾s ℝ ) |
77 |
76
|
subrgring |
⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) → ℝfld ∈ Ring ) |
78 |
10 77
|
ax-mp |
⊢ ℝfld ∈ Ring |
79 |
|
ringmnd |
⊢ ( ℝfld ∈ Ring → ℝfld ∈ Mnd ) |
80 |
78 79
|
mp1i |
⊢ ( 𝜑 → ℝfld ∈ Mnd ) |
81 |
1
|
oveq1i |
⊢ ( 𝑀 ↾s ℝ+ ) = ( ( mulGrp ‘ ℂfld ) ↾s ℝ+ ) |
82 |
81
|
reloggim |
⊢ ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) GrpIso ℝfld ) |
83 |
|
gimghm |
⊢ ( ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) GrpIso ℝfld ) → ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) GrpHom ℝfld ) ) |
84 |
82 83
|
ax-mp |
⊢ ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) GrpHom ℝfld ) |
85 |
|
ghmmhm |
⊢ ( ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) GrpHom ℝfld ) → ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) MndHom ℝfld ) ) |
86 |
84 85
|
mp1i |
⊢ ( 𝜑 → ( log ↾ ℝ+ ) ∈ ( ( 𝑀 ↾s ℝ+ ) MndHom ℝfld ) ) |
87 |
|
1ex |
⊢ 1 ∈ V |
88 |
87
|
a1i |
⊢ ( 𝜑 → 1 ∈ V ) |
89 |
4 2 88
|
fdmfifsupp |
⊢ ( 𝜑 → 𝐹 finSupp 1 ) |
90 |
64 68 75 80 2 86 4 89
|
gsummhm |
⊢ ( 𝜑 → ( ℝfld Σg ( ( log ↾ ℝ+ ) ∘ 𝐹 ) ) = ( ( log ↾ ℝ+ ) ‘ ( ( 𝑀 ↾s ℝ+ ) Σg 𝐹 ) ) ) |
91 |
|
subgsubm |
⊢ ( ℝ ∈ ( SubGrp ‘ ℂfld ) → ℝ ∈ ( SubMnd ‘ ℂfld ) ) |
92 |
12 91
|
syl |
⊢ ( 𝜑 → ℝ ∈ ( SubMnd ‘ ℂfld ) ) |
93 |
|
fco |
⊢ ( ( ( log ↾ ℝ+ ) : ℝ+ ⟶ ℝ ∧ 𝐹 : 𝐴 ⟶ ℝ+ ) → ( ( log ↾ ℝ+ ) ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
94 |
40 4 93
|
syl2anc |
⊢ ( 𝜑 → ( ( log ↾ ℝ+ ) ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
95 |
2 92 94 76
|
gsumsubm |
⊢ ( 𝜑 → ( ℂfld Σg ( ( log ↾ ℝ+ ) ∘ 𝐹 ) ) = ( ℝfld Σg ( ( log ↾ ℝ+ ) ∘ 𝐹 ) ) ) |
96 |
61
|
a1i |
⊢ ( 𝜑 → ℝ+ ∈ ( SubMnd ‘ 𝑀 ) ) |
97 |
2 96 4 62
|
gsumsubm |
⊢ ( 𝜑 → ( 𝑀 Σg 𝐹 ) = ( ( 𝑀 ↾s ℝ+ ) Σg 𝐹 ) ) |
98 |
97
|
fveq2d |
⊢ ( 𝜑 → ( ( log ↾ ℝ+ ) ‘ ( 𝑀 Σg 𝐹 ) ) = ( ( log ↾ ℝ+ ) ‘ ( ( 𝑀 ↾s ℝ+ ) Σg 𝐹 ) ) ) |
99 |
90 95 98
|
3eqtr4d |
⊢ ( 𝜑 → ( ℂfld Σg ( ( log ↾ ℝ+ ) ∘ 𝐹 ) ) = ( ( log ↾ ℝ+ ) ‘ ( 𝑀 Σg 𝐹 ) ) ) |
100 |
66 71 2 96 4 89
|
gsumsubmcl |
⊢ ( 𝜑 → ( 𝑀 Σg 𝐹 ) ∈ ℝ+ ) |
101 |
100
|
fvresd |
⊢ ( 𝜑 → ( ( log ↾ ℝ+ ) ‘ ( 𝑀 Σg 𝐹 ) ) = ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) |
102 |
48 99 101
|
3eqtrd |
⊢ ( 𝜑 → - ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) |
103 |
102
|
oveq1d |
⊢ ( 𝜑 → ( - ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) = ( ( log ‘ ( 𝑀 Σg 𝐹 ) ) / ( ♯ ‘ 𝐴 ) ) ) |
104 |
100
|
relogcld |
⊢ ( 𝜑 → ( log ‘ ( 𝑀 Σg 𝐹 ) ) ∈ ℝ ) |
105 |
104
|
recnd |
⊢ ( 𝜑 → ( log ‘ ( 𝑀 Σg 𝐹 ) ) ∈ ℂ ) |
106 |
105 25 26
|
divrec2d |
⊢ ( 𝜑 → ( ( log ‘ ( 𝑀 Σg 𝐹 ) ) / ( ♯ ‘ 𝐴 ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ) |
107 |
27 103 106
|
3eqtrd |
⊢ ( 𝜑 → - ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ) |
108 |
37
|
oveq2d |
⊢ ( 𝜑 → ( ℂfld Σg 𝐹 ) = ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
109 |
13
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
110 |
2 109
|
gsumfsum |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑘 ) ) |
111 |
108 110
|
eqtrd |
⊢ ( 𝜑 → ( ℂfld Σg 𝐹 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑘 ) ) |
112 |
2 3 13
|
fsumrpcl |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑘 ) ∈ ℝ+ ) |
113 |
111 112
|
eqeltrd |
⊢ ( 𝜑 → ( ℂfld Σg 𝐹 ) ∈ ℝ+ ) |
114 |
24
|
nnrpd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℝ+ ) |
115 |
113 114
|
rpdivcld |
⊢ ( 𝜑 → ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ∈ ℝ+ ) |
116 |
115
|
relogcld |
⊢ ( 𝜑 → ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ∈ ℝ ) |
117 |
20 24
|
nndivred |
⊢ ( 𝜑 → ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ∈ ℝ ) |
118 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
119 |
118
|
a1i |
⊢ ( 𝜑 → ℝ+ ⊆ ℝ ) |
120 |
|
relogcl |
⊢ ( 𝑤 ∈ ℝ+ → ( log ‘ 𝑤 ) ∈ ℝ ) |
121 |
120
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ+ ) → ( log ‘ 𝑤 ) ∈ ℝ ) |
122 |
121
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ+ ) → - ( log ‘ 𝑤 ) ∈ ℝ ) |
123 |
122
|
fmpttd |
⊢ ( 𝜑 → ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) : ℝ+ ⟶ ℝ ) |
124 |
|
ioorp |
⊢ ( 0 (,) +∞ ) = ℝ+ |
125 |
124
|
eleq2i |
⊢ ( 𝑎 ∈ ( 0 (,) +∞ ) ↔ 𝑎 ∈ ℝ+ ) |
126 |
124
|
eleq2i |
⊢ ( 𝑏 ∈ ( 0 (,) +∞ ) ↔ 𝑏 ∈ ℝ+ ) |
127 |
|
iccssioo2 |
⊢ ( ( 𝑎 ∈ ( 0 (,) +∞ ) ∧ 𝑏 ∈ ( 0 (,) +∞ ) ) → ( 𝑎 [,] 𝑏 ) ⊆ ( 0 (,) +∞ ) ) |
128 |
125 126 127
|
syl2anbr |
⊢ ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) → ( 𝑎 [,] 𝑏 ) ⊆ ( 0 (,) +∞ ) ) |
129 |
128 124
|
sseqtrdi |
⊢ ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) → ( 𝑎 [,] 𝑏 ) ⊆ ℝ+ ) |
130 |
129
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ) → ( 𝑎 [,] 𝑏 ) ⊆ ℝ+ ) |
131 |
24
|
nnrecred |
⊢ ( 𝜑 → ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℝ ) |
132 |
114
|
rpreccld |
⊢ ( 𝜑 → ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℝ+ ) |
133 |
132
|
rpge0d |
⊢ ( 𝜑 → 0 ≤ ( 1 / ( ♯ ‘ 𝐴 ) ) ) |
134 |
|
elrege0 |
⊢ ( ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ( 0 [,) +∞ ) ↔ ( ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℝ ∧ 0 ≤ ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
135 |
131 133 134
|
sylanbrc |
⊢ ( 𝜑 → ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ( 0 [,) +∞ ) ) |
136 |
|
fconst6g |
⊢ ( ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ( 0 [,) +∞ ) → ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) : 𝐴 ⟶ ( 0 [,) +∞ ) ) |
137 |
135 136
|
syl |
⊢ ( 𝜑 → ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) : 𝐴 ⟶ ( 0 [,) +∞ ) ) |
138 |
|
0lt1 |
⊢ 0 < 1 |
139 |
|
fconstmpt |
⊢ ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) = ( 𝑘 ∈ 𝐴 ↦ ( 1 / ( ♯ ‘ 𝐴 ) ) ) |
140 |
139
|
oveq2i |
⊢ ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) = ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
141 |
|
ringmnd |
⊢ ( ℂfld ∈ Ring → ℂfld ∈ Mnd ) |
142 |
6 141
|
mp1i |
⊢ ( 𝜑 → ℂfld ∈ Mnd ) |
143 |
131
|
recnd |
⊢ ( 𝜑 → ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℂ ) |
144 |
|
eqid |
⊢ ( .g ‘ ℂfld ) = ( .g ‘ ℂfld ) |
145 |
53 144
|
gsumconst |
⊢ ( ( ℂfld ∈ Mnd ∧ 𝐴 ∈ Fin ∧ ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℂ ) → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) = ( ( ♯ ‘ 𝐴 ) ( .g ‘ ℂfld ) ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
146 |
142 2 143 145
|
syl3anc |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) = ( ( ♯ ‘ 𝐴 ) ( .g ‘ ℂfld ) ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
147 |
24
|
nnzd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℤ ) |
148 |
|
cnfldmulg |
⊢ ( ( ( ♯ ‘ 𝐴 ) ∈ ℤ ∧ ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℂ ) → ( ( ♯ ‘ 𝐴 ) ( .g ‘ ℂfld ) ( 1 / ( ♯ ‘ 𝐴 ) ) ) = ( ( ♯ ‘ 𝐴 ) · ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
149 |
147 143 148
|
syl2anc |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) ( .g ‘ ℂfld ) ( 1 / ( ♯ ‘ 𝐴 ) ) ) = ( ( ♯ ‘ 𝐴 ) · ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
150 |
25 26
|
recidd |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) · ( 1 / ( ♯ ‘ 𝐴 ) ) ) = 1 ) |
151 |
146 149 150
|
3eqtrd |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) = 1 ) |
152 |
140 151
|
eqtrid |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) = 1 ) |
153 |
138 152
|
breqtrrid |
⊢ ( 𝜑 → 0 < ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) |
154 |
|
logccv |
⊢ ( ( ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) < ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ) |
155 |
154
|
3adant1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) < ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ) |
156 |
|
ioossre |
⊢ ( 0 (,) 1 ) ⊆ ℝ |
157 |
|
simp3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝑡 ∈ ( 0 (,) 1 ) ) |
158 |
156 157
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝑡 ∈ ℝ ) |
159 |
|
simp21 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝑥 ∈ ℝ+ ) |
160 |
159
|
relogcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( log ‘ 𝑥 ) ∈ ℝ ) |
161 |
158 160
|
remulcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 𝑡 · ( log ‘ 𝑥 ) ) ∈ ℝ ) |
162 |
|
1re |
⊢ 1 ∈ ℝ |
163 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ 𝑡 ∈ ℝ ) → ( 1 − 𝑡 ) ∈ ℝ ) |
164 |
162 158 163
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 1 − 𝑡 ) ∈ ℝ ) |
165 |
|
simp22 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝑦 ∈ ℝ+ ) |
166 |
165
|
relogcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( log ‘ 𝑦 ) ∈ ℝ ) |
167 |
164 166
|
remulcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ∈ ℝ ) |
168 |
161 167
|
readdcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) ∈ ℝ ) |
169 |
|
simp1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝜑 ) |
170 |
|
ioossicc |
⊢ ( 0 (,) 1 ) ⊆ ( 0 [,] 1 ) |
171 |
170 157
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝑡 ∈ ( 0 [,] 1 ) ) |
172 |
119 130
|
cvxcl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ∈ ℝ+ ) |
173 |
169 159 165 171 172
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ∈ ℝ+ ) |
174 |
173
|
relogcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ∈ ℝ ) |
175 |
168 174
|
ltnegd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) < ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ↔ - ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) < - ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) ) ) |
176 |
155 175
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → - ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) < - ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) ) |
177 |
|
fveq2 |
⊢ ( 𝑤 = ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) → ( log ‘ 𝑤 ) = ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ) |
178 |
177
|
negeqd |
⊢ ( 𝑤 = ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) → - ( log ‘ 𝑤 ) = - ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ) |
179 |
|
eqid |
⊢ ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) = ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) |
180 |
|
negex |
⊢ - ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ∈ V |
181 |
178 179 180
|
fvmpt |
⊢ ( ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ∈ ℝ+ → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) = - ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ) |
182 |
173 181
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) = - ( log ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) ) |
183 |
|
fveq2 |
⊢ ( 𝑤 = 𝑥 → ( log ‘ 𝑤 ) = ( log ‘ 𝑥 ) ) |
184 |
183
|
negeqd |
⊢ ( 𝑤 = 𝑥 → - ( log ‘ 𝑤 ) = - ( log ‘ 𝑥 ) ) |
185 |
|
negex |
⊢ - ( log ‘ 𝑥 ) ∈ V |
186 |
184 179 185
|
fvmpt |
⊢ ( 𝑥 ∈ ℝ+ → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) = - ( log ‘ 𝑥 ) ) |
187 |
159 186
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) = - ( log ‘ 𝑥 ) ) |
188 |
187
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 𝑡 · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) ) = ( 𝑡 · - ( log ‘ 𝑥 ) ) ) |
189 |
158
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → 𝑡 ∈ ℂ ) |
190 |
160
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( log ‘ 𝑥 ) ∈ ℂ ) |
191 |
189 190
|
mulneg2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 𝑡 · - ( log ‘ 𝑥 ) ) = - ( 𝑡 · ( log ‘ 𝑥 ) ) ) |
192 |
188 191
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 𝑡 · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) ) = - ( 𝑡 · ( log ‘ 𝑥 ) ) ) |
193 |
|
fveq2 |
⊢ ( 𝑤 = 𝑦 → ( log ‘ 𝑤 ) = ( log ‘ 𝑦 ) ) |
194 |
193
|
negeqd |
⊢ ( 𝑤 = 𝑦 → - ( log ‘ 𝑤 ) = - ( log ‘ 𝑦 ) ) |
195 |
|
negex |
⊢ - ( log ‘ 𝑦 ) ∈ V |
196 |
194 179 195
|
fvmpt |
⊢ ( 𝑦 ∈ ℝ+ → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) = - ( log ‘ 𝑦 ) ) |
197 |
165 196
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) = - ( log ‘ 𝑦 ) ) |
198 |
197
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 1 − 𝑡 ) · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) ) = ( ( 1 − 𝑡 ) · - ( log ‘ 𝑦 ) ) ) |
199 |
164
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 1 − 𝑡 ) ∈ ℂ ) |
200 |
166
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( log ‘ 𝑦 ) ∈ ℂ ) |
201 |
199 200
|
mulneg2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 1 − 𝑡 ) · - ( log ‘ 𝑦 ) ) = - ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) |
202 |
198 201
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 1 − 𝑡 ) · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) ) = - ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) |
203 |
192 202
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑡 · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) ) ) = ( - ( 𝑡 · ( log ‘ 𝑥 ) ) + - ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) ) |
204 |
161
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( 𝑡 · ( log ‘ 𝑥 ) ) ∈ ℂ ) |
205 |
167
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ∈ ℂ ) |
206 |
204 205
|
negdid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → - ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) = ( - ( 𝑡 · ( log ‘ 𝑥 ) ) + - ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) ) |
207 |
203 206
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑡 · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) ) ) = - ( ( 𝑡 · ( log ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( log ‘ 𝑦 ) ) ) ) |
208 |
176 182 207
|
3brtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑥 < 𝑦 ) ∧ 𝑡 ∈ ( 0 (,) 1 ) ) → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( 𝑡 · 𝑥 ) + ( ( 1 − 𝑡 ) · 𝑦 ) ) ) < ( ( 𝑡 · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑥 ) ) + ( ( 1 − 𝑡 ) · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑦 ) ) ) ) |
209 |
119 123 130 208
|
scvxcvx |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ℝ+ ∧ 𝑣 ∈ ℝ+ ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ) → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( 𝑠 · 𝑢 ) + ( ( 1 − 𝑠 ) · 𝑣 ) ) ) ≤ ( ( 𝑠 · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑢 ) ) + ( ( 1 − 𝑠 ) · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ 𝑣 ) ) ) ) |
210 |
119 123 130 2 137 4 153 209
|
jensen |
⊢ ( 𝜑 → ( ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ∈ ℝ+ ∧ ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ) ≤ ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ) ) |
211 |
210
|
simprd |
⊢ ( 𝜑 → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ) ≤ ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ) |
212 |
131
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 1 / ( ♯ ‘ 𝐴 ) ) ∈ ℝ ) |
213 |
139
|
a1i |
⊢ ( 𝜑 → ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) = ( 𝑘 ∈ 𝐴 ↦ ( 1 / ( ♯ ‘ 𝐴 ) ) ) ) |
214 |
2 212 13 213 37
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) = ( 𝑘 ∈ 𝐴 ↦ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( 𝐹 ‘ 𝑘 ) ) ) ) |
215 |
214
|
oveq2d |
⊢ ( 𝜑 → ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) = ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
216 |
|
cnfldadd |
⊢ + = ( +g ‘ ℂfld ) |
217 |
|
cnfldmul |
⊢ · = ( .r ‘ ℂfld ) |
218 |
6
|
a1i |
⊢ ( 𝜑 → ℂfld ∈ Ring ) |
219 |
109
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) : 𝐴 ⟶ ℂ ) |
220 |
219 2 18
|
fdmfifsupp |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) finSupp 0 ) |
221 |
53 5 216 217 218 2 143 109 220
|
gsummulc2 |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
222 |
|
fss |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℝ+ ∧ ℝ+ ⊆ ℝ ) → 𝐹 : 𝐴 ⟶ ℝ ) |
223 |
4 118 222
|
sylancl |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ ) |
224 |
4 2 18
|
fdmfifsupp |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
225 |
5 8 2 12 223 224
|
gsumsubgcl |
⊢ ( 𝜑 → ( ℂfld Σg 𝐹 ) ∈ ℝ ) |
226 |
225
|
recnd |
⊢ ( 𝜑 → ( ℂfld Σg 𝐹 ) ∈ ℂ ) |
227 |
226 25 26
|
divrec2d |
⊢ ( 𝜑 → ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg 𝐹 ) ) ) |
228 |
108
|
oveq2d |
⊢ ( 𝜑 → ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg 𝐹 ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
229 |
227 228
|
eqtr2d |
⊢ ( 𝜑 → ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) |
230 |
215 221 229
|
3eqtrd |
⊢ ( 𝜑 → ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) |
231 |
230 152
|
oveq12d |
⊢ ( 𝜑 → ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) = ( ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) / 1 ) ) |
232 |
225 24
|
nndivred |
⊢ ( 𝜑 → ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ∈ ℝ ) |
233 |
232
|
recnd |
⊢ ( 𝜑 → ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ∈ ℂ ) |
234 |
233
|
div1d |
⊢ ( 𝜑 → ( ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) / 1 ) = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) |
235 |
231 234
|
eqtrd |
⊢ ( 𝜑 → ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) |
236 |
235
|
fveq2d |
⊢ ( 𝜑 → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ) = ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
237 |
|
fveq2 |
⊢ ( 𝑤 = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) → ( log ‘ 𝑤 ) = ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
238 |
237
|
negeqd |
⊢ ( 𝑤 = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) → - ( log ‘ 𝑤 ) = - ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
239 |
|
negex |
⊢ - ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ∈ V |
240 |
238 179 239
|
fvmpt |
⊢ ( ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ∈ ℝ+ → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) = - ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
241 |
115 240
|
syl |
⊢ ( 𝜑 → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) = - ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
242 |
236 241
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ‘ ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · 𝐹 ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) ) = - ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
243 |
53 5 216 217 218 2 143 32 19
|
gsummulc2 |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
244 |
|
negex |
⊢ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ V |
245 |
244
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ V ) |
246 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) = ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ) |
247 |
|
fveq2 |
⊢ ( 𝑤 = ( 𝐹 ‘ 𝑘 ) → ( log ‘ 𝑤 ) = ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
248 |
247
|
negeqd |
⊢ ( 𝑤 = ( 𝐹 ‘ 𝑘 ) → - ( log ‘ 𝑤 ) = - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
249 |
13 37 246 248
|
fmptco |
⊢ ( 𝜑 → ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) = ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
250 |
2 212 245 213 249
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
251 |
250
|
oveq2d |
⊢ ( 𝜑 → ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) ) = ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
252 |
21 25 26
|
divrec2d |
⊢ ( 𝜑 → ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) = ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
253 |
243 251 252
|
3eqtr4d |
⊢ ( 𝜑 → ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) ) = ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ) |
254 |
253 152
|
oveq12d |
⊢ ( 𝜑 → ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) = ( ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) / 1 ) ) |
255 |
117
|
recnd |
⊢ ( 𝜑 → ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ∈ ℂ ) |
256 |
255
|
div1d |
⊢ ( 𝜑 → ( ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) / 1 ) = ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ) |
257 |
254 256
|
eqtrd |
⊢ ( 𝜑 → ( ( ℂfld Σg ( ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ∘f · ( ( 𝑤 ∈ ℝ+ ↦ - ( log ‘ 𝑤 ) ) ∘ 𝐹 ) ) ) / ( ℂfld Σg ( 𝐴 × { ( 1 / ( ♯ ‘ 𝐴 ) ) } ) ) ) = ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ) |
258 |
211 242 257
|
3brtr3d |
⊢ ( 𝜑 → - ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ≤ ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ) |
259 |
116 117 258
|
lenegcon1d |
⊢ ( 𝜑 → - ( ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ - ( log ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) / ( ♯ ‘ 𝐴 ) ) ≤ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
260 |
107 259
|
eqbrtrrd |
⊢ ( 𝜑 → ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ≤ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) |
261 |
131 104
|
remulcld |
⊢ ( 𝜑 → ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ∈ ℝ ) |
262 |
|
efle |
⊢ ( ( ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ∈ ℝ ∧ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ∈ ℝ ) → ( ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ≤ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ↔ ( exp ‘ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ) ≤ ( exp ‘ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) ) ) |
263 |
261 116 262
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ≤ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ↔ ( exp ‘ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ) ≤ ( exp ‘ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) ) ) |
264 |
260 263
|
mpbid |
⊢ ( 𝜑 → ( exp ‘ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ) ≤ ( exp ‘ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) ) |
265 |
100
|
rpcnd |
⊢ ( 𝜑 → ( 𝑀 Σg 𝐹 ) ∈ ℂ ) |
266 |
100
|
rpne0d |
⊢ ( 𝜑 → ( 𝑀 Σg 𝐹 ) ≠ 0 ) |
267 |
265 266 143
|
cxpefd |
⊢ ( 𝜑 → ( ( 𝑀 Σg 𝐹 ) ↑𝑐 ( 1 / ( ♯ ‘ 𝐴 ) ) ) = ( exp ‘ ( ( 1 / ( ♯ ‘ 𝐴 ) ) · ( log ‘ ( 𝑀 Σg 𝐹 ) ) ) ) ) |
268 |
115
|
reeflogd |
⊢ ( 𝜑 → ( exp ‘ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) = ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) |
269 |
268
|
eqcomd |
⊢ ( 𝜑 → ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) = ( exp ‘ ( log ‘ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) ) ) |
270 |
264 267 269
|
3brtr4d |
⊢ ( 𝜑 → ( ( 𝑀 Σg 𝐹 ) ↑𝑐 ( 1 / ( ♯ ‘ 𝐴 ) ) ) ≤ ( ( ℂfld Σg 𝐹 ) / ( ♯ ‘ 𝐴 ) ) ) |