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