Step | Hyp | Ref
| Expression |
1 | | 2fveq3 6509 |
. . . . 5
⊢ (𝑎 = (𝑇‘𝑏) → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘(𝑇‘𝑏)))) |
2 | | tpfi 8595 |
. . . . . 6
⊢ {0, 1, 2}
∈ Fin |
3 | 2 | a1i 11 |
. . . . 5
⊢ (𝜑 → {0, 1, 2} ∈
Fin) |
4 | | hgt750lemg.t |
. . . . . 6
⊢ (𝜑 → 𝑇:(0..^3)–1-1-onto→(0..^3)) |
5 | | fzo0to3tp 12944 |
. . . . . . 7
⊢ (0..^3) =
{0, 1, 2} |
6 | | f1oeq23 6441 |
. . . . . . 7
⊢ (((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})) |
7 | 5, 5, 6 | mp2an 680 |
. . . . . 6
⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) ↔ 𝑇:{0, 1, 2}–1-1-onto→{0,
1, 2}) |
8 | 4, 7 | sylib 210 |
. . . . 5
⊢ (𝜑 → 𝑇:{0, 1, 2}–1-1-onto→{0,
1, 2}) |
9 | | eqidd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝑇‘𝑏) = (𝑇‘𝑏)) |
10 | | hgt750lemg.l |
. . . . . . . 8
⊢ (𝜑 → 𝐿:ℕ⟶ℝ) |
11 | 10 | adantr 473 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝐿:ℕ⟶ℝ) |
12 | | hgt750lemg.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑁:(0..^3)⟶ℕ) |
13 | 12 | adantr 473 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑁:(0..^3)⟶ℕ) |
14 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑎 ∈ {0, 1, 2}) |
15 | 14, 5 | syl6eleqr 2879 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑎 ∈ (0..^3)) |
16 | 13, 15 | ffvelrnd 6683 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝑁‘𝑎) ∈ ℕ) |
17 | 11, 16 | ffvelrnd 6683 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘𝑎)) ∈ ℝ) |
18 | 17 | recnd 10474 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘𝑎)) ∈ ℂ) |
19 | 1, 3, 8, 9, 18 | fprodf1o 15166 |
. . . 4
⊢ (𝜑 → ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎)) = ∏𝑏 ∈ {0, 1, 2} (𝐿‘(𝑁‘(𝑇‘𝑏)))) |
20 | | hgt750lemg.f |
. . . . . . . . . . 11
⊢ 𝐹 = (𝑐 ∈ 𝑅 ↦ (𝑐 ∘ 𝑇)) |
21 | 20 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑐 ∈ 𝑅 ↦ (𝑐 ∘ 𝑇))) |
22 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 = 𝑁) → 𝑐 = 𝑁) |
23 | 22 | coeq1d 5586 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 = 𝑁) → (𝑐 ∘ 𝑇) = (𝑁 ∘ 𝑇)) |
24 | | hgt750lemg.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ 𝑅) |
25 | | f1of 6449 |
. . . . . . . . . . . . 13
⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) → 𝑇:(0..^3)⟶(0..^3)) |
26 | 4, 25 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇:(0..^3)⟶(0..^3)) |
27 | | ovexd 7016 |
. . . . . . . . . . . 12
⊢ (𝜑 → (0..^3) ∈
V) |
28 | | fex2 7459 |
. . . . . . . . . . . 12
⊢ ((𝑇:(0..^3)⟶(0..^3) ∧
(0..^3) ∈ V ∧ (0..^3) ∈ V) → 𝑇 ∈ V) |
29 | 26, 27, 27, 28 | syl3anc 1352 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ∈ V) |
30 | | coexg 7455 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ 𝑅 ∧ 𝑇 ∈ V) → (𝑁 ∘ 𝑇) ∈ V) |
31 | 24, 29, 30 | syl2anc 576 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 ∘ 𝑇) ∈ V) |
32 | 21, 23, 24, 31 | fvmptd 6607 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑁) = (𝑁 ∘ 𝑇)) |
33 | 32 | adantr 473 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝐹‘𝑁) = (𝑁 ∘ 𝑇)) |
34 | 33 | fveq1d 6506 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → ((𝐹‘𝑁)‘𝑏) = ((𝑁 ∘ 𝑇)‘𝑏)) |
35 | | f1ofun 6451 |
. . . . . . . . . 10
⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) → Fun 𝑇) |
36 | 4, 35 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝑇) |
37 | 36 | adantr 473 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → Fun 𝑇) |
38 | | f1odm 6453 |
. . . . . . . . . . 11
⊢ (𝑇:{0, 1, 2}–1-1-onto→{0,
1, 2} → dom 𝑇 = {0, 1,
2}) |
39 | 8, 38 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → dom 𝑇 = {0, 1, 2}) |
40 | 39 | eleq2d 2853 |
. . . . . . . . 9
⊢ (𝜑 → (𝑏 ∈ dom 𝑇 ↔ 𝑏 ∈ {0, 1, 2})) |
41 | 40 | biimpar 470 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → 𝑏 ∈ dom 𝑇) |
42 | | fvco 6593 |
. . . . . . . 8
⊢ ((Fun
𝑇 ∧ 𝑏 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘𝑏) = (𝑁‘(𝑇‘𝑏))) |
43 | 37, 41, 42 | syl2anc 576 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → ((𝑁 ∘ 𝑇)‘𝑏) = (𝑁‘(𝑇‘𝑏))) |
44 | 34, 43 | eqtr2d 2817 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝑁‘(𝑇‘𝑏)) = ((𝐹‘𝑁)‘𝑏)) |
45 | 44 | fveq2d 6508 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘(𝑇‘𝑏))) = (𝐿‘((𝐹‘𝑁)‘𝑏))) |
46 | 45 | prodeq2dv 15143 |
. . . 4
⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘(𝑁‘(𝑇‘𝑏))) = ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏))) |
47 | 19, 46 | eqtr2d 2817 |
. . 3
⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)) = ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎))) |
48 | | 2fveq3 6509 |
. . . 4
⊢ (𝑏 = 0 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘0))) |
49 | | 2fveq3 6509 |
. . . 4
⊢ (𝑏 = 1 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘1))) |
50 | | c0ex 10439 |
. . . . 5
⊢ 0 ∈
V |
51 | 50 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ∈
V) |
52 | | 1ex 10441 |
. . . . 5
⊢ 1 ∈
V |
53 | 52 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 ∈
V) |
54 | 32 | fveq1d 6506 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁)‘0) = ((𝑁 ∘ 𝑇)‘0)) |
55 | 50 | tpid1 4583 |
. . . . . . . . . 10
⊢ 0 ∈
{0, 1, 2} |
56 | 55, 39 | syl5eleqr 2875 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈ dom 𝑇) |
57 | | fvco 6593 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 0 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘0) = (𝑁‘(𝑇‘0))) |
58 | 36, 56, 57 | syl2anc 576 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘0) = (𝑁‘(𝑇‘0))) |
59 | 54, 58 | eqtrd 2816 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑁)‘0) = (𝑁‘(𝑇‘0))) |
60 | 55, 5 | eleqtrri 2867 |
. . . . . . . . . 10
⊢ 0 ∈
(0..^3) |
61 | 60 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈
(0..^3)) |
62 | 26, 61 | ffvelrnd 6683 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘0) ∈ (0..^3)) |
63 | 12, 62 | ffvelrnd 6683 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝑇‘0)) ∈ ℕ) |
64 | 59, 63 | eqeltrd 2868 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑁)‘0) ∈ ℕ) |
65 | 10, 64 | ffvelrnd 6683 |
. . . . 5
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘0)) ∈ ℝ) |
66 | 65 | recnd 10474 |
. . . 4
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘0)) ∈ ℂ) |
67 | 32 | fveq1d 6506 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁)‘1) = ((𝑁 ∘ 𝑇)‘1)) |
68 | 52 | tpid2 4585 |
. . . . . . . . . 10
⊢ 1 ∈
{0, 1, 2} |
69 | 68, 39 | syl5eleqr 2875 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈ dom 𝑇) |
70 | | fvco 6593 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 1 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘1) = (𝑁‘(𝑇‘1))) |
71 | 36, 69, 70 | syl2anc 576 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘1) = (𝑁‘(𝑇‘1))) |
72 | 67, 71 | eqtrd 2816 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑁)‘1) = (𝑁‘(𝑇‘1))) |
73 | 68, 5 | eleqtrri 2867 |
. . . . . . . . . 10
⊢ 1 ∈
(0..^3) |
74 | 73 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
(0..^3)) |
75 | 26, 74 | ffvelrnd 6683 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘1) ∈ (0..^3)) |
76 | 12, 75 | ffvelrnd 6683 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝑇‘1)) ∈ ℕ) |
77 | 72, 76 | eqeltrd 2868 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑁)‘1) ∈ ℕ) |
78 | 10, 77 | ffvelrnd 6683 |
. . . . 5
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘1)) ∈ ℝ) |
79 | 78 | recnd 10474 |
. . . 4
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘1)) ∈ ℂ) |
80 | | 0ne1 11517 |
. . . . 5
⊢ 0 ≠
1 |
81 | 80 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ≠ 1) |
82 | | 2fveq3 6509 |
. . . 4
⊢ (𝑏 = 2 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘2))) |
83 | | 2ex 11523 |
. . . . 5
⊢ 2 ∈
V |
84 | 83 | a1i 11 |
. . . 4
⊢ (𝜑 → 2 ∈
V) |
85 | 32 | fveq1d 6506 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁)‘2) = ((𝑁 ∘ 𝑇)‘2)) |
86 | 83 | tpid3 4588 |
. . . . . . . . . 10
⊢ 2 ∈
{0, 1, 2} |
87 | 86, 39 | syl5eleqr 2875 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈ dom 𝑇) |
88 | | fvco 6593 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 2 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘2) = (𝑁‘(𝑇‘2))) |
89 | 36, 87, 88 | syl2anc 576 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘2) = (𝑁‘(𝑇‘2))) |
90 | 85, 89 | eqtrd 2816 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑁)‘2) = (𝑁‘(𝑇‘2))) |
91 | 86, 5 | eleqtrri 2867 |
. . . . . . . . . 10
⊢ 2 ∈
(0..^3) |
92 | 91 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
(0..^3)) |
93 | 26, 92 | ffvelrnd 6683 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘2) ∈ (0..^3)) |
94 | 12, 93 | ffvelrnd 6683 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝑇‘2)) ∈ ℕ) |
95 | 90, 94 | eqeltrd 2868 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑁)‘2) ∈ ℕ) |
96 | 10, 95 | ffvelrnd 6683 |
. . . . 5
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘2)) ∈ ℝ) |
97 | 96 | recnd 10474 |
. . . 4
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘2)) ∈ ℂ) |
98 | | 0ne2 11660 |
. . . . 5
⊢ 0 ≠
2 |
99 | 98 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ≠ 2) |
100 | | 1ne2 11661 |
. . . . 5
⊢ 1 ≠
2 |
101 | 100 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 ≠ 2) |
102 | 48, 49, 51, 53, 66, 79, 81, 82, 84, 97, 99, 101 | prodtp 30313 |
. . 3
⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)) = (((𝐿‘((𝐹‘𝑁)‘0)) · (𝐿‘((𝐹‘𝑁)‘1))) · (𝐿‘((𝐹‘𝑁)‘2)))) |
103 | | 2fveq3 6509 |
. . . 4
⊢ (𝑎 = 0 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘0))) |
104 | | 2fveq3 6509 |
. . . 4
⊢ (𝑎 = 1 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘1))) |
105 | 12, 61 | ffvelrnd 6683 |
. . . . . 6
⊢ (𝜑 → (𝑁‘0) ∈ ℕ) |
106 | 10, 105 | ffvelrnd 6683 |
. . . . 5
⊢ (𝜑 → (𝐿‘(𝑁‘0)) ∈ ℝ) |
107 | 106 | recnd 10474 |
. . . 4
⊢ (𝜑 → (𝐿‘(𝑁‘0)) ∈ ℂ) |
108 | 12, 74 | ffvelrnd 6683 |
. . . . . 6
⊢ (𝜑 → (𝑁‘1) ∈ ℕ) |
109 | 10, 108 | ffvelrnd 6683 |
. . . . 5
⊢ (𝜑 → (𝐿‘(𝑁‘1)) ∈ ℝ) |
110 | 109 | recnd 10474 |
. . . 4
⊢ (𝜑 → (𝐿‘(𝑁‘1)) ∈ ℂ) |
111 | | 2fveq3 6509 |
. . . 4
⊢ (𝑎 = 2 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘2))) |
112 | 12, 92 | ffvelrnd 6683 |
. . . . . 6
⊢ (𝜑 → (𝑁‘2) ∈ ℕ) |
113 | 10, 112 | ffvelrnd 6683 |
. . . . 5
⊢ (𝜑 → (𝐿‘(𝑁‘2)) ∈ ℝ) |
114 | 113 | recnd 10474 |
. . . 4
⊢ (𝜑 → (𝐿‘(𝑁‘2)) ∈ ℂ) |
115 | 103, 104,
51, 53, 107, 110, 81, 111, 84, 114, 99, 101 | prodtp 30313 |
. . 3
⊢ (𝜑 → ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎)) = (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2)))) |
116 | 47, 102, 115 | 3eqtr3d 2824 |
. 2
⊢ (𝜑 → (((𝐿‘((𝐹‘𝑁)‘0)) · (𝐿‘((𝐹‘𝑁)‘1))) · (𝐿‘((𝐹‘𝑁)‘2))) = (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2)))) |
117 | 66, 79, 97 | mulassd 10469 |
. 2
⊢ (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))))) |
118 | 107, 110,
114 | mulassd 10469 |
. 2
⊢ (𝜑 → (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2))))) |
119 | 116, 117,
118 | 3eqtr3d 2824 |
1
⊢ (𝜑 → ((𝐿‘((𝐹‘𝑁)‘0)) · ((𝐿‘((𝐹‘𝑁)‘1)) · (𝐿‘((𝐹‘𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2))))) |