| Step | Hyp | Ref
| Expression |
| 1 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑎 = (𝑇‘𝑏) → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘(𝑇‘𝑏)))) |
| 2 | | tpfi 9365 |
. . . . . 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 13791 |
. . . . . . 7
⊢ (0..^3) =
{0, 1, 2} |
| 6 | | f1oeq23 6839 |
. . . . . . 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 692 |
. . . . . 6
⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) ↔ 𝑇:{0, 1, 2}–1-1-onto→{0,
1, 2}) |
| 8 | 4, 7 | sylib 218 |
. . . . 5
⊢ (𝜑 → 𝑇:{0, 1, 2}–1-1-onto→{0,
1, 2}) |
| 9 | | eqidd 2738 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝑇‘𝑏) = (𝑇‘𝑏)) |
| 10 | | hgt750lemg.l |
. . . . . . . 8
⊢ (𝜑 → 𝐿:ℕ⟶ℝ) |
| 11 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝐿:ℕ⟶ℝ) |
| 12 | | hgt750lemg.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑁:(0..^3)⟶ℕ) |
| 13 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑁:(0..^3)⟶ℕ) |
| 14 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑎 ∈ {0, 1, 2}) |
| 15 | 14, 5 | eleqtrrdi 2852 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑎 ∈ (0..^3)) |
| 16 | 13, 15 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝑁‘𝑎) ∈ ℕ) |
| 17 | 11, 16 | ffvelcdmd 7105 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘𝑎)) ∈ ℝ) |
| 18 | 17 | recnd 11289 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘𝑎)) ∈ ℂ) |
| 19 | 1, 3, 8, 9, 18 | fprodf1o 15982 |
. . . 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 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 = 𝑁) → 𝑐 = 𝑁) |
| 23 | 22 | coeq1d 5872 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 = 𝑁) → (𝑐 ∘ 𝑇) = (𝑁 ∘ 𝑇)) |
| 24 | | hgt750lemg.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ 𝑅) |
| 25 | | f1of 6848 |
. . . . . . . . . . . . 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 7466 |
. . . . . . . . . . . 12
⊢ (𝜑 → (0..^3) ∈
V) |
| 28 | 26, 27 | fexd 7247 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ∈ V) |
| 29 | | coexg 7951 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ 𝑅 ∧ 𝑇 ∈ V) → (𝑁 ∘ 𝑇) ∈ V) |
| 30 | 24, 28, 29 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 ∘ 𝑇) ∈ V) |
| 31 | 21, 23, 24, 30 | fvmptd 7023 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑁) = (𝑁 ∘ 𝑇)) |
| 32 | 31 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝐹‘𝑁) = (𝑁 ∘ 𝑇)) |
| 33 | 32 | fveq1d 6908 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → ((𝐹‘𝑁)‘𝑏) = ((𝑁 ∘ 𝑇)‘𝑏)) |
| 34 | | f1ofun 6850 |
. . . . . . . . . 10
⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) → Fun 𝑇) |
| 35 | 4, 34 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝑇) |
| 36 | 35 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → Fun 𝑇) |
| 37 | | f1odm 6852 |
. . . . . . . . . . 11
⊢ (𝑇:{0, 1, 2}–1-1-onto→{0,
1, 2} → dom 𝑇 = {0, 1,
2}) |
| 38 | 8, 37 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → dom 𝑇 = {0, 1, 2}) |
| 39 | 38 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝜑 → (𝑏 ∈ dom 𝑇 ↔ 𝑏 ∈ {0, 1, 2})) |
| 40 | 39 | biimpar 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → 𝑏 ∈ dom 𝑇) |
| 41 | | fvco 7007 |
. . . . . . . 8
⊢ ((Fun
𝑇 ∧ 𝑏 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘𝑏) = (𝑁‘(𝑇‘𝑏))) |
| 42 | 36, 40, 41 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → ((𝑁 ∘ 𝑇)‘𝑏) = (𝑁‘(𝑇‘𝑏))) |
| 43 | 33, 42 | eqtr2d 2778 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝑁‘(𝑇‘𝑏)) = ((𝐹‘𝑁)‘𝑏)) |
| 44 | 43 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘(𝑇‘𝑏))) = (𝐿‘((𝐹‘𝑁)‘𝑏))) |
| 45 | 44 | prodeq2dv 15958 |
. . . 4
⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘(𝑁‘(𝑇‘𝑏))) = ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏))) |
| 46 | 19, 45 | eqtr2d 2778 |
. . 3
⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)) = ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎))) |
| 47 | | 2fveq3 6911 |
. . . 4
⊢ (𝑏 = 0 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘0))) |
| 48 | | 2fveq3 6911 |
. . . 4
⊢ (𝑏 = 1 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘1))) |
| 49 | | c0ex 11255 |
. . . . 5
⊢ 0 ∈
V |
| 50 | 49 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ∈
V) |
| 51 | | 1ex 11257 |
. . . . 5
⊢ 1 ∈
V |
| 52 | 51 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 ∈
V) |
| 53 | 31 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁)‘0) = ((𝑁 ∘ 𝑇)‘0)) |
| 54 | 49 | tpid1 4768 |
. . . . . . . . . 10
⊢ 0 ∈
{0, 1, 2} |
| 55 | 54, 38 | eleqtrrid 2848 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈ dom 𝑇) |
| 56 | | fvco 7007 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 0 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘0) = (𝑁‘(𝑇‘0))) |
| 57 | 35, 55, 56 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘0) = (𝑁‘(𝑇‘0))) |
| 58 | 53, 57 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑁)‘0) = (𝑁‘(𝑇‘0))) |
| 59 | 54, 5 | eleqtrri 2840 |
. . . . . . . . . 10
⊢ 0 ∈
(0..^3) |
| 60 | 59 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈
(0..^3)) |
| 61 | 26, 60 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘0) ∈ (0..^3)) |
| 62 | 12, 61 | ffvelcdmd 7105 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝑇‘0)) ∈ ℕ) |
| 63 | 58, 62 | eqeltrd 2841 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑁)‘0) ∈ ℕ) |
| 64 | 10, 63 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘0)) ∈ ℝ) |
| 65 | 64 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘0)) ∈ ℂ) |
| 66 | 31 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁)‘1) = ((𝑁 ∘ 𝑇)‘1)) |
| 67 | 51 | tpid2 4770 |
. . . . . . . . . 10
⊢ 1 ∈
{0, 1, 2} |
| 68 | 67, 38 | eleqtrrid 2848 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈ dom 𝑇) |
| 69 | | fvco 7007 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 1 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘1) = (𝑁‘(𝑇‘1))) |
| 70 | 35, 68, 69 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘1) = (𝑁‘(𝑇‘1))) |
| 71 | 66, 70 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑁)‘1) = (𝑁‘(𝑇‘1))) |
| 72 | 67, 5 | eleqtrri 2840 |
. . . . . . . . . 10
⊢ 1 ∈
(0..^3) |
| 73 | 72 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
(0..^3)) |
| 74 | 26, 73 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘1) ∈ (0..^3)) |
| 75 | 12, 74 | ffvelcdmd 7105 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝑇‘1)) ∈ ℕ) |
| 76 | 71, 75 | eqeltrd 2841 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑁)‘1) ∈ ℕ) |
| 77 | 10, 76 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘1)) ∈ ℝ) |
| 78 | 77 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘1)) ∈ ℂ) |
| 79 | | 0ne1 12337 |
. . . . 5
⊢ 0 ≠
1 |
| 80 | 79 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ≠ 1) |
| 81 | | 2fveq3 6911 |
. . . 4
⊢ (𝑏 = 2 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘2))) |
| 82 | | 2ex 12343 |
. . . . 5
⊢ 2 ∈
V |
| 83 | 82 | a1i 11 |
. . . 4
⊢ (𝜑 → 2 ∈
V) |
| 84 | 31 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁)‘2) = ((𝑁 ∘ 𝑇)‘2)) |
| 85 | 82 | tpid3 4773 |
. . . . . . . . . 10
⊢ 2 ∈
{0, 1, 2} |
| 86 | 85, 38 | eleqtrrid 2848 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈ dom 𝑇) |
| 87 | | fvco 7007 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 2 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘2) = (𝑁‘(𝑇‘2))) |
| 88 | 35, 86, 87 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘2) = (𝑁‘(𝑇‘2))) |
| 89 | 84, 88 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑁)‘2) = (𝑁‘(𝑇‘2))) |
| 90 | 85, 5 | eleqtrri 2840 |
. . . . . . . . . 10
⊢ 2 ∈
(0..^3) |
| 91 | 90 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
(0..^3)) |
| 92 | 26, 91 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘2) ∈ (0..^3)) |
| 93 | 12, 92 | ffvelcdmd 7105 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝑇‘2)) ∈ ℕ) |
| 94 | 89, 93 | eqeltrd 2841 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑁)‘2) ∈ ℕ) |
| 95 | 10, 94 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘2)) ∈ ℝ) |
| 96 | 95 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘2)) ∈ ℂ) |
| 97 | | 0ne2 12473 |
. . . . 5
⊢ 0 ≠
2 |
| 98 | 97 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ≠ 2) |
| 99 | | 1ne2 12474 |
. . . . 5
⊢ 1 ≠
2 |
| 100 | 99 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 ≠ 2) |
| 101 | 47, 48, 50, 52, 65, 78, 80, 81, 83, 96, 98, 100 | prodtp 32829 |
. . 3
⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)) = (((𝐿‘((𝐹‘𝑁)‘0)) · (𝐿‘((𝐹‘𝑁)‘1))) · (𝐿‘((𝐹‘𝑁)‘2)))) |
| 102 | | 2fveq3 6911 |
. . . 4
⊢ (𝑎 = 0 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘0))) |
| 103 | | 2fveq3 6911 |
. . . 4
⊢ (𝑎 = 1 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘1))) |
| 104 | 12, 60 | ffvelcdmd 7105 |
. . . . . 6
⊢ (𝜑 → (𝑁‘0) ∈ ℕ) |
| 105 | 10, 104 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐿‘(𝑁‘0)) ∈ ℝ) |
| 106 | 105 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝐿‘(𝑁‘0)) ∈ ℂ) |
| 107 | 12, 73 | ffvelcdmd 7105 |
. . . . . 6
⊢ (𝜑 → (𝑁‘1) ∈ ℕ) |
| 108 | 10, 107 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐿‘(𝑁‘1)) ∈ ℝ) |
| 109 | 108 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝐿‘(𝑁‘1)) ∈ ℂ) |
| 110 | | 2fveq3 6911 |
. . . 4
⊢ (𝑎 = 2 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘2))) |
| 111 | 12, 91 | ffvelcdmd 7105 |
. . . . . 6
⊢ (𝜑 → (𝑁‘2) ∈ ℕ) |
| 112 | 10, 111 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐿‘(𝑁‘2)) ∈ ℝ) |
| 113 | 112 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝐿‘(𝑁‘2)) ∈ ℂ) |
| 114 | 102, 103,
50, 52, 106, 109, 80, 110, 83, 113, 98, 100 | prodtp 32829 |
. . 3
⊢ (𝜑 → ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎)) = (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2)))) |
| 115 | 46, 101, 114 | 3eqtr3d 2785 |
. 2
⊢ (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 11284 |
. 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))))) |
| 117 | 106, 109,
113 | mulassd 11284 |
. 2
⊢ (𝜑 → (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2))))) |
| 118 | 115, 116,
117 | 3eqtr3d 2785 |
1
⊢ (𝜑 → ((𝐿‘((𝐹‘𝑁)‘0)) · ((𝐿‘((𝐹‘𝑁)‘1)) · (𝐿‘((𝐹‘𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2))))) |