| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7440 | . . . . . . . 8
⊢ (𝑔 = ∅ → (𝐹 ++ 𝑔) = (𝐹 ++ ∅)) | 
| 2 | 1 | fveq2d 6909 | . . . . . . 7
⊢ (𝑔 = ∅ → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ ∅))) | 
| 3 | 2 | fveq1d 6907 | . . . . . 6
⊢ (𝑔 = ∅ → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ ∅))‘𝑁)) | 
| 4 | 3 | eqeq1d 2738 | . . . . 5
⊢ (𝑔 = ∅ → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))) | 
| 5 | 4 | imbi2d 340 | . . . 4
⊢ (𝑔 = ∅ → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))) | 
| 6 |  | oveq2 7440 | . . . . . . . 8
⊢ (𝑔 = 𝑒 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒)) | 
| 7 | 6 | fveq2d 6909 | . . . . . . 7
⊢ (𝑔 = 𝑒 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝑒))) | 
| 8 | 7 | fveq1d 6907 | . . . . . 6
⊢ (𝑔 = 𝑒 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁)) | 
| 9 | 8 | eqeq1d 2738 | . . . . 5
⊢ (𝑔 = 𝑒 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁))) | 
| 10 | 9 | imbi2d 340 | . . . 4
⊢ (𝑔 = 𝑒 → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))) | 
| 11 |  | oveq2 7440 | . . . . . . . 8
⊢ (𝑔 = (𝑒 ++ 〈“𝑘”〉) → (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ 〈“𝑘”〉))) | 
| 12 | 11 | fveq2d 6909 | . . . . . . 7
⊢ (𝑔 = (𝑒 ++ 〈“𝑘”〉) → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))) | 
| 13 | 12 | fveq1d 6907 | . . . . . 6
⊢ (𝑔 = (𝑒 ++ 〈“𝑘”〉) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁)) | 
| 14 | 13 | eqeq1d 2738 | . . . . 5
⊢ (𝑔 = (𝑒 ++ 〈“𝑘”〉) → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘𝐹)‘𝑁))) | 
| 15 | 14 | imbi2d 340 | . . . 4
⊢ (𝑔 = (𝑒 ++ 〈“𝑘”〉) → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))) | 
| 16 |  | oveq2 7440 | . . . . . . . 8
⊢ (𝑔 = 𝐺 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺)) | 
| 17 | 16 | fveq2d 6909 | . . . . . . 7
⊢ (𝑔 = 𝐺 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝐺))) | 
| 18 | 17 | fveq1d 6907 | . . . . . 6
⊢ (𝑔 = 𝐺 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝐺))‘𝑁)) | 
| 19 | 18 | eqeq1d 2738 | . . . . 5
⊢ (𝑔 = 𝐺 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))) | 
| 20 | 19 | imbi2d 340 | . . . 4
⊢ (𝑔 = 𝐺 → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))) | 
| 21 |  | ccatrid 14626 | . . . . . . 7
⊢ (𝐹 ∈ Word ℝ →
(𝐹 ++ ∅) = 𝐹) | 
| 22 | 21 | fveq2d 6909 | . . . . . 6
⊢ (𝐹 ∈ Word ℝ →
(𝑇‘(𝐹 ++ ∅)) = (𝑇‘𝐹)) | 
| 23 | 22 | fveq1d 6907 | . . . . 5
⊢ (𝐹 ∈ Word ℝ →
((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) | 
| 24 | 23 | adantr 480 | . . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) | 
| 25 |  | s1cl 14641 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℝ →
〈“𝑘”〉
∈ Word ℝ) | 
| 26 |  | ccatass 14627 | . . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧
〈“𝑘”〉
∈ Word ℝ) → ((𝐹 ++ 𝑒) ++ 〈“𝑘”〉) = (𝐹 ++ (𝑒 ++ 〈“𝑘”〉))) | 
| 27 | 25, 26 | syl3an3 1165 | . . . . . . . . . . . . 13
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ++ 𝑒) ++ 〈“𝑘”〉) = (𝐹 ++ (𝑒 ++ 〈“𝑘”〉))) | 
| 28 | 27 | 3expb 1120 | . . . . . . . . . . . 12
⊢ ((𝐹 ∈ Word ℝ ∧
(𝑒 ∈ Word ℝ
∧ 𝑘 ∈ ℝ))
→ ((𝐹 ++ 𝑒) ++ 〈“𝑘”〉) = (𝐹 ++ (𝑒 ++ 〈“𝑘”〉))) | 
| 29 | 28 | adantlr 715 | . . . . . . . . . . 11
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → ((𝐹 ++
𝑒) ++ 〈“𝑘”〉) = (𝐹 ++ (𝑒 ++ 〈“𝑘”〉))) | 
| 30 | 29 | fveq2d 6909 | . . . . . . . . . 10
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → (𝑇‘((𝐹 ++ 𝑒) ++ 〈“𝑘”〉)) = (𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))) | 
| 31 | 30 | fveq1d 6907 | . . . . . . . . 9
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ 〈“𝑘”〉))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁)) | 
| 32 |  | ccatcl 14613 | . . . . . . . . . . 11
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(𝐹 ++ 𝑒) ∈ Word ℝ) | 
| 33 | 32 | ad2ant2r 747 | . . . . . . . . . 10
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → (𝐹 ++
𝑒) ∈ Word
ℝ) | 
| 34 |  | simprr 772 | . . . . . . . . . 10
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → 𝑘 ∈
ℝ) | 
| 35 |  | lencl 14572 | . . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ Word ℝ →
(♯‘𝐹) ∈
ℕ0) | 
| 36 | 35 | nn0zd 12641 | . . . . . . . . . . . . . . 15
⊢ (𝐹 ∈ Word ℝ →
(♯‘𝐹) ∈
ℤ) | 
| 37 | 36 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘𝐹) ∈
ℤ) | 
| 38 |  | lencl 14572 | . . . . . . . . . . . . . . . 16
⊢ ((𝐹 ++ 𝑒) ∈ Word ℝ →
(♯‘(𝐹 ++ 𝑒)) ∈
ℕ0) | 
| 39 | 32, 38 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘(𝐹 ++ 𝑒)) ∈
ℕ0) | 
| 40 | 39 | nn0zd 12641 | . . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘(𝐹 ++ 𝑒)) ∈
ℤ) | 
| 41 | 35 | nn0red 12590 | . . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ Word ℝ →
(♯‘𝐹) ∈
ℝ) | 
| 42 |  | lencl 14572 | . . . . . . . . . . . . . . . 16
⊢ (𝑒 ∈ Word ℝ →
(♯‘𝑒) ∈
ℕ0) | 
| 43 |  | nn0addge1 12574 | . . . . . . . . . . . . . . . 16
⊢
(((♯‘𝐹)
∈ ℝ ∧ (♯‘𝑒) ∈ ℕ0) →
(♯‘𝐹) ≤
((♯‘𝐹) +
(♯‘𝑒))) | 
| 44 | 41, 42, 43 | syl2an 596 | . . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘𝐹) ≤
((♯‘𝐹) +
(♯‘𝑒))) | 
| 45 |  | ccatlen 14614 | . . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒))) | 
| 46 | 44, 45 | breqtrrd 5170 | . . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘𝐹) ≤
(♯‘(𝐹 ++ 𝑒))) | 
| 47 |  | eluz2 12885 | . . . . . . . . . . . . . 14
⊢
((♯‘(𝐹
++ 𝑒)) ∈
(ℤ≥‘(♯‘𝐹)) ↔ ((♯‘𝐹) ∈ ℤ ∧ (♯‘(𝐹 ++ 𝑒)) ∈ ℤ ∧ (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒)))) | 
| 48 | 37, 40, 46, 47 | syl3anbrc 1343 | . . . . . . . . . . . . 13
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(♯‘(𝐹 ++ 𝑒)) ∈
(ℤ≥‘(♯‘𝐹))) | 
| 49 |  | fzoss2 13728 | . . . . . . . . . . . . 13
⊢
((♯‘(𝐹
++ 𝑒)) ∈
(ℤ≥‘(♯‘𝐹)) → (0..^(♯‘𝐹)) ⊆
(0..^(♯‘(𝐹 ++
𝑒)))) | 
| 50 | 48, 49 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) →
(0..^(♯‘𝐹))
⊆ (0..^(♯‘(𝐹 ++ 𝑒)))) | 
| 51 | 50 | ad2ant2r 747 | . . . . . . . . . . 11
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒)))) | 
| 52 |  | simplr 768 | . . . . . . . . . . 11
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → 𝑁 ∈
(0..^(♯‘𝐹))) | 
| 53 | 51, 52 | sseldd 3983 | . . . . . . . . . 10
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → 𝑁 ∈
(0..^(♯‘(𝐹 ++
𝑒)))) | 
| 54 |  | signsv.p | . . . . . . . . . . 11
⊢  ⨣ =
(𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦
if(𝑏 = 0, 𝑎, 𝑏)) | 
| 55 |  | signsv.w | . . . . . . . . . . 11
⊢ 𝑊 = {〈(Base‘ndx), {-1,
0, 1}〉, 〈(+g‘ndx), ⨣
〉} | 
| 56 |  | signsv.t | . . . . . . . . . . 11
⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈
(0..^(♯‘𝑓))
↦ (𝑊
Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) | 
| 57 |  | signsv.v | . . . . . . . . . . 11
⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈
(1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) | 
| 58 | 54, 55, 56, 57 | signstfvp 34587 | . . . . . . . . . 10
⊢ (((𝐹 ++ 𝑒) ∈ Word ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑁 ∈
(0..^(♯‘(𝐹 ++
𝑒)))) → ((𝑇‘((𝐹 ++ 𝑒) ++ 〈“𝑘”〉))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁)) | 
| 59 | 33, 34, 53, 58 | syl3anc 1372 | . . . . . . . . 9
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ 〈“𝑘”〉))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁)) | 
| 60 | 31, 59 | eqtr3d 2778 | . . . . . . . 8
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁)) | 
| 61 |  | id 22 | . . . . . . . 8
⊢ (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) | 
| 62 | 60, 61 | sylan9eq 2796 | . . . . . . 7
⊢ ((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) ∧ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) → ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) | 
| 63 | 62 | ex 412 | . . . . . 6
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
∧ (𝑒 ∈ Word
ℝ ∧ 𝑘 ∈
ℝ)) → (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘𝐹)‘𝑁))) | 
| 64 | 63 | expcom 413 | . . . . 5
⊢ ((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))) | 
| 65 | 64 | a2d 29 | . . . 4
⊢ ((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ (𝑒 ++ 〈“𝑘”〉)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))) | 
| 66 | 5, 10, 15, 20, 24, 65 | wrdind 14761 | . . 3
⊢ (𝐺 ∈ Word ℝ →
((𝐹 ∈ Word ℝ
∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))) | 
| 67 | 66 | 3impib 1116 | . 2
⊢ ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) | 
| 68 | 67 | 3com12 1123 | 1
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) |