Step | Hyp | Ref
| Expression |
1 | | fveq2 6722 |
. . . . . . . 8
⊢ (𝑤 = ∅ →
(♯‘𝑤) =
(♯‘∅)) |
2 | 1 | oveq2d 7234 |
. . . . . . 7
⊢ (𝑤 = ∅ →
(0..^(♯‘𝑤)) =
(0..^(♯‘∅))) |
3 | 2 | adantr 484 |
. . . . . 6
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(0..^(♯‘𝑤)) =
(0..^(♯‘∅))) |
4 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑤 = ∅ → (𝑤‘𝑖) = (∅‘𝑖)) |
5 | 4 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑤 = ∅ → ((𝑤‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛)) |
6 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑢 = ∅ → (𝑢‘𝑖) = (∅‘𝑖)) |
7 | 6 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑢 = ∅ → ((𝑢‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛)) |
8 | 5, 7 | eqeqan12d 2751 |
. . . . . . 7
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) → (((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛))) |
9 | 8 | ralbidv 3118 |
. . . . . 6
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛))) |
10 | 3, 9 | raleqbidv 3318 |
. . . . 5
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑖 ∈
(0..^(♯‘∅))∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛))) |
11 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑤 = ∅ → (𝑆 Σg
𝑤) = (𝑆 Σg
∅)) |
12 | 11 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑤 = ∅ → ((𝑆 Σg
𝑤)‘𝑛) = ((𝑆 Σg
∅)‘𝑛)) |
13 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑢 = ∅ → (𝑍 Σg
𝑢) = (𝑍 Σg
∅)) |
14 | 13 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑢 = ∅ → ((𝑍 Σg
𝑢)‘𝑛) = ((𝑍 Σg
∅)‘𝑛)) |
15 | 12, 14 | eqeqan12d 2751 |
. . . . . 6
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) → (((𝑆 Σg
𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛))) |
16 | 15 | ralbidv 3118 |
. . . . 5
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛))) |
17 | 10, 16 | imbi12d 348 |
. . . 4
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
((∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛)) ↔ (∀𝑖 ∈
(0..^(♯‘∅))∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛)))) |
18 | 17 | imbi2d 344 |
. . 3
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) →
(∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛))) ↔ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘∅))∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛))))) |
19 | | fveq2 6722 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥)) |
20 | 19 | oveq2d 7234 |
. . . . . . 7
⊢ (𝑤 = 𝑥 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑥))) |
21 | 20 | adantr 484 |
. . . . . 6
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (0..^(♯‘𝑤)) = (0..^(♯‘𝑥))) |
22 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑤 = 𝑥 → (𝑤‘𝑖) = (𝑥‘𝑖)) |
23 | 22 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → ((𝑤‘𝑖)‘𝑛) = ((𝑥‘𝑖)‘𝑛)) |
24 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑢 = 𝑦 → (𝑢‘𝑖) = (𝑦‘𝑖)) |
25 | 24 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑢 = 𝑦 → ((𝑢‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛)) |
26 | 23, 25 | eqeqan12d 2751 |
. . . . . . 7
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛))) |
27 | 26 | ralbidv 3118 |
. . . . . 6
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛))) |
28 | 21, 27 | raleqbidv 3318 |
. . . . 5
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(♯‘𝑥))∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛))) |
29 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → (𝑆 Σg 𝑤) = (𝑆 Σg 𝑥)) |
30 | 29 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑤 = 𝑥 → ((𝑆 Σg 𝑤)‘𝑛) = ((𝑆 Σg 𝑥)‘𝑛)) |
31 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑢 = 𝑦 → (𝑍 Σg 𝑢) = (𝑍 Σg 𝑦)) |
32 | 31 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑢 = 𝑦 → ((𝑍 Σg 𝑢)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛)) |
33 | 30, 32 | eqeqan12d 2751 |
. . . . . 6
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛))) |
34 | 33 | ralbidv 3118 |
. . . . 5
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛))) |
35 | 28, 34 | imbi12d 348 |
. . . 4
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → ((∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛)) ↔ (∀𝑖 ∈ (0..^(♯‘𝑥))∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛)))) |
36 | 35 | imbi2d 344 |
. . 3
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛))) ↔ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑥))∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛))))) |
37 | | fveq2 6722 |
. . . . . . . 8
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → (♯‘𝑤) = (♯‘(𝑥 ++ 〈“𝑏”〉))) |
38 | 37 | oveq2d 7234 |
. . . . . . 7
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) →
(0..^(♯‘𝑤)) =
(0..^(♯‘(𝑥 ++
〈“𝑏”〉)))) |
39 | 38 | adantr 484 |
. . . . . 6
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) →
(0..^(♯‘𝑤)) =
(0..^(♯‘(𝑥 ++
〈“𝑏”〉)))) |
40 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → (𝑤‘𝑖) = ((𝑥 ++ 〈“𝑏”〉)‘𝑖)) |
41 | 40 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → ((𝑤‘𝑖)‘𝑛) = (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛)) |
42 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑢 = (𝑦 ++ 〈“𝑝”〉) → (𝑢‘𝑖) = ((𝑦 ++ 〈“𝑝”〉)‘𝑖)) |
43 | 42 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑢 = (𝑦 ++ 〈“𝑝”〉) → ((𝑢‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛)) |
44 | 41, 43 | eqeqan12d 2751 |
. . . . . . 7
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → (((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛))) |
45 | 44 | ralbidv 3118 |
. . . . . 6
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → (∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛))) |
46 | 39, 45 | raleqbidv 3318 |
. . . . 5
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → (∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(♯‘(𝑥 ++ 〈“𝑏”〉)))∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛))) |
47 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → (𝑆 Σg 𝑤) = (𝑆 Σg (𝑥 ++ 〈“𝑏”〉))) |
48 | 47 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → ((𝑆 Σg 𝑤)‘𝑛) = ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛)) |
49 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑢 = (𝑦 ++ 〈“𝑝”〉) → (𝑍 Σg 𝑢) = (𝑍 Σg (𝑦 ++ 〈“𝑝”〉))) |
50 | 49 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑢 = (𝑦 ++ 〈“𝑝”〉) → ((𝑍 Σg 𝑢)‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛)) |
51 | 48, 50 | eqeqan12d 2751 |
. . . . . 6
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → (((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛))) |
52 | 51 | ralbidv 3118 |
. . . . 5
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛))) |
53 | 46, 52 | imbi12d 348 |
. . . 4
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → ((∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛)) ↔ (∀𝑖 ∈ (0..^(♯‘(𝑥 ++ 〈“𝑏”〉)))∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛)))) |
54 | 53 | imbi2d 344 |
. . 3
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛))) ↔ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘(𝑥 ++
〈“𝑏”〉)))∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛))))) |
55 | | fveq2 6722 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) |
56 | 55 | oveq2d 7234 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊))) |
57 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖)) |
58 | 57 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ((𝑤‘𝑖)‘𝑛) = ((𝑊‘𝑖)‘𝑛)) |
59 | 58 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → (((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) |
60 | 59 | ralbidv 3118 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) |
61 | 56, 60 | raleqbidv 3318 |
. . . . 5
⊢ (𝑤 = 𝑊 → (∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) |
62 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑆 Σg 𝑤) = (𝑆 Σg 𝑊)) |
63 | 62 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ((𝑆 Σg 𝑤)‘𝑛) = ((𝑆 Σg 𝑊)‘𝑛)) |
64 | 63 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛) ↔ ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) |
65 | 64 | ralbidv 3118 |
. . . . 5
⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) |
66 | 61, 65 | imbi12d 348 |
. . . 4
⊢ (𝑤 = 𝑊 → ((∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)) ↔ (∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))) |
67 | 66 | imbi2d 344 |
. . 3
⊢ (𝑤 = 𝑊 → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) ↔ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑊))∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))))) |
68 | | fveq1 6721 |
. . . . . . . . 9
⊢ (𝑢 = 𝑈 → (𝑢‘𝑖) = (𝑈‘𝑖)) |
69 | 68 | fveq1d 6724 |
. . . . . . . 8
⊢ (𝑢 = 𝑈 → ((𝑢‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) |
70 | 69 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑢 = 𝑈 → (((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) |
71 | 70 | ralbidv 3118 |
. . . . . 6
⊢ (𝑢 = 𝑈 → (∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) |
72 | 71 | ralbidv 3118 |
. . . . 5
⊢ (𝑢 = 𝑈 → (∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) |
73 | | oveq2 7226 |
. . . . . . . 8
⊢ (𝑢 = 𝑈 → (𝑍 Σg 𝑢) = (𝑍 Σg 𝑈)) |
74 | 73 | fveq1d 6724 |
. . . . . . 7
⊢ (𝑢 = 𝑈 → ((𝑍 Σg 𝑢)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)) |
75 | 74 | eqeq2d 2748 |
. . . . . 6
⊢ (𝑢 = 𝑈 → (((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) |
76 | 75 | ralbidv 3118 |
. . . . 5
⊢ (𝑢 = 𝑈 → (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) |
77 | 72, 76 | imbi12d 348 |
. . . 4
⊢ (𝑢 = 𝑈 → ((∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛)) ↔ (∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))) |
78 | 77 | imbi2d 344 |
. . 3
⊢ (𝑢 = 𝑈 → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑢)‘𝑛))) ↔ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑤)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))))) |
79 | | gsmsymgreq.i |
. . . . . . . . . 10
⊢ 𝐼 = (𝑁 ∩ 𝑀) |
80 | | eleq2 2826 |
. . . . . . . . . . . 12
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 ↔ 𝑛 ∈ (𝑁 ∩ 𝑀))) |
81 | | elin 3887 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (𝑁 ∩ 𝑀) ↔ (𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀)) |
82 | 80, 81 | bitrdi 290 |
. . . . . . . . . . 11
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 ↔ (𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀))) |
83 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀) → 𝑛 ∈ 𝑁) |
84 | 82, 83 | syl6bi 256 |
. . . . . . . . . 10
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑁)) |
85 | 79, 84 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑁) |
86 | 85 | adantl 485 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝑁) |
87 | | fvresi 6993 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑁 → (( I ↾ 𝑁)‘𝑛) = 𝑛) |
88 | 86, 87 | syl 17 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → (( I ↾ 𝑁)‘𝑛) = 𝑛) |
89 | | simpr 488 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀) → 𝑛 ∈ 𝑀) |
90 | 82, 89 | syl6bi 256 |
. . . . . . . . . 10
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑀)) |
91 | 79, 90 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑀) |
92 | 91 | adantl 485 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝑀) |
93 | | fvresi 6993 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑀 → (( I ↾ 𝑀)‘𝑛) = 𝑛) |
94 | 92, 93 | syl 17 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → (( I ↾ 𝑀)‘𝑛) = 𝑛) |
95 | 88, 94 | eqtr4d 2780 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → (( I ↾ 𝑁)‘𝑛) = (( I ↾ 𝑀)‘𝑛)) |
96 | 95 | ralrimiva 3105 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) →
∀𝑛 ∈ 𝐼 (( I ↾ 𝑁)‘𝑛) = (( I ↾ 𝑀)‘𝑛)) |
97 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑆) = (0g‘𝑆) |
98 | 97 | gsum0 18161 |
. . . . . . . . 9
⊢ (𝑆 Σg
∅) = (0g‘𝑆) |
99 | | gsmsymgrfix.s |
. . . . . . . . . . 11
⊢ 𝑆 = (SymGrp‘𝑁) |
100 | 99 | symgid 18798 |
. . . . . . . . . 10
⊢ (𝑁 ∈ Fin → ( I ↾
𝑁) =
(0g‘𝑆)) |
101 | 100 | adantr 484 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ( I ↾
𝑁) =
(0g‘𝑆)) |
102 | 98, 101 | eqtr4id 2797 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑆 Σg
∅) = ( I ↾ 𝑁)) |
103 | 102 | fveq1d 6724 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ((𝑆 Σg
∅)‘𝑛) = (( I
↾ 𝑁)‘𝑛)) |
104 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑍) = (0g‘𝑍) |
105 | 104 | gsum0 18161 |
. . . . . . . . 9
⊢ (𝑍 Σg
∅) = (0g‘𝑍) |
106 | | gsmsymgreq.z |
. . . . . . . . . . 11
⊢ 𝑍 = (SymGrp‘𝑀) |
107 | 106 | symgid 18798 |
. . . . . . . . . 10
⊢ (𝑀 ∈ Fin → ( I ↾
𝑀) =
(0g‘𝑍)) |
108 | 107 | adantl 485 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ( I ↾
𝑀) =
(0g‘𝑍)) |
109 | 105, 108 | eqtr4id 2797 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑍 Σg
∅) = ( I ↾ 𝑀)) |
110 | 109 | fveq1d 6724 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ((𝑍 Σg
∅)‘𝑛) = (( I
↾ 𝑀)‘𝑛)) |
111 | 103, 110 | eqeq12d 2753 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛) ↔ ((
I ↾ 𝑁)‘𝑛) = (( I ↾ 𝑀)‘𝑛))) |
112 | 111 | ralbidv 3118 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) →
(∀𝑛 ∈ 𝐼 ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛) ↔
∀𝑛 ∈ 𝐼 (( I ↾ 𝑁)‘𝑛) = (( I ↾ 𝑀)‘𝑛))) |
113 | 96, 112 | mpbird 260 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) →
∀𝑛 ∈ 𝐼 ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛)) |
114 | 113 | a1d 25 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) →
(∀𝑖 ∈
(0..^(♯‘∅))∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg
∅)‘𝑛) = ((𝑍 Σg
∅)‘𝑛))) |
115 | | gsmsymgrfix.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑆) |
116 | | gsmsymgreq.p |
. . . . . 6
⊢ 𝑃 = (Base‘𝑍) |
117 | 99, 115, 106, 116, 79 | gsmsymgreqlem2 18828 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑥 ∈ Word 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑦 ∈ Word 𝑃 ∧ 𝑝 ∈ 𝑃) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((∀𝑖 ∈ (0..^(♯‘𝑥))∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑥 ++ 〈“𝑏”〉)))∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛)))) |
118 | 117 | expcom 417 |
. . . 4
⊢ (((𝑥 ∈ Word 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑦 ∈ Word 𝑃 ∧ 𝑝 ∈ 𝑃) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ((∀𝑖 ∈
(0..^(♯‘𝑥))∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑥 ++ 〈“𝑏”〉)))∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛))))) |
119 | 118 | a2d 29 |
. . 3
⊢ (((𝑥 ∈ Word 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑦 ∈ Word 𝑃 ∧ 𝑝 ∈ 𝑃) ∧ (♯‘𝑥) = (♯‘𝑦)) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑥))∀𝑛 ∈ 𝐼 ((𝑥‘𝑖)‘𝑛) = ((𝑦‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑥)‘𝑛) = ((𝑍 Σg 𝑦)‘𝑛))) → ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘(𝑥 ++
〈“𝑏”〉)))∀𝑛 ∈ 𝐼 (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛) = (((𝑦 ++ 〈“𝑝”〉)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛) = ((𝑍 Σg (𝑦 ++ 〈“𝑝”〉))‘𝑛))))) |
120 | 18, 36, 54, 67, 78, 114, 119 | wrd2ind 14293 |
. 2
⊢ ((𝑊 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝑃 ∧ (♯‘𝑊) = (♯‘𝑈)) → ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (∀𝑖 ∈
(0..^(♯‘𝑊))∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))) |
121 | 120 | impcom 411 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝑊 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝑃 ∧ (♯‘𝑊) = (♯‘𝑈))) → (∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) |