| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑤 = ∅ →
(♯‘𝑤) =
(♯‘∅)) | 
| 2 | 1 | oveq2d 7447 | . . . . . . 7
⊢ (𝑤 = ∅ →
(0..^(♯‘𝑤)) =
(0..^(♯‘∅))) | 
| 3 | 2 | adantr 480 | . . . . . 6
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(0..^(♯‘𝑤)) =
(0..^(♯‘∅))) | 
| 4 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑤 = ∅ → (𝑤‘𝑖) = (∅‘𝑖)) | 
| 5 | 4 | fveq1d 6908 | . . . . . . . 8
⊢ (𝑤 = ∅ → ((𝑤‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛)) | 
| 6 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑢 = ∅ → (𝑢‘𝑖) = (∅‘𝑖)) | 
| 7 | 6 | fveq1d 6908 | . . . . . . . 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 3178 | . . . . . 6
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛))) | 
| 10 | 3, 9 | raleqbidv 3346 | . . . . 5
⊢ ((𝑤 = ∅ ∧ 𝑢 = ∅) →
(∀𝑖 ∈
(0..^(♯‘𝑤))∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑖 ∈
(0..^(♯‘∅))∀𝑛 ∈ 𝐼 ((∅‘𝑖)‘𝑛) = ((∅‘𝑖)‘𝑛))) | 
| 11 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑤 = ∅ → (𝑆 Σg
𝑤) = (𝑆 Σg
∅)) | 
| 12 | 11 | fveq1d 6908 | . . . . . . 7
⊢ (𝑤 = ∅ → ((𝑆 Σg
𝑤)‘𝑛) = ((𝑆 Σg
∅)‘𝑛)) | 
| 13 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑢 = ∅ → (𝑍 Σg
𝑢) = (𝑍 Σg
∅)) | 
| 14 | 13 | fveq1d 6908 | . . . . . . 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 3178 | . . . . 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 344 | . . . 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 340 | . . 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 6906 | . . . . . . . 8
⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥)) | 
| 20 | 19 | oveq2d 7447 | . . . . . . 7
⊢ (𝑤 = 𝑥 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑥))) | 
| 21 | 20 | adantr 480 | . . . . . 6
⊢ ((𝑤 = 𝑥 ∧ 𝑢 = 𝑦) → (0..^(♯‘𝑤)) = (0..^(♯‘𝑥))) | 
| 22 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑤 = 𝑥 → (𝑤‘𝑖) = (𝑥‘𝑖)) | 
| 23 | 22 | fveq1d 6908 | . . . . . . . 8
⊢ (𝑤 = 𝑥 → ((𝑤‘𝑖)‘𝑛) = ((𝑥‘𝑖)‘𝑛)) | 
| 24 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑢 = 𝑦 → (𝑢‘𝑖) = (𝑦‘𝑖)) | 
| 25 | 24 | fveq1d 6908 | . . . . . . . 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 3178 | . . . . . 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 3346 | . . . . 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 7439 | . . . . . . . 8
⊢ (𝑤 = 𝑥 → (𝑆 Σg 𝑤) = (𝑆 Σg 𝑥)) | 
| 30 | 29 | fveq1d 6908 | . . . . . . 7
⊢ (𝑤 = 𝑥 → ((𝑆 Σg 𝑤)‘𝑛) = ((𝑆 Σg 𝑥)‘𝑛)) | 
| 31 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑢 = 𝑦 → (𝑍 Σg 𝑢) = (𝑍 Σg 𝑦)) | 
| 32 | 31 | fveq1d 6908 | . . . . . . 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 3178 | . . . . 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 344 | . . . 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 340 | . . 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 6906 | . . . . . . . 8
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → (♯‘𝑤) = (♯‘(𝑥 ++ 〈“𝑏”〉))) | 
| 38 | 37 | oveq2d 7447 | . . . . . . 7
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) →
(0..^(♯‘𝑤)) =
(0..^(♯‘(𝑥 ++
〈“𝑏”〉)))) | 
| 39 | 38 | adantr 480 | . . . . . 6
⊢ ((𝑤 = (𝑥 ++ 〈“𝑏”〉) ∧ 𝑢 = (𝑦 ++ 〈“𝑝”〉)) →
(0..^(♯‘𝑤)) =
(0..^(♯‘(𝑥 ++
〈“𝑏”〉)))) | 
| 40 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → (𝑤‘𝑖) = ((𝑥 ++ 〈“𝑏”〉)‘𝑖)) | 
| 41 | 40 | fveq1d 6908 | . . . . . . . 8
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → ((𝑤‘𝑖)‘𝑛) = (((𝑥 ++ 〈“𝑏”〉)‘𝑖)‘𝑛)) | 
| 42 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑢 = (𝑦 ++ 〈“𝑝”〉) → (𝑢‘𝑖) = ((𝑦 ++ 〈“𝑝”〉)‘𝑖)) | 
| 43 | 42 | fveq1d 6908 | . . . . . . . 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 3178 | . . . . . 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 3346 | . . . . 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 7439 | . . . . . . . 8
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → (𝑆 Σg 𝑤) = (𝑆 Σg (𝑥 ++ 〈“𝑏”〉))) | 
| 48 | 47 | fveq1d 6908 | . . . . . . 7
⊢ (𝑤 = (𝑥 ++ 〈“𝑏”〉) → ((𝑆 Σg 𝑤)‘𝑛) = ((𝑆 Σg (𝑥 ++ 〈“𝑏”〉))‘𝑛)) | 
| 49 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑢 = (𝑦 ++ 〈“𝑝”〉) → (𝑍 Σg 𝑢) = (𝑍 Σg (𝑦 ++ 〈“𝑝”〉))) | 
| 50 | 49 | fveq1d 6908 | . . . . . . 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 3178 | . . . . 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 344 | . . . 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 340 | . . 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 6906 | . . . . . . 7
⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | 
| 56 | 55 | oveq2d 7447 | . . . . . 6
⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊))) | 
| 57 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑤 = 𝑊 → (𝑤‘𝑖) = (𝑊‘𝑖)) | 
| 58 | 57 | fveq1d 6908 | . . . . . . . 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 3178 | . . . . . 6
⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) | 
| 61 | 56, 60 | raleqbidv 3346 | . . . . 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 7439 | . . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑆 Σg 𝑤) = (𝑆 Σg 𝑊)) | 
| 63 | 62 | fveq1d 6908 | . . . . . . 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 3178 | . . . . 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 344 | . . . 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 340 | . . 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 6905 | . . . . . . . . 9
⊢ (𝑢 = 𝑈 → (𝑢‘𝑖) = (𝑈‘𝑖)) | 
| 69 | 68 | fveq1d 6908 | . . . . . . . 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 3178 | . . . . . 6
⊢ (𝑢 = 𝑈 → (∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑢‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑤‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) | 
| 72 | 71 | ralbidv 3178 | . . . . 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 7439 | . . . . . . . 8
⊢ (𝑢 = 𝑈 → (𝑍 Σg 𝑢) = (𝑍 Σg 𝑈)) | 
| 74 | 73 | fveq1d 6908 | . . . . . . 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 3178 | . . . . 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 344 | . . . 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 340 | . . 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 2830 | . . . . . . . . . . . 12
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 ↔ 𝑛 ∈ (𝑁 ∩ 𝑀))) | 
| 81 |  | elin 3967 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ (𝑁 ∩ 𝑀) ↔ (𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀)) | 
| 82 | 80, 81 | bitrdi 287 | . . . . . . . . . . 11
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 ↔ (𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀))) | 
| 83 |  | simpl 482 | . . . . . . . . . . 11
⊢ ((𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀) → 𝑛 ∈ 𝑁) | 
| 84 | 82, 83 | biimtrdi 253 | . . . . . . . . . 10
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑁)) | 
| 85 | 79, 84 | ax-mp 5 | . . . . . . . . 9
⊢ (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑁) | 
| 86 | 85 | adantl 481 | . . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝑁) | 
| 87 |  | fvresi 7193 | . . . . . . . 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 484 | . . . . . . . . . . 11
⊢ ((𝑛 ∈ 𝑁 ∧ 𝑛 ∈ 𝑀) → 𝑛 ∈ 𝑀) | 
| 90 | 82, 89 | biimtrdi 253 | . . . . . . . . . 10
⊢ (𝐼 = (𝑁 ∩ 𝑀) → (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑀)) | 
| 91 | 79, 90 | ax-mp 5 | . . . . . . . . 9
⊢ (𝑛 ∈ 𝐼 → 𝑛 ∈ 𝑀) | 
| 92 | 91 | adantl 481 | . . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝑀) | 
| 93 |  | fvresi 7193 | . . . . . . . 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 3146 | . . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) →
∀𝑛 ∈ 𝐼 (( I ↾ 𝑁)‘𝑛) = (( I ↾ 𝑀)‘𝑛)) | 
| 97 |  | eqid 2737 | . . . . . . . . . 10
⊢
(0g‘𝑆) = (0g‘𝑆) | 
| 98 | 97 | gsum0 18697 | . . . . . . . . 9
⊢ (𝑆 Σg
∅) = (0g‘𝑆) | 
| 99 |  | gsmsymgrfix.s | . . . . . . . . . . 11
⊢ 𝑆 = (SymGrp‘𝑁) | 
| 100 | 99 | symgid 19419 | . . . . . . . . . 10
⊢ (𝑁 ∈ Fin → ( I ↾
𝑁) =
(0g‘𝑆)) | 
| 101 | 100 | adantr 480 | . . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ( I ↾
𝑁) =
(0g‘𝑆)) | 
| 102 | 98, 101 | eqtr4id 2796 | . . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑆 Σg
∅) = ( I ↾ 𝑁)) | 
| 103 | 102 | fveq1d 6908 | . . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ((𝑆 Σg
∅)‘𝑛) = (( I
↾ 𝑁)‘𝑛)) | 
| 104 |  | eqid 2737 | . . . . . . . . . 10
⊢
(0g‘𝑍) = (0g‘𝑍) | 
| 105 | 104 | gsum0 18697 | . . . . . . . . 9
⊢ (𝑍 Σg
∅) = (0g‘𝑍) | 
| 106 |  | gsmsymgreq.z | . . . . . . . . . . 11
⊢ 𝑍 = (SymGrp‘𝑀) | 
| 107 | 106 | symgid 19419 | . . . . . . . . . 10
⊢ (𝑀 ∈ Fin → ( I ↾
𝑀) =
(0g‘𝑍)) | 
| 108 | 107 | adantl 481 | . . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → ( I ↾
𝑀) =
(0g‘𝑍)) | 
| 109 | 105, 108 | eqtr4id 2796 | . . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑍 Σg
∅) = ( I ↾ 𝑀)) | 
| 110 | 109 | fveq1d 6908 | . . . . . . 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 3178 | . . . . 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 257 | . . . 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 19449 | . . . . 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 413 | . . . 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 14761 | . 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 407 | 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))) |