| Step | Hyp | Ref
| Expression |
| 1 | | gsumzcl.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 2 | | gsumzcl.a |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 3 | | inex1g 5319 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝑊) ∈ V) |
| 4 | 2, 3 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∩ 𝑊) ∈ V) |
| 5 | | gsumzcl.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝐺) |
| 6 | 5 | gsumz 18849 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∩ 𝑊) ∈ V) → (𝐺 Σg (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = 0 ) |
| 7 | 1, 4, 6 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = 0 ) |
| 8 | 5 | gsumz 18849 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 9 | 1, 2, 8 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 10 | 7, 9 | eqtr4d 2780 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
| 11 | 10 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
| 12 | | resres 6010 |
. . . . . . . 8
⊢ ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴 ∩ 𝑊)) |
| 13 | | gsumzcl.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 14 | | ffn 6736 |
. . . . . . . . . 10
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) |
| 15 | | fnresdm 6687 |
. . . . . . . . . 10
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
| 16 | 13, 14, 15 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹) |
| 17 | 16 | reseq1d 5996 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ 𝑊)) |
| 18 | 12, 17 | eqtr3id 2791 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊)) |
| 19 | 18 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊)) |
| 20 | 5 | fvexi 6920 |
. . . . . . . . . 10
⊢ 0 ∈
V |
| 21 | 20 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈ V) |
| 22 | | ssid 4006 |
. . . . . . . . . 10
⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ) |
| 23 | 22 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
| 24 | 13, 2, 21, 23 | gsumcllem 19926 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
| 25 | 24 | reseq1d 5996 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊))) |
| 26 | | inss1 4237 |
. . . . . . . 8
⊢ (𝐴 ∩ 𝑊) ⊆ 𝐴 |
| 27 | | resmpt 6055 |
. . . . . . . 8
⊢ ((𝐴 ∩ 𝑊) ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) |
| 28 | 26, 27 | ax-mp 5 |
. . . . . . 7
⊢ ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ) |
| 29 | 25, 28 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) |
| 30 | 19, 29 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ 𝑊) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) |
| 31 | 30 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))) |
| 32 | 24 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
| 33 | 11, 31, 32 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹)) |
| 34 | 33 | ex 412 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))) |
| 35 | | f1ofo 6855 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 )) |
| 36 | | forn 6823 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
| 37 | 35, 36 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
| 38 | 37 | ad2antll 729 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 )) |
| 39 | | gsumzres.s |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊) |
| 40 | 39 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝑊) |
| 41 | 38, 40 | eqsstrd 4018 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 ⊆ 𝑊) |
| 42 | | cores 6269 |
. . . . . . . . 9
⊢ (ran
𝑓 ⊆ 𝑊 → ((𝐹 ↾ 𝑊) ∘ 𝑓) = (𝐹 ∘ 𝑓)) |
| 43 | 41, 42 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ↾ 𝑊) ∘ 𝑓) = (𝐹 ∘ 𝑓)) |
| 44 | 43 | seqeq3d 14050 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
seq1((+g‘𝐺), ((𝐹 ↾ 𝑊) ∘ 𝑓)) = seq1((+g‘𝐺), (𝐹 ∘ 𝑓))) |
| 45 | 44 | fveq1d 6908 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(seq1((+g‘𝐺), ((𝐹 ↾ 𝑊) ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))) =
(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 )))) |
| 46 | | gsumzcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐺) |
| 47 | | eqid 2737 |
. . . . . . 7
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 48 | | gsumzcl.z |
. . . . . . 7
⊢ 𝑍 = (Cntz‘𝐺) |
| 49 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd) |
| 50 | 4 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴 ∩ 𝑊) ∈ V) |
| 51 | 13 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵) |
| 52 | | fssres 6774 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑊) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵) |
| 53 | 51, 26, 52 | sylancl 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵) |
| 54 | 18 | feq1d 6720 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵 ↔ (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵)) |
| 55 | 54 | biimpa 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵) → (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵) |
| 56 | 53, 55 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵) |
| 57 | | gsumzcl.c |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 58 | | resss 6019 |
. . . . . . . . . 10
⊢ (𝐹 ↾ 𝑊) ⊆ 𝐹 |
| 59 | 58 | rnssi 5951 |
. . . . . . . . 9
⊢ ran
(𝐹 ↾ 𝑊) ⊆ ran 𝐹 |
| 60 | 48 | cntzidss 19358 |
. . . . . . . . 9
⊢ ((ran
𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹) → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊))) |
| 61 | 57, 59, 60 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊))) |
| 62 | 61 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊))) |
| 63 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(♯‘(𝐹 supp
0 ))
∈ ℕ) |
| 64 | | f1of1 6847 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 )) |
| 65 | 64 | ad2antll 729 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 )) |
| 66 | | suppssdm 8202 |
. . . . . . . . . . 11
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
| 67 | 66, 13 | fssdm 6755 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴) |
| 68 | 67, 39 | ssind 4241 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊)) |
| 69 | 68 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊)) |
| 70 | | f1ss 6809 |
. . . . . . . 8
⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊)) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐴 ∩ 𝑊)) |
| 71 | 65, 69, 70 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐴 ∩ 𝑊)) |
| 72 | 13, 2 | fexd 7247 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 ∈ V) |
| 73 | | ressuppss 8208 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 )) |
| 74 | 72, 20, 73 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 )) |
| 75 | | sseq2 4010 |
. . . . . . . . . . 11
⊢ (ran
𝑓 = (𝐹 supp 0 ) → (((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓 ↔ ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))) |
| 76 | 74, 75 | imbitrrid 246 |
. . . . . . . . . 10
⊢ (ran
𝑓 = (𝐹 supp 0 ) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓)) |
| 77 | 35, 36, 76 | 3syl 18 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓)) |
| 78 | 77 | adantl 481 |
. . . . . . . 8
⊢
(((♯‘(𝐹
supp 0 ))
∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓)) |
| 79 | 78 | impcom 407 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓) |
| 80 | | eqid 2737 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝑊) ∘ 𝑓) supp 0 ) = (((𝐹 ↾ 𝑊) ∘ 𝑓) supp 0 ) |
| 81 | 46, 5, 47, 48, 49, 50, 56, 62, 63, 71, 79, 80 | gsumval3 19925 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
(𝐹 ↾ 𝑊)) =
(seq1((+g‘𝐺), ((𝐹 ↾ 𝑊) ∘ 𝑓))‘(♯‘(𝐹 supp 0 )))) |
| 82 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉) |
| 83 | 57 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 84 | 67 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴) |
| 85 | | f1ss 6809 |
. . . . . . . 8
⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) |
| 86 | 65, 84, 85 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) |
| 87 | 22, 38 | sseqtrrid 4027 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
| 88 | | eqid 2737 |
. . . . . . 7
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
| 89 | 46, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88 | gsumval3 19925 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
𝐹) =
(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 )))) |
| 90 | 45, 81, 89 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹)) |
| 91 | 90 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))) |
| 92 | 91 | exlimdv 1933 |
. . 3
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))) |
| 93 | 92 | expimpd 453 |
. 2
⊢ (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg
(𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))) |
| 94 | | gsumzres.w |
. . 3
⊢ (𝜑 → 𝐹 finSupp 0 ) |
| 95 | | fsuppimp 9408 |
. . . 4
⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈
Fin)) |
| 96 | 95 | simprd 495 |
. . 3
⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈
Fin) |
| 97 | | fz1f1o 15746 |
. . 3
⊢ ((𝐹 supp 0 ) ∈ Fin →
((𝐹 supp 0 ) = ∅ ∨
((♯‘(𝐹 supp
0 ))
∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))) |
| 98 | 94, 96, 97 | 3syl 18 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨
((♯‘(𝐹 supp
0 ))
∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))) |
| 99 | 34, 93, 98 | mpjaod 861 |
1
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹)) |