| Step | Hyp | Ref
| Expression |
| 1 | | gsumzcl.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 2 | | gsumzcl.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 3 | | gsumzcl.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝐺) |
| 4 | 3 | gsumz 18849 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 5 | 1, 2, 4 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 6 | | gsumzf1o.h |
. . . . . . . . 9
⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) |
| 7 | | f1of1 6847 |
. . . . . . . . 9
⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–1-1→𝐴) |
| 8 | 6, 7 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐻:𝐶–1-1→𝐴) |
| 9 | | f1dmex 7981 |
. . . . . . . 8
⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V) |
| 10 | 8, 2, 9 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ V) |
| 11 | 3 | gsumz 18849 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ 𝐶 ∈ V) → (𝐺 Σg
(𝑥 ∈ 𝐶 ↦ 0 )) = 0 ) |
| 12 | 1, 10, 11 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )) = 0 ) |
| 13 | 5, 12 | eqtr4d 2780 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 ))) |
| 14 | 13 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 ))) |
| 15 | | gsumzcl.f |
. . . . . 6
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 16 | 3 | fvexi 6920 |
. . . . . . 7
⊢ 0 ∈
V |
| 17 | 16 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈ V) |
| 18 | | ssidd 4007 |
. . . . . 6
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
| 19 | 15, 2, 17, 18 | gsumcllem 19926 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
| 20 | 19 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
| 21 | | f1of 6848 |
. . . . . . . . 9
⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴) |
| 22 | 6, 21 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐻:𝐶⟶𝐴) |
| 23 | 22 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻:𝐶⟶𝐴) |
| 24 | 23 | ffvelcdmda 7104 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴) |
| 25 | 23 | feqmptd 6977 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥))) |
| 26 | | eqidd 2738 |
. . . . . 6
⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 ) |
| 27 | 24, 25, 19, 26 | fmptco 7149 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 )) |
| 28 | 27 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝐹 ∘ 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 ))) |
| 29 | 14, 20, 28 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |
| 30 | 29 | ex 412 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg
𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))) |
| 31 | 6 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐻:𝐶–1-1-onto→𝐴) |
| 32 | | f1ococnv2 6875 |
. . . . . . . . . . . . . 14
⊢ (𝐻:𝐶–1-1-onto→𝐴 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴)) |
| 33 | 31, 32 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴)) |
| 34 | 33 | coeq1d 5872 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (( I ↾ 𝐴) ∘ 𝑓)) |
| 35 | | f1of1 6847 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 )) |
| 36 | 35 | ad2antll 729 |
. . . . . . . . . . . . . 14
⊢ ((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 )) |
| 37 | | suppssdm 8202 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
| 38 | 37, 15 | fssdm 6755 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴) |
| 39 | 38 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴) |
| 40 | | f1ss 6809 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) |
| 41 | 36, 39, 40 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) |
| 42 | | f1f 6804 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴) |
| 43 | | fcoi2 6783 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴 → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓) |
| 44 | 41, 42, 43 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (( I ↾
𝐴) ∘ 𝑓) = 𝑓) |
| 45 | 34, 44 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = 𝑓) |
| 46 | | coass 6285 |
. . . . . . . . . . 11
⊢ ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (𝐻 ∘ (◡𝐻 ∘ 𝑓)) |
| 47 | 45, 46 | eqtr3di 2792 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (◡𝐻 ∘ 𝑓))) |
| 48 | 47 | coeq2d 5873 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓)))) |
| 49 | | coass 6285 |
. . . . . . . . 9
⊢ ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓))) |
| 50 | 48, 49 | eqtr4di 2795 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓) = ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓))) |
| 51 | 50 | seqeq3d 14050 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) = seq1((+g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)))) |
| 52 | 51 | fveq1d 6908 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))) =
(seq1((+g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 )))) |
| 53 | | gsumzcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐺) |
| 54 | | eqid 2737 |
. . . . . . 7
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 55 | | gsumzcl.z |
. . . . . . 7
⊢ 𝑍 = (Cntz‘𝐺) |
| 56 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd) |
| 57 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉) |
| 58 | 15 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵) |
| 59 | | gsumzcl.c |
. . . . . . . 8
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 60 | 59 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 61 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(♯‘(𝐹 supp
0 ))
∈ ℕ) |
| 62 | | ssid 4006 |
. . . . . . . 8
⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ) |
| 63 | | f1ofo 6855 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 )) |
| 64 | | forn 6823 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
| 65 | 63, 64 | syl 17 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
| 66 | 65 | ad2antll 729 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 )) |
| 67 | 62, 66 | sseqtrrid 4027 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
| 68 | | eqid 2737 |
. . . . . . 7
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
| 69 | 53, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68 | 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 )))) |
| 70 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐶 ∈ V) |
| 71 | | fco 6760 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵) |
| 72 | 15, 22, 71 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵) |
| 73 | 72 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝐻):𝐶⟶𝐵) |
| 74 | | rncoss 5986 |
. . . . . . . . 9
⊢ ran
(𝐹 ∘ 𝐻) ⊆ ran 𝐹 |
| 75 | 55 | cntzidss 19358 |
. . . . . . . . 9
⊢ ((ran
𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻))) |
| 76 | 59, 74, 75 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻))) |
| 77 | 76 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻))) |
| 78 | | f1ocnv 6860 |
. . . . . . . . . 10
⊢ (𝐻:𝐶–1-1-onto→𝐴 → ◡𝐻:𝐴–1-1-onto→𝐶) |
| 79 | | f1of1 6847 |
. . . . . . . . . 10
⊢ (◡𝐻:𝐴–1-1-onto→𝐶 → ◡𝐻:𝐴–1-1→𝐶) |
| 80 | 6, 78, 79 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝐻:𝐴–1-1→𝐶) |
| 81 | 80 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ◡𝐻:𝐴–1-1→𝐶) |
| 82 | | f1co 6815 |
. . . . . . . 8
⊢ ((◡𝐻:𝐴–1-1→𝐶 ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) → (◡𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1→𝐶) |
| 83 | 81, 41, 82 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1→𝐶) |
| 84 | | ssidd 4007 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
| 85 | 15, 2 | fexd 7247 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 ∈ V) |
| 86 | | suppimacnv 8199 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
| 87 | 85, 16, 86 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
| 88 | 87 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 )) |
| 89 | 88 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 )) |
| 90 | 84, 89, 66 | 3sstr4d 4039 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓) |
| 91 | | imass2 6120 |
. . . . . . . . . 10
⊢ ((◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓 → (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) ⊆ (◡𝐻 “ ran 𝑓)) |
| 92 | 90, 91 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) ⊆ (◡𝐻 “ ran 𝑓)) |
| 93 | | cnvco 5896 |
. . . . . . . . . . 11
⊢ ◡(𝐹 ∘ 𝐻) = (◡𝐻 ∘ ◡𝐹) |
| 94 | 93 | imaeq1i 6075 |
. . . . . . . . . 10
⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 })) |
| 95 | | imaco 6271 |
. . . . . . . . . 10
⊢ ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) |
| 96 | 94, 95 | eqtri 2765 |
. . . . . . . . 9
⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) |
| 97 | | rnco2 6273 |
. . . . . . . . 9
⊢ ran
(◡𝐻 ∘ 𝑓) = (◡𝐻 “ ran 𝑓) |
| 98 | 92, 96, 97 | 3sstr4g 4037 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓)) |
| 99 | | f1oexrnex 7949 |
. . . . . . . . . . . . 13
⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V) |
| 100 | 6, 2, 99 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐻 ∈ V) |
| 101 | | coexg 7951 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V) |
| 102 | 85, 100, 101 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V) |
| 103 | | suppimacnv 8199 |
. . . . . . . . . . 11
⊢ (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐹 ∘ 𝐻) supp 0 ) = (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 }))) |
| 104 | 102, 16, 103 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐹 ∘ 𝐻) supp 0 ) = (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 }))) |
| 105 | 104 | sseq1d 4015 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓) ↔ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓))) |
| 106 | 105 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓) ↔ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓))) |
| 107 | 98, 106 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓)) |
| 108 | | eqid 2737 |
. . . . . . 7
⊢ (((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) supp 0 ) = (((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) supp 0 ) |
| 109 | 53, 3, 54, 55, 56, 70, 73, 77, 61, 83, 107, 108 | gsumval3 19925 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
(𝐹 ∘ 𝐻)) =
(seq1((+g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 )))) |
| 110 | 52, 69, 109 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |
| 111 | 110 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))) |
| 112 | 111 | exlimdv 1933 |
. . 3
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))) |
| 113 | 112 | expimpd 453 |
. 2
⊢ (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))) |
| 114 | | gsumzcl.w |
. . 3
⊢ (𝜑 → 𝐹 finSupp 0 ) |
| 115 | | fsuppimp 9408 |
. . . 4
⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈
Fin)) |
| 116 | 115 | simprd 495 |
. . 3
⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈
Fin) |
| 117 | | 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 )))) |
| 118 | 114, 116,
117 | 3syl 18 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨
((♯‘(𝐹 supp
0 ))
∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))) |
| 119 | 30, 113, 118 | mpjaod 861 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |