| Step | Hyp | Ref
| Expression |
| 1 | | gsumzcl.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 2 | | gsumzcl.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 3 | | gsumzcl.0 |
. . . . . . . . 9
⊢ 0 =
(0g‘𝐺) |
| 4 | 3 | fvexi 6920 |
. . . . . . . 8
⊢ 0 ∈
V |
| 5 | 4 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ V) |
| 6 | | ssidd 4007 |
. . . . . . 7
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
| 7 | 1, 2, 5, 6 | gsumcllem 19926 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
| 8 | 7 | oveq2d 7447 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
| 9 | | gsumzcl.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 10 | 3 | gsumz 18849 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 11 | 9, 2, 10 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 12 | 11 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 13 | 8, 12 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = 0 ) |
| 14 | | gsumzcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐺) |
| 15 | 14, 3 | mndidcl 18762 |
. . . . . 6
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
| 16 | 9, 15 | syl 17 |
. . . . 5
⊢ (𝜑 → 0 ∈ 𝐵) |
| 17 | 16 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 0 ∈ 𝐵) |
| 18 | 13, 17 | eqeltrd 2841 |
. . 3
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) ∈ 𝐵) |
| 19 | 18 | ex 412 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg
𝐹) ∈ 𝐵)) |
| 20 | | eqid 2737 |
. . . . . . 7
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 21 | | gsumzcl.z |
. . . . . . 7
⊢ 𝑍 = (Cntz‘𝐺) |
| 22 | 9 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd) |
| 23 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉) |
| 24 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵) |
| 25 | | gsumzcl.c |
. . . . . . . 8
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 26 | 25 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 27 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(♯‘(𝐹 supp
0 ))
∈ ℕ) |
| 28 | | f1of1 6847 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 )) |
| 29 | 28 | 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 )) |
| 30 | | suppssdm 8202 |
. . . . . . . . . 10
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
| 31 | 30, 1 | fssdm 6755 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴) |
| 32 | 31 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴) |
| 33 | | 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) |
| 34 | 29, 32, 33 | 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) |
| 35 | | ssid 4006 |
. . . . . . . 8
⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ) |
| 36 | | f1ofo 6855 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 )) |
| 37 | | forn 6823 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
| 38 | 36, 37 | syl 17 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
| 39 | 38 | ad2antll 729 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 )) |
| 40 | 35, 39 | sseqtrrid 4027 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
| 41 | | eqid 2737 |
. . . . . . 7
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
| 42 | 14, 3, 20, 21, 22, 23, 24, 26, 27, 34, 40, 41 | 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 )))) |
| 43 | | nnuz 12921 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
| 44 | 27, 43 | eleqtrdi 2851 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(♯‘(𝐹 supp
0 ))
∈ (ℤ≥‘1)) |
| 45 | | f1f 6804 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴) |
| 46 | 34, 45 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴) |
| 47 | | fco 6760 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))⟶𝐵) |
| 48 | 24, 46, 47 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))⟶𝐵) |
| 49 | 48 | ffvelcdmda 7104 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) ∧ 𝑘 ∈
(1...(♯‘(𝐹 supp
0 ))))
→ ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 50 | 14, 20 | mndcl 18755 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
| 51 | 50 | 3expb 1121 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
| 52 | 22, 51 | sylan 580 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
| 53 | 44, 49, 52 | seqcl 14063 |
. . . . . 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 ))) ∈ 𝐵) |
| 54 | 42, 53 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
𝐹) ∈ 𝐵) |
| 55 | 54 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
| 56 | 55 | exlimdv 1933 |
. . 3
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
| 57 | 56 | expimpd 453 |
. 2
⊢ (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
| 58 | | gsumzcl2.w |
. . 3
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
| 59 | | 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 )))) |
| 60 | 58, 59 | syl 17 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨
((♯‘(𝐹 supp
0 ))
∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))) |
| 61 | 19, 57, 60 | mpjaod 861 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |