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 | | fvex 6342 |
. . . . . . . . 9
⊢
(0g‘𝐺) ∈ V |
5 | 3, 4 | eqeltri 2846 |
. . . . . . . 8
⊢ 0 ∈
V |
6 | 5 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ V) |
7 | | ssid 3773 |
. . . . . . . 8
⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ) |
8 | 7 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
9 | 1, 2, 6, 8 | gsumcllem 18515 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
10 | 9 | oveq2d 6808 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
11 | | gsumzcl.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
12 | 3 | gsumz 17581 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
13 | 11, 2, 12 | syl2anc 565 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
14 | 13 | adantr 466 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
15 | 10, 14 | eqtrd 2805 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = 0 ) |
16 | | gsumzcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐺) |
17 | 16, 3 | mndidcl 17515 |
. . . . . 6
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
18 | 11, 17 | syl 17 |
. . . . 5
⊢ (𝜑 → 0 ∈ 𝐵) |
19 | 18 | adantr 466 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 0 ∈ 𝐵) |
20 | 15, 19 | eqeltrd 2850 |
. . 3
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) ∈ 𝐵) |
21 | 20 | ex 397 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg
𝐹) ∈ 𝐵)) |
22 | | eqid 2771 |
. . . . . . 7
⊢
(+g‘𝐺) = (+g‘𝐺) |
23 | | gsumzcl.z |
. . . . . . 7
⊢ 𝑍 = (Cntz‘𝐺) |
24 | 11 | adantr 466 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd) |
25 | 2 | adantr 466 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉) |
26 | 1 | adantr 466 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵) |
27 | | gsumzcl.c |
. . . . . . . 8
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
28 | 27 | adantr 466 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
29 | | simprl 746 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(♯‘(𝐹 supp
0 ))
∈ ℕ) |
30 | | f1of1 6277 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 )) |
31 | 30 | ad2antll 700 |
. . . . . . . 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 )) |
32 | | suppssdm 7458 |
. . . . . . . . . 10
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
33 | | fdm 6191 |
. . . . . . . . . . 11
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) |
34 | 1, 33 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → dom 𝐹 = 𝐴) |
35 | 32, 34 | syl5sseq 3802 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴) |
36 | 35 | adantr 466 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴) |
37 | | f1ss 6246 |
. . . . . . . 8
⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) |
38 | 31, 36, 37 | syl2anc 565 |
. . . . . . 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) |
39 | | f1ofo 6285 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 )) |
40 | | forn 6259 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
41 | 39, 40 | syl 17 |
. . . . . . . . 9
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 )) |
42 | 41 | ad2antll 700 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 )) |
43 | 7, 42 | syl5sseqr 3803 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
44 | | eqid 2771 |
. . . . . . 7
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
45 | 16, 3, 22, 23, 24, 25, 26, 28, 29, 38, 43, 44 | gsumval3 18514 |
. . . . . 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 )))) |
46 | | nnuz 11924 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
47 | 29, 46 | syl6eleq 2860 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) →
(♯‘(𝐹 supp
0 ))
∈ (ℤ≥‘1)) |
48 | | f1f 6241 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴) |
49 | 38, 48 | 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) |
50 | | fco 6198 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))⟶𝐵) |
51 | 26, 49, 50 | syl2anc 565 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))⟶𝐵) |
52 | 51 | ffvelrnda 6502 |
. . . . . . 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) |
53 | 16, 22 | mndcl 17508 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
54 | 53 | 3expb 1113 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
55 | 24, 54 | sylan 561 |
. . . . . . 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) |
56 | 47, 52, 55 | seqcl 13027 |
. . . . . 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) |
57 | 45, 56 | eqeltrd 2850 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
𝐹) ∈ 𝐵) |
58 | 57 | expr 444 |
. . . 4
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
59 | 58 | exlimdv 2013 |
. . 3
⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
60 | 59 | expimpd 441 |
. 2
⊢ (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
61 | | gsumzcl2.w |
. . 3
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
62 | | fz1f1o 14648 |
. . 3
⊢ ((𝐹 supp 0 ) ∈ Fin →
((𝐹 supp 0 ) = ∅ ∨
((♯‘(𝐹 supp
0 ))
∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))) |
63 | 61, 62 | syl 17 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨
((♯‘(𝐹 supp
0 ))
∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))) |
64 | 21, 60, 63 | mpjaod 839 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |