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 6201 |
. . . . . . . . 9
⊢
(0g‘𝐺) ∈ V |
5 | 3, 4 | eqeltri 2697 |
. . . . . . . 8
⊢ 0 ∈
V |
6 | 5 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ V) |
7 | | ssid 3624 |
. . . . . . . 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 18309 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
10 | 9 | oveq2d 6666 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
11 | | gsumzcl.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
12 | 3 | gsumz 17374 |
. . . . . . 7
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
13 | 11, 2, 12 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
14 | 13 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
(𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
15 | 10, 14 | eqtrd 2656 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) = 0 ) |
16 | | gsumzcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐺) |
17 | 16, 3 | mndidcl 17308 |
. . . . . 6
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
18 | 11, 17 | syl 17 |
. . . . 5
⊢ (𝜑 → 0 ∈ 𝐵) |
19 | 18 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 0 ∈ 𝐵) |
20 | 15, 19 | eqeltrd 2701 |
. . 3
⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg
𝐹) ∈ 𝐵) |
21 | 20 | ex 450 |
. 2
⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg
𝐹) ∈ 𝐵)) |
22 | | eqid 2622 |
. . . . . . 7
⊢
(+g‘𝐺) = (+g‘𝐺) |
23 | | gsumzcl.z |
. . . . . . 7
⊢ 𝑍 = (Cntz‘𝐺) |
24 | 11 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd) |
25 | 2 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉) |
26 | 1 | adantr 481 |
. . . . . . 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 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
29 | | simprl 794 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈
ℕ) |
30 | | f1of1 6136 |
. . . . . . . . 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 765 |
. . . . . . . 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 7308 |
. . . . . . . . . 10
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
33 | | fdm 6051 |
. . . . . . . . . . 11
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) |
34 | 1, 33 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → dom 𝐹 = 𝐴) |
35 | 32, 34 | syl5sseq 3653 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴) |
36 | 35 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴) |
37 | | f1ss 6106 |
. . . . . . . 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 693 |
. . . . . . 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 6144 |
. . . . . . . . . 10
⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 )) |
40 | | forn 6118 |
. . . . . . . . . 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 765 |
. . . . . . . 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 3654 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
44 | | eqid 2622 |
. . . . . . 7
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
45 | 16, 3, 22, 23, 24, 25, 26, 28, 29, 38, 43, 44 | gsumval3 18308 |
. . . . . 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 11723 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
47 | 29, 46 | syl6eleq 2711 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈
(ℤ≥‘1)) |
48 | | f1f 6101 |
. . . . . . . . . 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 6058 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))⟶𝐵) |
51 | 26, 49, 50 | syl2anc 693 |
. . . . . . . 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 6359 |
. . . . . . 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 17301 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
54 | 53 | 3expb 1266 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘(+g‘𝐺)𝑥) ∈ 𝐵) |
55 | 24, 54 | sylan 488 |
. . . . . . 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 12821 |
. . . . . 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 2701 |
. . . . 5
⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧
𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg
𝐹) ∈ 𝐵) |
58 | 57 | expr 643 |
. . . 4
⊢ ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) →
(𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg
𝐹) ∈ 𝐵)) |
59 | 58 | exlimdv 1861 |
. . 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 629 |
. 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 14441 |
. . 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 396 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |