Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzcl.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumzcl.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsumzcl.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
4 |
|
gsumzcl.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
5 |
|
gsumzcl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
6 |
|
gsumzcl.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
7 |
|
gsumzcl.c |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
8 |
|
gsumzcl2.w |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin ) |
9 |
2
|
fvexi |
⊢ 0 ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
11 |
|
ssidd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
12 |
6 5 10 11
|
gsumcllem |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) = ∅ ) → 𝐹 = ( 𝑘 ∈ 𝐴 ↦ 0 ) ) |
13 |
12
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) = ∅ ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) ) |
14 |
2
|
gsumz |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉 ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
15 |
4 5 14
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) = ∅ ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
17 |
13 16
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) = ∅ ) → ( 𝐺 Σg 𝐹 ) = 0 ) |
18 |
1 2
|
mndidcl |
⊢ ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 ) |
19 |
4 18
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) = ∅ ) → 0 ∈ 𝐵 ) |
21 |
17 20
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) = ∅ ) → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) |
22 |
21
|
ex |
⊢ ( 𝜑 → ( ( 𝐹 supp 0 ) = ∅ → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) ) |
23 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
24 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → 𝐺 ∈ Mnd ) |
25 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → 𝐴 ∈ 𝑉 ) |
26 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
27 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
28 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ) |
29 |
|
f1of1 |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) → 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1→ ( 𝐹 supp 0 ) ) |
30 |
29
|
ad2antll |
⊢ ( ( 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 ) ) |
31 |
|
suppssdm |
⊢ ( 𝐹 supp 0 ) ⊆ dom 𝐹 |
32 |
31 6
|
fssdm |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝐴 ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( 𝐹 supp 0 ) ⊆ 𝐴 ) |
34 |
|
f1ss |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1→ ( 𝐹 supp 0 ) ∧ ( 𝐹 supp 0 ) ⊆ 𝐴 ) → 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1→ 𝐴 ) |
35 |
30 33 34
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1→ 𝐴 ) |
36 |
|
ssid |
⊢ ( 𝐹 supp 0 ) ⊆ ( 𝐹 supp 0 ) |
37 |
|
f1ofo |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) → 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –onto→ ( 𝐹 supp 0 ) ) |
38 |
|
forn |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –onto→ ( 𝐹 supp 0 ) → ran 𝑓 = ( 𝐹 supp 0 ) ) |
39 |
37 38
|
syl |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) → ran 𝑓 = ( 𝐹 supp 0 ) ) |
40 |
39
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ran 𝑓 = ( 𝐹 supp 0 ) ) |
41 |
36 40
|
sseqtrrid |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( 𝐹 supp 0 ) ⊆ ran 𝑓 ) |
42 |
|
eqid |
⊢ ( ( 𝐹 ∘ 𝑓 ) supp 0 ) = ( ( 𝐹 ∘ 𝑓 ) supp 0 ) |
43 |
1 2 23 3 24 25 26 27 28 35 41 42
|
gsumval3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ) |
44 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
45 |
28 44
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ( ℤ≥ ‘ 1 ) ) |
46 |
|
f1f |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ⟶ 𝐴 ) |
47 |
35 46
|
syl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ⟶ 𝐴 ) |
48 |
|
fco |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ⟶ 𝐴 ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ⟶ 𝐵 ) |
49 |
26 47 48
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ⟶ 𝐵 ) |
50 |
49
|
ffvelrnda |
⊢ ( ( ( 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 ) |
51 |
1 23
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑘 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝐵 ) |
52 |
51
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑘 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝐵 ) |
53 |
24 52
|
sylan |
⊢ ( ( ( 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 ) |
54 |
45 50 53
|
seqcl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( 𝐹 supp 0 ) ) ) ∈ 𝐵 ) |
55 |
43 54
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) |
56 |
55
|
expr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) ) |
57 |
56
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) ) |
58 |
57
|
expimpd |
⊢ ( 𝜑 → ( ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) ) |
59 |
|
fz1f1o |
⊢ ( ( 𝐹 supp 0 ) ∈ Fin → ( ( 𝐹 supp 0 ) = ∅ ∨ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) ) |
60 |
8 59
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 supp 0 ) = ∅ ∨ ( ( ♯ ‘ ( 𝐹 supp 0 ) ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( 𝐹 supp 0 ) ) ) –1-1-onto→ ( 𝐹 supp 0 ) ) ) ) |
61 |
22 58 60
|
mpjaod |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) |