Step |
Hyp |
Ref |
Expression |
1 |
|
gsumconst.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumconst.m |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → 𝑋 ∈ 𝐵 ) |
4 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
5 |
1 4 2
|
mulg0 |
⊢ ( 𝑋 ∈ 𝐵 → ( 0 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
6 |
3 5
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( 0 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
7 |
|
fveq2 |
⊢ ( 𝐴 = ∅ → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ ∅ ) ) |
8 |
7
|
adantl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ ∅ ) ) |
9 |
|
hash0 |
⊢ ( ♯ ‘ ∅ ) = 0 |
10 |
8 9
|
eqtrdi |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( ♯ ‘ 𝐴 ) = 0 ) |
11 |
10
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( ( ♯ ‘ 𝐴 ) · 𝑋 ) = ( 0 · 𝑋 ) ) |
12 |
|
mpteq1 |
⊢ ( 𝐴 = ∅ → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) = ( 𝑘 ∈ ∅ ↦ 𝑋 ) ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) = ( 𝑘 ∈ ∅ ↦ 𝑋 ) ) |
14 |
|
mpt0 |
⊢ ( 𝑘 ∈ ∅ ↦ 𝑋 ) = ∅ |
15 |
13 14
|
eqtrdi |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) = ∅ ) |
16 |
15
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( 𝐺 Σg ∅ ) ) |
17 |
4
|
gsum0 |
⊢ ( 𝐺 Σg ∅ ) = ( 0g ‘ 𝐺 ) |
18 |
16 17
|
eqtrdi |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( 0g ‘ 𝐺 ) ) |
19 |
6 11 18
|
3eqtr4rd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ 𝐴 = ∅ ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) |
20 |
19
|
ex |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) → ( 𝐴 = ∅ → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) ) |
21 |
|
simprl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
22 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
23 |
21 22
|
eleqtrdi |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 1 ) ) |
24 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) |
25 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑋 ∈ 𝐵 ) |
26 |
25
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → 𝑋 ∈ 𝐵 ) |
27 |
|
eqid |
⊢ ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) = ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) |
28 |
27
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) ‘ 𝑥 ) = 𝑋 ) |
29 |
24 26 28
|
syl2anc |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) ‘ 𝑥 ) = 𝑋 ) |
30 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
31 |
30
|
ad2antll |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
32 |
31
|
ffvelrnda |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝐴 ) |
33 |
31
|
feqmptd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 = ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ ( 𝑓 ‘ 𝑥 ) ) ) |
34 |
|
eqidd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) = ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) |
35 |
|
eqidd |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → 𝑋 = 𝑋 ) |
36 |
32 33 34 35
|
fmptco |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) = ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) ) |
37 |
36
|
fveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) ‘ 𝑥 ) ) |
38 |
37
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ↦ 𝑋 ) ‘ 𝑥 ) ) |
39 |
|
elfznn |
⊢ ( 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) → 𝑥 ∈ ℕ ) |
40 |
|
fvconst2g |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ ℕ ) → ( ( ℕ × { 𝑋 } ) ‘ 𝑥 ) = 𝑋 ) |
41 |
25 39 40
|
syl2an |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ℕ × { 𝑋 } ) ‘ 𝑥 ) = 𝑋 ) |
42 |
29 38 41
|
3eqtr4d |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( ℕ × { 𝑋 } ) ‘ 𝑥 ) ) |
43 |
23 42
|
seqfveq |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( seq 1 ( ( +g ‘ 𝐺 ) , ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
44 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
45 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
46 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝐺 ∈ Mnd ) |
47 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝐴 ∈ Fin ) |
48 |
25
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 ) |
49 |
48
|
fmpttd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) : 𝐴 ⟶ 𝐵 ) |
50 |
|
eqidd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑋 ( +g ‘ 𝐺 ) 𝑋 ) = ( 𝑋 ( +g ‘ 𝐺 ) 𝑋 ) ) |
51 |
1 44 45
|
elcntzsn |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝑋 ∈ ( ( Cntz ‘ 𝐺 ) ‘ { 𝑋 } ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 ( +g ‘ 𝐺 ) 𝑋 ) = ( 𝑋 ( +g ‘ 𝐺 ) 𝑋 ) ) ) ) |
52 |
25 51
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑋 ∈ ( ( Cntz ‘ 𝐺 ) ‘ { 𝑋 } ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 ( +g ‘ 𝐺 ) 𝑋 ) = ( 𝑋 ( +g ‘ 𝐺 ) 𝑋 ) ) ) ) |
53 |
25 50 52
|
mpbir2and |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑋 ∈ ( ( Cntz ‘ 𝐺 ) ‘ { 𝑋 } ) ) |
54 |
53
|
snssd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → { 𝑋 } ⊆ ( ( Cntz ‘ 𝐺 ) ‘ { 𝑋 } ) ) |
55 |
|
snidg |
⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ { 𝑋 } ) |
56 |
25 55
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑋 ∈ { 𝑋 } ) |
57 |
56
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ { 𝑋 } ) |
58 |
57
|
fmpttd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) : 𝐴 ⟶ { 𝑋 } ) |
59 |
58
|
frnd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ran ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ⊆ { 𝑋 } ) |
60 |
45
|
cntzidss |
⊢ ( ( { 𝑋 } ⊆ ( ( Cntz ‘ 𝐺 ) ‘ { 𝑋 } ) ∧ ran ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ⊆ { 𝑋 } ) → ran ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) |
61 |
54 59 60
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ran ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) |
62 |
|
f1of1 |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1→ 𝐴 ) |
63 |
62
|
ad2antll |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1→ 𝐴 ) |
64 |
|
suppssdm |
⊢ ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) supp ( 0g ‘ 𝐺 ) ) ⊆ dom ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) |
65 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) = ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) |
66 |
65
|
dmmptss |
⊢ dom ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ⊆ 𝐴 |
67 |
66
|
a1i |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → dom ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ⊆ 𝐴 ) |
68 |
64 67
|
sstrid |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) supp ( 0g ‘ 𝐺 ) ) ⊆ 𝐴 ) |
69 |
|
f1ofo |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 ) |
70 |
|
forn |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 → ran 𝑓 = 𝐴 ) |
71 |
69 70
|
syl |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ran 𝑓 = 𝐴 ) |
72 |
71
|
ad2antll |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ran 𝑓 = 𝐴 ) |
73 |
68 72
|
sseqtrrd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) supp ( 0g ‘ 𝐺 ) ) ⊆ ran 𝑓 ) |
74 |
|
eqid |
⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) supp ( 0g ‘ 𝐺 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) supp ( 0g ‘ 𝐺 ) ) |
75 |
1 4 44 45 46 47 49 61 21 63 73 74
|
gsumval3 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
76 |
|
eqid |
⊢ seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) = seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) |
77 |
1 44 2 76
|
mulgnn |
⊢ ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( ( ♯ ‘ 𝐴 ) · 𝑋 ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
78 |
21 25 77
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( ♯ ‘ 𝐴 ) · 𝑋 ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
79 |
43 75 78
|
3eqtr4d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) |
80 |
79
|
expr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) ) |
81 |
80
|
exlimdv |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) ) |
82 |
81
|
expimpd |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) ) |
83 |
|
fz1f1o |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
84 |
83
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
85 |
20 82 84
|
mpjaod |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( ♯ ‘ 𝐴 ) · 𝑋 ) ) |