Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzmhm.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumzmhm.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
3 |
|
gsumzmhm.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
4 |
|
gsumzmhm.h |
⊢ ( 𝜑 → 𝐻 ∈ Mnd ) |
5 |
|
gsumzmhm.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
6 |
|
gsumzmhm.k |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) ) |
7 |
|
gsumzmhm.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
8 |
|
gsumzmhm.c |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
9 |
|
gsumzmhm.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
10 |
|
gsumzmhm.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
11 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
12 |
11
|
gsumz |
⊢ ( ( 𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉 ) → ( 𝐻 Σg ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) = ( 0g ‘ 𝐻 ) ) |
13 |
4 5 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝐻 Σg ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) = ( 0g ‘ 𝐻 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐻 Σg ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) = ( 0g ‘ 𝐻 ) ) |
15 |
9 11
|
mhm0 |
⊢ ( 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) → ( 𝐾 ‘ 0 ) = ( 0g ‘ 𝐻 ) ) |
16 |
6 15
|
syl |
⊢ ( 𝜑 → ( 𝐾 ‘ 0 ) = ( 0g ‘ 𝐻 ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐾 ‘ 0 ) = ( 0g ‘ 𝐻 ) ) |
18 |
14 17
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐻 Σg ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) = ( 𝐾 ‘ 0 ) ) |
19 |
1 9
|
mndidcl |
⊢ ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 ) |
20 |
3 19
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) ∧ 𝑘 ∈ 𝐴 ) → 0 ∈ 𝐵 ) |
22 |
9
|
fvexi |
⊢ 0 ∈ V |
23 |
22
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
24 |
|
fex |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝑉 ) → 𝐹 ∈ V ) |
25 |
7 5 24
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
26 |
|
suppimacnv |
⊢ ( ( 𝐹 ∈ V ∧ 0 ∈ V ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
27 |
25 23 26
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
28 |
|
ssid |
⊢ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ⊆ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) |
29 |
27 28
|
eqsstrdi |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
30 |
7 5 23 29
|
gsumcllem |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → 𝐹 = ( 𝑘 ∈ 𝐴 ↦ 0 ) ) |
31 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
32 |
1 31
|
mhmf |
⊢ ( 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) → 𝐾 : 𝐵 ⟶ ( Base ‘ 𝐻 ) ) |
33 |
6 32
|
syl |
⊢ ( 𝜑 → 𝐾 : 𝐵 ⟶ ( Base ‘ 𝐻 ) ) |
34 |
33
|
feqmptd |
⊢ ( 𝜑 → 𝐾 = ( 𝑥 ∈ 𝐵 ↦ ( 𝐾 ‘ 𝑥 ) ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → 𝐾 = ( 𝑥 ∈ 𝐵 ↦ ( 𝐾 ‘ 𝑥 ) ) ) |
36 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( 𝐾 ‘ 𝑥 ) = ( 𝐾 ‘ 0 ) ) |
37 |
21 30 35 36
|
fmptco |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐾 ∘ 𝐹 ) = ( 𝑘 ∈ 𝐴 ↦ ( 𝐾 ‘ 0 ) ) ) |
38 |
16
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐾 ‘ 0 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝑘 ∈ 𝐴 ↦ ( 𝐾 ‘ 0 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) |
40 |
37 39
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐾 ∘ 𝐹 ) = ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) |
41 |
40
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐻 Σg ( 𝑘 ∈ 𝐴 ↦ ( 0g ‘ 𝐻 ) ) ) ) |
42 |
30
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) ) |
43 |
9
|
gsumz |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉 ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
44 |
3 5 43
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
46 |
42 45
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐺 Σg 𝐹 ) = 0 ) |
47 |
46
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) = ( 𝐾 ‘ 0 ) ) |
48 |
18 41 47
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) |
49 |
48
|
ex |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) ) |
50 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝐺 ∈ Mnd ) |
51 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
52 |
1 51
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
53 |
52
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
54 |
50 53
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
55 |
|
f1of1 |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
56 |
55
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
57 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ⊆ dom 𝐹 |
58 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
59 |
57 58
|
fssdm |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ⊆ 𝐴 ) |
60 |
|
f1ss |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ⊆ 𝐴 ) → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1→ 𝐴 ) |
61 |
56 59 60
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1→ 𝐴 ) |
62 |
|
f1f |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ⟶ 𝐴 ) |
63 |
61 62
|
syl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ⟶ 𝐴 ) |
64 |
|
fco |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ⟶ 𝐴 ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ⟶ 𝐵 ) |
65 |
7 63 64
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ⟶ 𝐵 ) |
66 |
65
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑥 ) ∈ 𝐵 ) |
67 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ) |
68 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
69 |
67 68
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ( ℤ≥ ‘ 1 ) ) |
70 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) ) |
71 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
72 |
1 51 71
|
mhmlin |
⊢ ( ( 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐾 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = ( ( 𝐾 ‘ 𝑥 ) ( +g ‘ 𝐻 ) ( 𝐾 ‘ 𝑦 ) ) ) |
73 |
72
|
3expb |
⊢ ( ( 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐾 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = ( ( 𝐾 ‘ 𝑥 ) ( +g ‘ 𝐻 ) ( 𝐾 ‘ 𝑦 ) ) ) |
74 |
70 73
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐾 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = ( ( 𝐾 ‘ 𝑥 ) ( +g ‘ 𝐻 ) ( 𝐾 ‘ 𝑦 ) ) ) |
75 |
|
coass |
⊢ ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) = ( 𝐾 ∘ ( 𝐹 ∘ 𝑓 ) ) |
76 |
75
|
fveq1i |
⊢ ( ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝐾 ∘ ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑥 ) |
77 |
|
fvco3 |
⊢ ( ( ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ⟶ 𝐵 ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) → ( ( 𝐾 ∘ ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑥 ) = ( 𝐾 ‘ ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑥 ) ) ) |
78 |
65 77
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) → ( ( 𝐾 ∘ ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑥 ) = ( 𝐾 ‘ ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑥 ) ) ) |
79 |
76 78
|
syl5req |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) → ( 𝐾 ‘ ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑥 ) ) = ( ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) ‘ 𝑥 ) ) |
80 |
54 66 69 74 79
|
seqhomo |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐾 ‘ ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) = ( seq 1 ( ( +g ‘ 𝐻 ) , ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) |
81 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝐴 ∈ 𝑉 ) |
82 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
83 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐹 supp 0 ) ⊆ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
84 |
|
f1ofo |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
85 |
|
forn |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) → ran 𝑓 = ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
86 |
84 85
|
syl |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) → ran 𝑓 = ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
87 |
86
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ran 𝑓 = ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
88 |
83 87
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐹 supp 0 ) ⊆ ran 𝑓 ) |
89 |
|
eqid |
⊢ ( ( 𝐹 ∘ 𝑓 ) supp 0 ) = ( ( 𝐹 ∘ 𝑓 ) supp 0 ) |
90 |
1 9 51 2 50 81 58 82 67 61 88 89
|
gsumval3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) |
91 |
90
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) = ( 𝐾 ‘ ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) |
92 |
|
eqid |
⊢ ( Cntz ‘ 𝐻 ) = ( Cntz ‘ 𝐻 ) |
93 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 𝐻 ∈ Mnd ) |
94 |
|
fco |
⊢ ( ( 𝐾 : 𝐵 ⟶ ( Base ‘ 𝐻 ) ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( 𝐾 ∘ 𝐹 ) : 𝐴 ⟶ ( Base ‘ 𝐻 ) ) |
95 |
33 58 94
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐾 ∘ 𝐹 ) : 𝐴 ⟶ ( Base ‘ 𝐻 ) ) |
96 |
2 92
|
cntzmhm2 |
⊢ ( ( 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) ∧ ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) → ( 𝐾 “ ran 𝐹 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝐾 “ ran 𝐹 ) ) ) |
97 |
6 82 96
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐾 “ ran 𝐹 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝐾 “ ran 𝐹 ) ) ) |
98 |
|
rnco2 |
⊢ ran ( 𝐾 ∘ 𝐹 ) = ( 𝐾 “ ran 𝐹 ) |
99 |
98
|
fveq2i |
⊢ ( ( Cntz ‘ 𝐻 ) ‘ ran ( 𝐾 ∘ 𝐹 ) ) = ( ( Cntz ‘ 𝐻 ) ‘ ( 𝐾 “ ran 𝐹 ) ) |
100 |
97 98 99
|
3sstr4g |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ran ( 𝐾 ∘ 𝐹 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ran ( 𝐾 ∘ 𝐹 ) ) ) |
101 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) → 𝑥 ∈ 𝐴 ) |
102 |
|
fvco3 |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐾 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐾 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
103 |
58 101 102
|
syl2an |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ( 𝐾 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐾 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
104 |
22
|
a1i |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → 0 ∈ V ) |
105 |
58 83 81 104
|
suppssr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
106 |
105
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐾 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐾 ‘ 0 ) ) |
107 |
16
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐾 ‘ 0 ) = ( 0g ‘ 𝐻 ) ) |
108 |
103 106 107
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ( 𝐾 ∘ 𝐹 ) ‘ 𝑥 ) = ( 0g ‘ 𝐻 ) ) |
109 |
95 108
|
suppss |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ( 𝐾 ∘ 𝐹 ) supp ( 0g ‘ 𝐻 ) ) ⊆ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) |
110 |
109 87
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( ( 𝐾 ∘ 𝐹 ) supp ( 0g ‘ 𝐻 ) ) ⊆ ran 𝑓 ) |
111 |
|
eqid |
⊢ ( ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) supp ( 0g ‘ 𝐻 ) ) = ( ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) supp ( 0g ‘ 𝐻 ) ) |
112 |
31 11 71 92 93 81 95 100 67 61 110 111
|
gsumval3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( seq 1 ( ( +g ‘ 𝐻 ) , ( ( 𝐾 ∘ 𝐹 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) |
113 |
80 91 112
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) |
114 |
113
|
expr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) ) |
115 |
114
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) ) |
116 |
115
|
expimpd |
⊢ ( 𝜑 → ( ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) ) |
117 |
10
|
fsuppimpd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin ) |
118 |
27 117
|
eqeltrrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∈ Fin ) |
119 |
|
fz1f1o |
⊢ ( ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∈ Fin → ( ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ∨ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) |
120 |
118 119
|
syl |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ∅ ∨ ( ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) |
121 |
49 116 120
|
mpjaod |
⊢ ( 𝜑 → ( 𝐻 Σg ( 𝐾 ∘ 𝐹 ) ) = ( 𝐾 ‘ ( 𝐺 Σg 𝐹 ) ) ) |