Step |
Hyp |
Ref |
Expression |
1 |
|
gsumress.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumress.o |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
gsumress.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝑆 ) |
4 |
|
gsumress.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑉 ) |
5 |
|
gsumress.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
6 |
|
gsumress.s |
⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 ) |
7 |
|
gsumress.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
8 |
|
gsumress.z |
⊢ ( 𝜑 → 0 ∈ 𝑆 ) |
9 |
|
gsumress.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) |
10 |
|
oveq1 |
⊢ ( 𝑦 = 0 → ( 𝑦 + 𝑥 ) = ( 0 + 𝑥 ) ) |
11 |
10
|
eqeq1d |
⊢ ( 𝑦 = 0 → ( ( 𝑦 + 𝑥 ) = 𝑥 ↔ ( 0 + 𝑥 ) = 𝑥 ) ) |
12 |
11
|
ovanraleqv |
⊢ ( 𝑦 = 0 → ( ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) ) |
13 |
6 8
|
sseldd |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
14 |
9
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) |
15 |
12 13 14
|
elrabd |
⊢ ( 𝜑 → 0 ∈ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ) |
16 |
15
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ) |
17 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
18 |
|
eqid |
⊢ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } = { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } |
19 |
1 17 2 18
|
mgmidsssn0 |
⊢ ( 𝐺 ∈ 𝑉 → { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ⊆ { ( 0g ‘ 𝐺 ) } ) |
20 |
4 19
|
syl |
⊢ ( 𝜑 → { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ⊆ { ( 0g ‘ 𝐺 ) } ) |
21 |
20 15
|
sseldd |
⊢ ( 𝜑 → 0 ∈ { ( 0g ‘ 𝐺 ) } ) |
22 |
|
elsni |
⊢ ( 0 ∈ { ( 0g ‘ 𝐺 ) } → 0 = ( 0g ‘ 𝐺 ) ) |
23 |
21 22
|
syl |
⊢ ( 𝜑 → 0 = ( 0g ‘ 𝐺 ) ) |
24 |
23
|
sneqd |
⊢ ( 𝜑 → { 0 } = { ( 0g ‘ 𝐺 ) } ) |
25 |
20 24
|
sseqtrrd |
⊢ ( 𝜑 → { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ⊆ { 0 } ) |
26 |
16 25
|
eqssd |
⊢ ( 𝜑 → { 0 } = { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ) |
27 |
11
|
ovanraleqv |
⊢ ( 𝑦 = 0 → ( ∀ 𝑥 ∈ 𝑆 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑆 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) ) |
28 |
6
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝐵 ) |
29 |
28 9
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) |
30 |
29
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) |
31 |
27 8 30
|
elrabd |
⊢ ( 𝜑 → 0 ∈ { 𝑦 ∈ 𝑆 ∣ ∀ 𝑥 ∈ 𝑆 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ) |
32 |
3 1
|
ressbas2 |
⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 = ( Base ‘ 𝐻 ) ) |
33 |
6 32
|
syl |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ 𝐻 ) ) |
34 |
|
fvex |
⊢ ( Base ‘ 𝐻 ) ∈ V |
35 |
33 34
|
eqeltrdi |
⊢ ( 𝜑 → 𝑆 ∈ V ) |
36 |
3 2
|
ressplusg |
⊢ ( 𝑆 ∈ V → + = ( +g ‘ 𝐻 ) ) |
37 |
35 36
|
syl |
⊢ ( 𝜑 → + = ( +g ‘ 𝐻 ) ) |
38 |
37
|
oveqd |
⊢ ( 𝜑 → ( 𝑦 + 𝑥 ) = ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) ) |
39 |
38
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑦 + 𝑥 ) = 𝑥 ↔ ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ) ) |
40 |
37
|
oveqd |
⊢ ( 𝜑 → ( 𝑥 + 𝑦 ) = ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) ) |
41 |
40
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑥 + 𝑦 ) = 𝑥 ↔ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) ) |
42 |
39 41
|
anbi12d |
⊢ ( 𝜑 → ( ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) ↔ ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) ) ) |
43 |
33 42
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑆 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) ) ) |
44 |
33 43
|
rabeqbidv |
⊢ ( 𝜑 → { 𝑦 ∈ 𝑆 ∣ ∀ 𝑥 ∈ 𝑆 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } = { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) |
45 |
31 44
|
eleqtrd |
⊢ ( 𝜑 → 0 ∈ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) |
46 |
45
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) |
47 |
3
|
ovexi |
⊢ 𝐻 ∈ V |
48 |
47
|
a1i |
⊢ ( 𝜑 → 𝐻 ∈ V ) |
49 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
50 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
51 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
52 |
|
eqid |
⊢ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } = { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } |
53 |
49 50 51 52
|
mgmidsssn0 |
⊢ ( 𝐻 ∈ V → { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ⊆ { ( 0g ‘ 𝐻 ) } ) |
54 |
48 53
|
syl |
⊢ ( 𝜑 → { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ⊆ { ( 0g ‘ 𝐻 ) } ) |
55 |
54 45
|
sseldd |
⊢ ( 𝜑 → 0 ∈ { ( 0g ‘ 𝐻 ) } ) |
56 |
|
elsni |
⊢ ( 0 ∈ { ( 0g ‘ 𝐻 ) } → 0 = ( 0g ‘ 𝐻 ) ) |
57 |
55 56
|
syl |
⊢ ( 𝜑 → 0 = ( 0g ‘ 𝐻 ) ) |
58 |
57
|
sneqd |
⊢ ( 𝜑 → { 0 } = { ( 0g ‘ 𝐻 ) } ) |
59 |
54 58
|
sseqtrrd |
⊢ ( 𝜑 → { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ⊆ { 0 } ) |
60 |
46 59
|
eqssd |
⊢ ( 𝜑 → { 0 } = { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) |
61 |
26 60
|
eqtr3d |
⊢ ( 𝜑 → { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } = { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) |
62 |
61
|
sseq2d |
⊢ ( 𝜑 → ( ran 𝐹 ⊆ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ↔ ran 𝐹 ⊆ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) ) |
63 |
23 57
|
eqtr3d |
⊢ ( 𝜑 → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
64 |
37
|
seqeq2d |
⊢ ( 𝜑 → seq 𝑚 ( + , 𝐹 ) = seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ) |
65 |
64
|
fveq1d |
⊢ ( 𝜑 → ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) |
66 |
65
|
eqeq2d |
⊢ ( 𝜑 → ( 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ↔ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) |
67 |
66
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ↔ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) ) |
68 |
67
|
rexbidv |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) ) |
69 |
68
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ↔ ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) ) |
70 |
69
|
iotabidv |
⊢ ( 𝜑 → ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) = ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) ) |
71 |
37
|
seqeq2d |
⊢ ( 𝜑 → seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) = seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ) |
72 |
71
|
fveq1d |
⊢ ( 𝜑 → ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) |
73 |
72
|
eqeq2d |
⊢ ( 𝜑 → ( 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ↔ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) |
74 |
73
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ↔ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) |
75 |
74
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) |
76 |
75
|
iotabidv |
⊢ ( 𝜑 → ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) = ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) |
77 |
70 76
|
ifeq12d |
⊢ ( 𝜑 → if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) = if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) ) |
78 |
62 63 77
|
ifbieq12d |
⊢ ( 𝜑 → if ( ran 𝐹 ⊆ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } , ( 0g ‘ 𝐺 ) , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) ) = if ( ran 𝐹 ⊆ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } , ( 0g ‘ 𝐻 ) , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) ) ) |
79 |
26
|
difeq2d |
⊢ ( 𝜑 → ( V ∖ { 0 } ) = ( V ∖ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ) ) |
80 |
79
|
imaeq2d |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } ) ) ) |
81 |
7 6
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
82 |
1 17 2 18 80 4 5 81
|
gsumval |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = if ( ran 𝐹 ⊆ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑥 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) } , ( 0g ‘ 𝐺 ) , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) ) ) |
83 |
60
|
difeq2d |
⊢ ( 𝜑 → ( V ∖ { 0 } ) = ( V ∖ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) ) |
84 |
83
|
imaeq2d |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } ) ) ) |
85 |
33
|
feq3d |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝑆 ↔ 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐻 ) ) ) |
86 |
7 85
|
mpbid |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐻 ) ) |
87 |
49 50 51 52 84 48 5 86
|
gsumval |
⊢ ( 𝜑 → ( 𝐻 Σg 𝐹 ) = if ( ran 𝐹 ⊆ { 𝑦 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐻 ) ( ( 𝑦 ( +g ‘ 𝐻 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) = 𝑥 ) } , ( 0g ‘ 𝐻 ) , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ∧ 𝑧 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 0 } ) ) ) ) ) ) ) ) ) |
88 |
78 82 87
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐻 Σg 𝐹 ) ) |