Step |
Hyp |
Ref |
Expression |
1 |
|
gsumval3.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumval3.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsumval3.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
gsumval3.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
5 |
|
gsumval3.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
6 |
|
gsumval3.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
7 |
|
gsumval3.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
8 |
|
gsumval3.c |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
9 |
|
gsumval3.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
10 |
|
gsumval3.h |
⊢ ( 𝜑 → 𝐻 : ( 1 ... 𝑀 ) –1-1→ 𝐴 ) |
11 |
|
gsumval3.n |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ran 𝐻 ) |
12 |
|
gsumval3.w |
⊢ 𝑊 = ( ( 𝐹 ∘ 𝐻 ) supp 0 ) |
13 |
2
|
gsumz |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉 ) → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
14 |
5 6 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
16 |
7
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
18 |
|
f1f |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) –1-1→ 𝐴 → 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 ) |
19 |
10 18
|
syl |
⊢ ( 𝜑 → 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 ) |
20 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 ) |
21 |
|
f1f1orn |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) –1-1→ 𝐴 → 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) |
22 |
10 21
|
syl |
⊢ ( 𝜑 → 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) |
24 |
|
f1ocnv |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 → ◡ 𝐻 : ran 𝐻 –1-1-onto→ ( 1 ... 𝑀 ) ) |
25 |
|
f1of |
⊢ ( ◡ 𝐻 : ran 𝐻 –1-1-onto→ ( 1 ... 𝑀 ) → ◡ 𝐻 : ran 𝐻 ⟶ ( 1 ... 𝑀 ) ) |
26 |
23 24 25
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ◡ 𝐻 : ran 𝐻 ⟶ ( 1 ... 𝑀 ) ) |
27 |
26
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( ◡ 𝐻 ‘ 𝑥 ) ∈ ( 1 ... 𝑀 ) ) |
28 |
|
fvco3 |
⊢ ( ( 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 ∧ ( ◡ 𝐻 ‘ 𝑥 ) ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝐻 ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) ) ) |
29 |
20 27 28
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( ( 𝐹 ∘ 𝐻 ) ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝐻 ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) ) ) |
30 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝑊 = ∅ ) |
31 |
30
|
difeq2d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( ( 1 ... 𝑀 ) ∖ 𝑊 ) = ( ( 1 ... 𝑀 ) ∖ ∅ ) ) |
32 |
|
dif0 |
⊢ ( ( 1 ... 𝑀 ) ∖ ∅ ) = ( 1 ... 𝑀 ) |
33 |
31 32
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( ( 1 ... 𝑀 ) ∖ 𝑊 ) = ( 1 ... 𝑀 ) ) |
34 |
33
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( ( 1 ... 𝑀 ) ∖ 𝑊 ) = ( 1 ... 𝑀 ) ) |
35 |
27 34
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( ◡ 𝐻 ‘ 𝑥 ) ∈ ( ( 1 ... 𝑀 ) ∖ 𝑊 ) ) |
36 |
|
fco |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 ) → ( 𝐹 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ 𝐵 ) |
37 |
7 19 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ 𝐵 ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐹 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ 𝐵 ) |
39 |
12
|
eqimss2i |
⊢ ( ( 𝐹 ∘ 𝐻 ) supp 0 ) ⊆ 𝑊 |
40 |
39
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( ( 𝐹 ∘ 𝐻 ) supp 0 ) ⊆ 𝑊 ) |
41 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 1 ... 𝑀 ) ∈ V ) |
42 |
2
|
fvexi |
⊢ 0 ∈ V |
43 |
42
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 0 ∈ V ) |
44 |
38 40 41 43
|
suppssr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ ( ◡ 𝐻 ‘ 𝑥 ) ∈ ( ( 1 ... 𝑀 ) ∖ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) = 0 ) |
45 |
35 44
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( ( 𝐹 ∘ 𝐻 ) ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) = 0 ) |
46 |
|
f1ocnvfv2 |
⊢ ( ( 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ∧ 𝑥 ∈ ran 𝐻 ) → ( 𝐻 ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) = 𝑥 ) |
47 |
23 46
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( 𝐻 ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) = 𝑥 ) |
48 |
47
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( 𝐹 ‘ ( 𝐻 ‘ ( ◡ 𝐻 ‘ 𝑥 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
49 |
29 45 48
|
3eqtr3rd |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
50 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
51 |
50
|
elsn |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ↔ ( 𝐹 ‘ 𝑥 ) = 0 ) |
52 |
49 51
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ran 𝐻 ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
53 |
52
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ∈ ran 𝐻 ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
54 |
|
eldif |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ ran 𝐻 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻 ) ) |
55 |
42
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
56 |
7 11 6 55
|
suppssr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝐻 ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
57 |
56 51
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝐻 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
58 |
54 57
|
sylan2br |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
59 |
58
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
60 |
59
|
anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ 𝐴 ) ∧ ¬ 𝑥 ∈ ran 𝐻 ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
61 |
53 60
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
62 |
61 51
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
63 |
62
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
64 |
17 63
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
65 |
64
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) ) |
66 |
1 2
|
mndidcl |
⊢ ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 ) |
67 |
1 3 2
|
mndlid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 0 ∈ 𝐵 ) → ( 0 + 0 ) = 0 ) |
68 |
5 66 67
|
syl2anc2 |
⊢ ( 𝜑 → ( 0 + 0 ) = 0 ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 0 + 0 ) = 0 ) |
70 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
71 |
9 70
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 1 ) ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝑀 ∈ ( ℤ≥ ‘ 1 ) ) |
73 |
33
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝑥 ∈ ( ( 1 ... 𝑀 ) ∖ 𝑊 ) ↔ 𝑥 ∈ ( 1 ... 𝑀 ) ) ) |
74 |
73
|
biimpar |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ( 1 ... 𝑀 ) ) → 𝑥 ∈ ( ( 1 ... 𝑀 ) ∖ 𝑊 ) ) |
75 |
38 40 41 43
|
suppssr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ( ( 1 ... 𝑀 ) ∖ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑥 ) = 0 ) |
76 |
74 75
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑥 ) = 0 ) |
77 |
69 72 76
|
seqid3 |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = 0 ) |
78 |
15 65 77
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) |
79 |
|
fzf |
⊢ ... : ( ℤ × ℤ ) ⟶ 𝒫 ℤ |
80 |
|
ffn |
⊢ ( ... : ( ℤ × ℤ ) ⟶ 𝒫 ℤ → ... Fn ( ℤ × ℤ ) ) |
81 |
|
ovelrn |
⊢ ( ... Fn ( ℤ × ℤ ) → ( 𝐴 ∈ ran ... ↔ ∃ 𝑚 ∈ ℤ ∃ 𝑛 ∈ ℤ 𝐴 = ( 𝑚 ... 𝑛 ) ) ) |
82 |
79 80 81
|
mp2b |
⊢ ( 𝐴 ∈ ran ... ↔ ∃ 𝑚 ∈ ℤ ∃ 𝑛 ∈ ℤ 𝐴 = ( 𝑚 ... 𝑛 ) ) |
83 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → 𝐺 ∈ Mnd ) |
84 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → 𝐴 = ( 𝑚 ... 𝑛 ) ) |
85 |
|
frel |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → Rel 𝐹 ) |
86 |
|
reldm0 |
⊢ ( Rel 𝐹 → ( 𝐹 = ∅ ↔ dom 𝐹 = ∅ ) ) |
87 |
7 85 86
|
3syl |
⊢ ( 𝜑 → ( 𝐹 = ∅ ↔ dom 𝐹 = ∅ ) ) |
88 |
7
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝐴 ) |
89 |
88
|
eqeq1d |
⊢ ( 𝜑 → ( dom 𝐹 = ∅ ↔ 𝐴 = ∅ ) ) |
90 |
87 89
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 = ∅ ↔ 𝐴 = ∅ ) ) |
91 |
|
coeq1 |
⊢ ( 𝐹 = ∅ → ( 𝐹 ∘ 𝐻 ) = ( ∅ ∘ 𝐻 ) ) |
92 |
|
co01 |
⊢ ( ∅ ∘ 𝐻 ) = ∅ |
93 |
91 92
|
eqtrdi |
⊢ ( 𝐹 = ∅ → ( 𝐹 ∘ 𝐻 ) = ∅ ) |
94 |
93
|
oveq1d |
⊢ ( 𝐹 = ∅ → ( ( 𝐹 ∘ 𝐻 ) supp 0 ) = ( ∅ supp 0 ) ) |
95 |
|
supp0 |
⊢ ( 0 ∈ V → ( ∅ supp 0 ) = ∅ ) |
96 |
42 95
|
ax-mp |
⊢ ( ∅ supp 0 ) = ∅ |
97 |
94 96
|
eqtrdi |
⊢ ( 𝐹 = ∅ → ( ( 𝐹 ∘ 𝐻 ) supp 0 ) = ∅ ) |
98 |
12 97
|
eqtrid |
⊢ ( 𝐹 = ∅ → 𝑊 = ∅ ) |
99 |
90 98
|
syl6bir |
⊢ ( 𝜑 → ( 𝐴 = ∅ → 𝑊 = ∅ ) ) |
100 |
99
|
necon3d |
⊢ ( 𝜑 → ( 𝑊 ≠ ∅ → 𝐴 ≠ ∅ ) ) |
101 |
100
|
imp |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → 𝐴 ≠ ∅ ) |
102 |
101
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → 𝐴 ≠ ∅ ) |
103 |
84 102
|
eqnetrrd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( 𝑚 ... 𝑛 ) ≠ ∅ ) |
104 |
|
fzn0 |
⊢ ( ( 𝑚 ... 𝑛 ) ≠ ∅ ↔ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) |
105 |
103 104
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) |
106 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
107 |
84
|
feq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐹 : ( 𝑚 ... 𝑛 ) ⟶ 𝐵 ) ) |
108 |
106 107
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → 𝐹 : ( 𝑚 ... 𝑛 ) ⟶ 𝐵 ) |
109 |
1 3 83 105 108
|
gsumval2 |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) |
110 |
|
frn |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 → ran 𝐻 ⊆ 𝐴 ) |
111 |
10 18 110
|
3syl |
⊢ ( 𝜑 → ran 𝐻 ⊆ 𝐴 ) |
112 |
111
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ran 𝐻 ⊆ 𝐴 ) |
113 |
112 84
|
sseqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ran 𝐻 ⊆ ( 𝑚 ... 𝑛 ) ) |
114 |
|
fzssuz |
⊢ ( 𝑚 ... 𝑛 ) ⊆ ( ℤ≥ ‘ 𝑚 ) |
115 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑚 ) ⊆ ℤ |
116 |
|
zssre |
⊢ ℤ ⊆ ℝ |
117 |
115 116
|
sstri |
⊢ ( ℤ≥ ‘ 𝑚 ) ⊆ ℝ |
118 |
114 117
|
sstri |
⊢ ( 𝑚 ... 𝑛 ) ⊆ ℝ |
119 |
113 118
|
sstrdi |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ran 𝐻 ⊆ ℝ ) |
120 |
|
ltso |
⊢ < Or ℝ |
121 |
|
soss |
⊢ ( ran 𝐻 ⊆ ℝ → ( < Or ℝ → < Or ran 𝐻 ) ) |
122 |
119 120 121
|
mpisyl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → < Or ran 𝐻 ) |
123 |
|
fzfi |
⊢ ( 1 ... 𝑀 ) ∈ Fin |
124 |
123
|
a1i |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ Fin ) |
125 |
19 124
|
fexd |
⊢ ( 𝜑 → 𝐻 ∈ V ) |
126 |
|
f1oen3g |
⊢ ( ( 𝐻 ∈ V ∧ 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) → ( 1 ... 𝑀 ) ≈ ran 𝐻 ) |
127 |
125 22 126
|
syl2anc |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ≈ ran 𝐻 ) |
128 |
|
enfi |
⊢ ( ( 1 ... 𝑀 ) ≈ ran 𝐻 → ( ( 1 ... 𝑀 ) ∈ Fin ↔ ran 𝐻 ∈ Fin ) ) |
129 |
127 128
|
syl |
⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∈ Fin ↔ ran 𝐻 ∈ Fin ) ) |
130 |
123 129
|
mpbii |
⊢ ( 𝜑 → ran 𝐻 ∈ Fin ) |
131 |
130
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ran 𝐻 ∈ Fin ) |
132 |
|
fz1iso |
⊢ ( ( < Or ran 𝐻 ∧ ran 𝐻 ∈ Fin ) → ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) |
133 |
122 131 132
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) |
134 |
9
|
nnnn0d |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
135 |
|
hashfz1 |
⊢ ( 𝑀 ∈ ℕ0 → ( ♯ ‘ ( 1 ... 𝑀 ) ) = 𝑀 ) |
136 |
134 135
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 1 ... 𝑀 ) ) = 𝑀 ) |
137 |
124 22
|
hasheqf1od |
⊢ ( 𝜑 → ( ♯ ‘ ( 1 ... 𝑀 ) ) = ( ♯ ‘ ran 𝐻 ) ) |
138 |
136 137
|
eqtr3d |
⊢ ( 𝜑 → 𝑀 = ( ♯ ‘ ran 𝐻 ) ) |
139 |
138
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑀 = ( ♯ ‘ ran 𝐻 ) ) |
140 |
139
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑀 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ran 𝐻 ) ) ) |
141 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐺 ∈ Mnd ) |
142 |
1 3
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
143 |
142
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
144 |
141 143
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
145 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
146 |
145
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ran 𝐹 ) → 𝑥 ∈ ( 𝑍 ‘ ran 𝐹 ) ) |
147 |
3 4
|
cntzi |
⊢ ( ( 𝑥 ∈ ( 𝑍 ‘ ran 𝐹 ) ∧ 𝑦 ∈ ran 𝐹 ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
148 |
146 147
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ran 𝐹 ) ∧ 𝑦 ∈ ran 𝐹 ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
149 |
148
|
anasss |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ ( 𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
150 |
1 3
|
mndass |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
151 |
141 150
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
152 |
71
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑀 ∈ ( ℤ≥ ‘ 1 ) ) |
153 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
154 |
153
|
frnd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ran 𝐹 ⊆ 𝐵 ) |
155 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) |
156 |
|
isof1o |
⊢ ( 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) → 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) –1-1-onto→ ran 𝐻 ) |
157 |
155 156
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) –1-1-onto→ ran 𝐻 ) |
158 |
139
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 1 ... 𝑀 ) = ( 1 ... ( ♯ ‘ ran 𝐻 ) ) ) |
159 |
158
|
f1oeq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ↔ 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) –1-1-onto→ ran 𝐻 ) ) |
160 |
157 159
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) |
161 |
|
f1ocnv |
⊢ ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 → ◡ 𝑓 : ran 𝐻 –1-1-onto→ ( 1 ... 𝑀 ) ) |
162 |
160 161
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ◡ 𝑓 : ran 𝐻 –1-1-onto→ ( 1 ... 𝑀 ) ) |
163 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) |
164 |
|
f1oco |
⊢ ( ( ◡ 𝑓 : ran 𝐻 –1-1-onto→ ( 1 ... 𝑀 ) ∧ 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 ) → ( ◡ 𝑓 ∘ 𝐻 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ) |
165 |
162 163 164
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( ◡ 𝑓 ∘ 𝐻 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ) |
166 |
|
ffn |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 ) |
167 |
|
dffn4 |
⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 ) |
168 |
166 167
|
sylib |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 : 𝐴 –onto→ ran 𝐹 ) |
169 |
|
fof |
⊢ ( 𝐹 : 𝐴 –onto→ ran 𝐹 → 𝐹 : 𝐴 ⟶ ran 𝐹 ) |
170 |
153 168 169
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐹 : 𝐴 ⟶ ran 𝐹 ) |
171 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 → 𝑓 : ( 1 ... 𝑀 ) ⟶ ran 𝐻 ) |
172 |
160 171
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑓 : ( 1 ... 𝑀 ) ⟶ ran 𝐻 ) |
173 |
111
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ran 𝐻 ⊆ 𝐴 ) |
174 |
172 173
|
fssd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑓 : ( 1 ... 𝑀 ) ⟶ 𝐴 ) |
175 |
|
fco |
⊢ ( ( 𝐹 : 𝐴 ⟶ ran 𝐹 ∧ 𝑓 : ( 1 ... 𝑀 ) ⟶ 𝐴 ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... 𝑀 ) ⟶ ran 𝐹 ) |
176 |
170 174 175
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... 𝑀 ) ⟶ ran 𝐹 ) |
177 |
176
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑥 ) ∈ ran 𝐹 ) |
178 |
|
f1ococnv2 |
⊢ ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 → ( 𝑓 ∘ ◡ 𝑓 ) = ( I ↾ ran 𝐻 ) ) |
179 |
160 178
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝑓 ∘ ◡ 𝑓 ) = ( I ↾ ran 𝐻 ) ) |
180 |
179
|
coeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( ( 𝑓 ∘ ◡ 𝑓 ) ∘ 𝐻 ) = ( ( I ↾ ran 𝐻 ) ∘ 𝐻 ) ) |
181 |
|
f1of |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐻 → 𝐻 : ( 1 ... 𝑀 ) ⟶ ran 𝐻 ) |
182 |
|
fcoi2 |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) ⟶ ran 𝐻 → ( ( I ↾ ran 𝐻 ) ∘ 𝐻 ) = 𝐻 ) |
183 |
163 181 182
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( ( I ↾ ran 𝐻 ) ∘ 𝐻 ) = 𝐻 ) |
184 |
180 183
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐻 = ( ( 𝑓 ∘ ◡ 𝑓 ) ∘ 𝐻 ) ) |
185 |
|
coass |
⊢ ( ( 𝑓 ∘ ◡ 𝑓 ) ∘ 𝐻 ) = ( 𝑓 ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) |
186 |
184 185
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐻 = ( 𝑓 ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ) |
187 |
186
|
coeq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝐹 ∘ 𝐻 ) = ( 𝐹 ∘ ( 𝑓 ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ) ) |
188 |
|
coass |
⊢ ( ( 𝐹 ∘ 𝑓 ) ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) = ( 𝐹 ∘ ( 𝑓 ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ) |
189 |
187 188
|
eqtr4di |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝐹 ∘ 𝐻 ) = ( ( 𝐹 ∘ 𝑓 ) ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ) |
190 |
189
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑘 ) = ( ( ( 𝐹 ∘ 𝑓 ) ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ‘ 𝑘 ) ) |
191 |
190
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑘 ) = ( ( ( 𝐹 ∘ 𝑓 ) ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ‘ 𝑘 ) ) |
192 |
|
f1of |
⊢ ( ◡ 𝑓 : ran 𝐻 –1-1-onto→ ( 1 ... 𝑀 ) → ◡ 𝑓 : ran 𝐻 ⟶ ( 1 ... 𝑀 ) ) |
193 |
160 161 192
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ◡ 𝑓 : ran 𝐻 ⟶ ( 1 ... 𝑀 ) ) |
194 |
163 181
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐻 : ( 1 ... 𝑀 ) ⟶ ran 𝐻 ) |
195 |
|
fco |
⊢ ( ( ◡ 𝑓 : ran 𝐻 ⟶ ( 1 ... 𝑀 ) ∧ 𝐻 : ( 1 ... 𝑀 ) ⟶ ran 𝐻 ) → ( ◡ 𝑓 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ ( 1 ... 𝑀 ) ) |
196 |
193 194 195
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( ◡ 𝑓 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ ( 1 ... 𝑀 ) ) |
197 |
|
fvco3 |
⊢ ( ( ( ◡ 𝑓 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ ( 1 ... 𝑀 ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 𝐹 ∘ 𝑓 ) ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ‘ 𝑘 ) = ( ( 𝐹 ∘ 𝑓 ) ‘ ( ( ◡ 𝑓 ∘ 𝐻 ) ‘ 𝑘 ) ) ) |
198 |
196 197
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 𝐹 ∘ 𝑓 ) ∘ ( ◡ 𝑓 ∘ 𝐻 ) ) ‘ 𝑘 ) = ( ( 𝐹 ∘ 𝑓 ) ‘ ( ( ◡ 𝑓 ∘ 𝐻 ) ‘ 𝑘 ) ) ) |
199 |
191 198
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑘 ) = ( ( 𝐹 ∘ 𝑓 ) ‘ ( ( ◡ 𝑓 ∘ 𝐻 ) ‘ 𝑘 ) ) ) |
200 |
144 149 151 152 154 165 177 199
|
seqf1o |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑀 ) ) |
201 |
1 3 2
|
mndlid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = 𝑥 ) |
202 |
141 201
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = 𝑥 ) |
203 |
1 3 2
|
mndrid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = 𝑥 ) |
204 |
141 203
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = 𝑥 ) |
205 |
141 66
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 0 ∈ 𝐵 ) |
206 |
|
fdm |
⊢ ( 𝐻 : ( 1 ... 𝑀 ) ⟶ 𝐴 → dom 𝐻 = ( 1 ... 𝑀 ) ) |
207 |
10 18 206
|
3syl |
⊢ ( 𝜑 → dom 𝐻 = ( 1 ... 𝑀 ) ) |
208 |
|
eluzfz1 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑀 ) ) |
209 |
|
ne0i |
⊢ ( 1 ∈ ( 1 ... 𝑀 ) → ( 1 ... 𝑀 ) ≠ ∅ ) |
210 |
71 208 209
|
3syl |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ≠ ∅ ) |
211 |
207 210
|
eqnetrd |
⊢ ( 𝜑 → dom 𝐻 ≠ ∅ ) |
212 |
|
dm0rn0 |
⊢ ( dom 𝐻 = ∅ ↔ ran 𝐻 = ∅ ) |
213 |
212
|
necon3bii |
⊢ ( dom 𝐻 ≠ ∅ ↔ ran 𝐻 ≠ ∅ ) |
214 |
211 213
|
sylib |
⊢ ( 𝜑 → ran 𝐻 ≠ ∅ ) |
215 |
214
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ran 𝐻 ≠ ∅ ) |
216 |
113
|
adantrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ran 𝐻 ⊆ ( 𝑚 ... 𝑛 ) ) |
217 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝐴 = ( 𝑚 ... 𝑛 ) ) |
218 |
217
|
eleq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ( 𝑚 ... 𝑛 ) ) ) |
219 |
218
|
biimpar |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ( 𝑚 ... 𝑛 ) ) → 𝑥 ∈ 𝐴 ) |
220 |
153
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
221 |
219 220
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ( 𝑚 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
222 |
217
|
difeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝐴 ∖ ran 𝐻 ) = ( ( 𝑚 ... 𝑛 ) ∖ ran 𝐻 ) ) |
223 |
222
|
eleq2d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( 𝑥 ∈ ( 𝐴 ∖ ran 𝐻 ) ↔ 𝑥 ∈ ( ( 𝑚 ... 𝑛 ) ∖ ran 𝐻 ) ) ) |
224 |
223
|
biimpar |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ( ( 𝑚 ... 𝑛 ) ∖ ran 𝐻 ) ) → 𝑥 ∈ ( 𝐴 ∖ ran 𝐻 ) ) |
225 |
56
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝐻 ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
226 |
224 225
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑥 ∈ ( ( 𝑚 ... 𝑛 ) ∖ ran 𝐻 ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
227 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) –1-1-onto→ ran 𝐻 → 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) ⟶ ran 𝐻 ) |
228 |
155 156 227
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) ⟶ ran 𝐻 ) |
229 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ ran 𝐻 ) ) ⟶ ran 𝐻 ∧ 𝑦 ∈ ( 1 ... ( ♯ ‘ ran 𝐻 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ‘ 𝑦 ) ) ) |
230 |
228 229
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) ∧ 𝑦 ∈ ( 1 ... ( ♯ ‘ ran 𝐻 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ‘ 𝑦 ) ) ) |
231 |
202 204 144 205 155 215 216 221 226 230
|
seqcoll2 |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ran 𝐻 ) ) ) |
232 |
140 200 231
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) ) ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) |
233 |
232
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) |
234 |
233
|
exlimdv |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ ran 𝐻 ) ) , ran 𝐻 ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) |
235 |
133 234
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) |
236 |
109 235
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ 𝐴 = ( 𝑚 ... 𝑛 ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) |
237 |
236
|
ex |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → ( 𝐴 = ( 𝑚 ... 𝑛 ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
238 |
237
|
rexlimdvw |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → ( ∃ 𝑛 ∈ ℤ 𝐴 = ( 𝑚 ... 𝑛 ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
239 |
238
|
rexlimdvw |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → ( ∃ 𝑚 ∈ ℤ ∃ 𝑛 ∈ ℤ 𝐴 = ( 𝑚 ... 𝑛 ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
240 |
82 239
|
syl5bi |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → ( 𝐴 ∈ ran ... → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
241 |
|
suppssdm |
⊢ ( ( 𝐹 ∘ 𝐻 ) supp 0 ) ⊆ dom ( 𝐹 ∘ 𝐻 ) |
242 |
12 241
|
eqsstri |
⊢ 𝑊 ⊆ dom ( 𝐹 ∘ 𝐻 ) |
243 |
242 37
|
fssdm |
⊢ ( 𝜑 → 𝑊 ⊆ ( 1 ... 𝑀 ) ) |
244 |
|
fz1ssnn |
⊢ ( 1 ... 𝑀 ) ⊆ ℕ |
245 |
|
nnssre |
⊢ ℕ ⊆ ℝ |
246 |
244 245
|
sstri |
⊢ ( 1 ... 𝑀 ) ⊆ ℝ |
247 |
243 246
|
sstrdi |
⊢ ( 𝜑 → 𝑊 ⊆ ℝ ) |
248 |
|
soss |
⊢ ( 𝑊 ⊆ ℝ → ( < Or ℝ → < Or 𝑊 ) ) |
249 |
247 120 248
|
mpisyl |
⊢ ( 𝜑 → < Or 𝑊 ) |
250 |
|
ssfi |
⊢ ( ( ( 1 ... 𝑀 ) ∈ Fin ∧ 𝑊 ⊆ ( 1 ... 𝑀 ) ) → 𝑊 ∈ Fin ) |
251 |
123 243 250
|
sylancr |
⊢ ( 𝜑 → 𝑊 ∈ Fin ) |
252 |
|
fz1iso |
⊢ ( ( < Or 𝑊 ∧ 𝑊 ∈ Fin ) → ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) |
253 |
249 251 252
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) |
254 |
253
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ¬ 𝐴 ∈ ran ... ) → ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) |
255 |
1 2 3 4 5 6 7 8 9 10 11 12
|
gsumval3lem2 |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ ( 𝐻 ∘ 𝑓 ) ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) |
256 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 𝐺 ∈ Mnd ) |
257 |
256 201
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = 𝑥 ) |
258 |
256 203
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = 𝑥 ) |
259 |
256 143
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
260 |
256 66
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 0 ∈ 𝐵 ) |
261 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) |
262 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 𝑊 ≠ ∅ ) |
263 |
243
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 𝑊 ⊆ ( 1 ... 𝑀 ) ) |
264 |
37
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → ( 𝐹 ∘ 𝐻 ) : ( 1 ... 𝑀 ) ⟶ 𝐵 ) |
265 |
264
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ 𝑥 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑥 ) ∈ 𝐵 ) |
266 |
39
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝐻 ) supp 0 ) ⊆ 𝑊 ) |
267 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → ( 1 ... 𝑀 ) ∈ V ) |
268 |
42
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 0 ∈ V ) |
269 |
264 266 267 268
|
suppssr |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ 𝑥 ∈ ( ( 1 ... 𝑀 ) ∖ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑥 ) = 0 ) |
270 |
|
coass |
⊢ ( ( 𝐹 ∘ 𝐻 ) ∘ 𝑓 ) = ( 𝐹 ∘ ( 𝐻 ∘ 𝑓 ) ) |
271 |
270
|
fveq1i |
⊢ ( ( ( 𝐹 ∘ 𝐻 ) ∘ 𝑓 ) ‘ 𝑦 ) = ( ( 𝐹 ∘ ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑦 ) |
272 |
|
isof1o |
⊢ ( 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) |
273 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝑊 ) |
274 |
261 272 273
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝑊 ) |
275 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝑊 ∧ 𝑦 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹 ∘ 𝐻 ) ∘ 𝑓 ) ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝐻 ) ‘ ( 𝑓 ‘ 𝑦 ) ) ) |
276 |
274 275
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ 𝑦 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹 ∘ 𝐻 ) ∘ 𝑓 ) ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝐻 ) ‘ ( 𝑓 ‘ 𝑦 ) ) ) |
277 |
271 276
|
eqtr3id |
⊢ ( ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) ∧ 𝑦 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝐻 ) ‘ ( 𝑓 ‘ 𝑦 ) ) ) |
278 |
257 258 259 260 261 262 263 265 269 277
|
seqcoll2 |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) = ( seq 1 ( + , ( 𝐹 ∘ ( 𝐻 ∘ 𝑓 ) ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) |
279 |
255 278
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ( ¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) |
280 |
279
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ¬ 𝐴 ∈ ran ... ) → ( 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
281 |
280
|
exlimdv |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ¬ 𝐴 ∈ ran ... ) → ( ∃ 𝑓 𝑓 Isom < , < ( ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑊 ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
282 |
254 281
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) ∧ ¬ 𝐴 ∈ ran ... ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) |
283 |
282
|
ex |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → ( ¬ 𝐴 ∈ ran ... → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) ) |
284 |
240 283
|
pm2.61d |
⊢ ( ( 𝜑 ∧ 𝑊 ≠ ∅ ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) |
285 |
78 284
|
pm2.61dane |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝐻 ) ) ‘ 𝑀 ) ) |