Step |
Hyp |
Ref |
Expression |
1 |
|
efgval.w |
⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) ) |
2 |
|
efgval.r |
⊢ ∼ = ( ~FG ‘ 𝐼 ) |
3 |
|
efgval2.m |
⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ 〈 𝑦 , ( 1o ∖ 𝑧 ) 〉 ) |
4 |
|
efgval2.t |
⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) ) |
5 |
|
efgred.d |
⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) ) |
6 |
|
efgred.s |
⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ) |
7 |
1 2 3 4 5 6
|
efgsdm |
⊢ ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑎 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑎 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑎 − 1 ) ) ) ) ) |
8 |
7
|
simp1bi |
⊢ ( 𝐹 ∈ dom 𝑆 → 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ) |
9 |
|
eldifsn |
⊢ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ↔ ( 𝐹 ∈ Word 𝑊 ∧ 𝐹 ≠ ∅ ) ) |
10 |
|
lennncl |
⊢ ( ( 𝐹 ∈ Word 𝑊 ∧ 𝐹 ≠ ∅ ) → ( ♯ ‘ 𝐹 ) ∈ ℕ ) |
11 |
9 10
|
sylbi |
⊢ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ♯ ‘ 𝐹 ) ∈ ℕ ) |
12 |
|
fzo0end |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
13 |
8 11 12
|
3syl |
⊢ ( 𝐹 ∈ dom 𝑆 → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
14 |
|
nnm1nn0 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ℕ0 ) |
15 |
8 11 14
|
3syl |
⊢ ( 𝐹 ∈ dom 𝑆 → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ℕ0 ) |
16 |
|
eleq1 |
⊢ ( 𝑎 = 0 → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
17 |
|
fveq2 |
⊢ ( 𝑎 = 0 → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 0 ) ) |
18 |
17
|
breq2d |
⊢ ( 𝑎 = 0 → ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ↔ ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 0 ) ) ) |
19 |
16 18
|
imbi12d |
⊢ ( 𝑎 = 0 → ( ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ↔ ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 0 ) ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑎 = 0 → ( ( 𝐹 ∈ dom 𝑆 → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ) ↔ ( 𝐹 ∈ dom 𝑆 → ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 0 ) ) ) ) ) |
21 |
|
eleq1 |
⊢ ( 𝑎 = 𝑖 → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑎 = 𝑖 → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑖 ) ) |
23 |
22
|
breq2d |
⊢ ( 𝑎 = 𝑖 → ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ↔ ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) ) |
24 |
21 23
|
imbi12d |
⊢ ( 𝑎 = 𝑖 → ( ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ↔ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑎 = 𝑖 → ( ( 𝐹 ∈ dom 𝑆 → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ) ↔ ( 𝐹 ∈ dom 𝑆 → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
26 |
|
eleq1 |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
27 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) |
28 |
27
|
breq2d |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ↔ ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) |
29 |
26 28
|
imbi12d |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) |
30 |
29
|
imbi2d |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( ( 𝐹 ∈ dom 𝑆 → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ) ↔ ( 𝐹 ∈ dom 𝑆 → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
31 |
|
eleq1 |
⊢ ( 𝑎 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
32 |
|
fveq2 |
⊢ ( 𝑎 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) |
33 |
32
|
breq2d |
⊢ ( 𝑎 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ↔ ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) |
34 |
31 33
|
imbi12d |
⊢ ( 𝑎 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
35 |
34
|
imbi2d |
⊢ ( 𝑎 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( ( 𝐹 ∈ dom 𝑆 → ( 𝑎 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑎 ) ) ) ↔ ( 𝐹 ∈ dom 𝑆 → ( ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) ) |
36 |
1 2
|
efger |
⊢ ∼ Er 𝑊 |
37 |
36
|
a1i |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ∼ Er 𝑊 ) |
38 |
|
eldifi |
⊢ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) → 𝐹 ∈ Word 𝑊 ) |
39 |
|
wrdf |
⊢ ( 𝐹 ∈ Word 𝑊 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ 𝑊 ) |
40 |
8 38 39
|
3syl |
⊢ ( 𝐹 ∈ dom 𝑆 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ 𝑊 ) |
41 |
40
|
ffvelrnda |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 0 ) ∈ 𝑊 ) |
42 |
37 41
|
erref |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 0 ) ) |
43 |
42
|
ex |
⊢ ( 𝐹 ∈ dom 𝑆 → ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 0 ) ) ) |
44 |
|
elnn0uz |
⊢ ( 𝑖 ∈ ℕ0 ↔ 𝑖 ∈ ( ℤ≥ ‘ 0 ) ) |
45 |
|
peano2fzor |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 0 ) ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
46 |
44 45
|
sylanb |
⊢ ( ( 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
47 |
46
|
3adant1 |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
48 |
47
|
3expia |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
49 |
48
|
imim1d |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) ) ) |
50 |
40
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ 𝑊 ) |
51 |
50 47
|
ffvelrnd |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑖 ) ∈ 𝑊 ) |
52 |
|
fvoveq1 |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( 𝐹 ‘ ( 𝑎 − 1 ) ) = ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) |
53 |
52
|
fveq2d |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑎 − 1 ) ) ) = ( 𝑇 ‘ ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) ) |
54 |
53
|
rneqd |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑎 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) ) |
55 |
27 54
|
eleq12d |
⊢ ( 𝑎 = ( 𝑖 + 1 ) → ( ( 𝐹 ‘ 𝑎 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑎 − 1 ) ) ) ↔ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) ) ) |
56 |
7
|
simp3bi |
⊢ ( 𝐹 ∈ dom 𝑆 → ∀ 𝑎 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑎 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑎 − 1 ) ) ) ) |
57 |
56
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ∀ 𝑎 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑎 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑎 − 1 ) ) ) ) |
58 |
|
nn0p1nn |
⊢ ( 𝑖 ∈ ℕ0 → ( 𝑖 + 1 ) ∈ ℕ ) |
59 |
58
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑖 + 1 ) ∈ ℕ ) |
60 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
61 |
59 60
|
eleqtrdi |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑖 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
62 |
|
elfzolt2b |
⊢ ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝑖 + 1 ) ∈ ( ( 𝑖 + 1 ) ..^ ( ♯ ‘ 𝐹 ) ) ) |
63 |
62
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑖 + 1 ) ∈ ( ( 𝑖 + 1 ) ..^ ( ♯ ‘ 𝐹 ) ) ) |
64 |
|
elfzo3 |
⊢ ( ( 𝑖 + 1 ) ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( ( 𝑖 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ ( 𝑖 + 1 ) ∈ ( ( 𝑖 + 1 ) ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
65 |
61 63 64
|
sylanbrc |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑖 + 1 ) ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) |
66 |
55 57 65
|
rspcdva |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ ( 𝑖 + 1 ) ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) ) |
67 |
|
nn0cn |
⊢ ( 𝑖 ∈ ℕ0 → 𝑖 ∈ ℂ ) |
68 |
67
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑖 ∈ ℂ ) |
69 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
70 |
|
pncan |
⊢ ( ( 𝑖 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑖 + 1 ) − 1 ) = 𝑖 ) |
71 |
68 69 70
|
sylancl |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑖 + 1 ) − 1 ) = 𝑖 ) |
72 |
71
|
fveq2d |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) = ( 𝐹 ‘ 𝑖 ) ) |
73 |
72
|
fveq2d |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑇 ‘ ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) = ( 𝑇 ‘ ( 𝐹 ‘ 𝑖 ) ) ) |
74 |
73
|
rneqd |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ran ( 𝑇 ‘ ( 𝐹 ‘ ( ( 𝑖 + 1 ) − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝐹 ‘ 𝑖 ) ) ) |
75 |
66 74
|
eleqtrd |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ ( 𝑖 + 1 ) ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ 𝑖 ) ) ) |
76 |
1 2 3 4
|
efgi2 |
⊢ ( ( ( 𝐹 ‘ 𝑖 ) ∈ 𝑊 ∧ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( 𝐹 ‘ 𝑖 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) |
77 |
51 75 76
|
syl2anc |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑖 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) |
78 |
36
|
a1i |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ∼ Er 𝑊 ) |
79 |
78
|
ertr |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ∧ ( 𝐹 ‘ 𝑖 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) |
80 |
77 79
|
mpan2d |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ∧ ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) |
81 |
80
|
3expia |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) |
82 |
81
|
a2d |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ) → ( ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) |
83 |
49 82
|
syld |
⊢ ( ( 𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) |
84 |
83
|
expcom |
⊢ ( 𝑖 ∈ ℕ0 → ( 𝐹 ∈ dom 𝑆 → ( ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
85 |
84
|
a2d |
⊢ ( 𝑖 ∈ ℕ0 → ( ( 𝐹 ∈ dom 𝑆 → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ 𝑖 ) ) ) → ( 𝐹 ∈ dom 𝑆 → ( ( 𝑖 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
86 |
20 25 30 35 43 85
|
nn0ind |
⊢ ( ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ℕ0 → ( 𝐹 ∈ dom 𝑆 → ( ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
87 |
15 86
|
mpcom |
⊢ ( 𝐹 ∈ dom 𝑆 → ( ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) |
88 |
13 87
|
mpd |
⊢ ( 𝐹 ∈ dom 𝑆 → ( 𝐹 ‘ 0 ) ∼ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) |
89 |
1 2 3 4 5 6
|
efgsval |
⊢ ( 𝐹 ∈ dom 𝑆 → ( 𝑆 ‘ 𝐹 ) = ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) |
90 |
88 89
|
breqtrrd |
⊢ ( 𝐹 ∈ dom 𝑆 → ( 𝐹 ‘ 0 ) ∼ ( 𝑆 ‘ 𝐹 ) ) |