Step |
Hyp |
Ref |
Expression |
1 |
|
c0ex |
⊢ 0 ∈ V |
2 |
1
|
tpid1 |
⊢ 0 ∈ { 0 , 1 , 2 } |
3 |
|
fzo0to3tp |
⊢ ( 0 ..^ 3 ) = { 0 , 1 , 2 } |
4 |
2 3
|
eleqtrri |
⊢ 0 ∈ ( 0 ..^ 3 ) |
5 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑊 ) = 3 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 3 ) ) |
6 |
4 5
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝑊 ) = 3 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
7 |
|
wrdsymbcl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 0 ) ∈ 𝑉 ) |
8 |
6 7
|
sylan2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( 𝑊 ‘ 0 ) ∈ 𝑉 ) |
9 |
|
1ex |
⊢ 1 ∈ V |
10 |
9
|
tpid2 |
⊢ 1 ∈ { 0 , 1 , 2 } |
11 |
10 3
|
eleqtrri |
⊢ 1 ∈ ( 0 ..^ 3 ) |
12 |
11 5
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝑊 ) = 3 → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
13 |
|
wrdsymbcl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 1 ) ∈ 𝑉 ) |
14 |
12 13
|
sylan2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( 𝑊 ‘ 1 ) ∈ 𝑉 ) |
15 |
|
2ex |
⊢ 2 ∈ V |
16 |
15
|
tpid3 |
⊢ 2 ∈ { 0 , 1 , 2 } |
17 |
16 3
|
eleqtrri |
⊢ 2 ∈ ( 0 ..^ 3 ) |
18 |
17 5
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝑊 ) = 3 → 2 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
19 |
|
wrdsymbcl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 2 ) ∈ 𝑉 ) |
20 |
18 19
|
sylan2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( 𝑊 ‘ 2 ) ∈ 𝑉 ) |
21 |
|
simpr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( ♯ ‘ 𝑊 ) = 3 ) |
22 |
|
eqid |
⊢ ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) |
23 |
|
eqid |
⊢ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) |
24 |
|
eqid |
⊢ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) |
25 |
22 23 24
|
3pm3.2i |
⊢ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) ) |
26 |
21 25
|
jctir |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) ) ) ) |
27 |
|
eqeq2 |
⊢ ( 𝑎 = ( 𝑊 ‘ 0 ) → ( ( 𝑊 ‘ 0 ) = 𝑎 ↔ ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ) ) |
28 |
27
|
3anbi1d |
⊢ ( 𝑎 = ( 𝑊 ‘ 0 ) → ( ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ↔ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) |
29 |
28
|
anbi2d |
⊢ ( 𝑎 = ( 𝑊 ‘ 0 ) → ( ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) |
30 |
|
eqeq2 |
⊢ ( 𝑏 = ( 𝑊 ‘ 1 ) → ( ( 𝑊 ‘ 1 ) = 𝑏 ↔ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ) ) |
31 |
30
|
3anbi2d |
⊢ ( 𝑏 = ( 𝑊 ‘ 1 ) → ( ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ↔ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) |
32 |
31
|
anbi2d |
⊢ ( 𝑏 = ( 𝑊 ‘ 1 ) → ( ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) |
33 |
|
eqeq2 |
⊢ ( 𝑐 = ( 𝑊 ‘ 2 ) → ( ( 𝑊 ‘ 2 ) = 𝑐 ↔ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) ) ) |
34 |
33
|
3anbi3d |
⊢ ( 𝑐 = ( 𝑊 ‘ 2 ) → ( ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ↔ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) ) ) ) |
35 |
34
|
anbi2d |
⊢ ( 𝑐 = ( 𝑊 ‘ 2 ) → ( ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) ) ) ) ) |
36 |
29 32 35
|
rspc3ev |
⊢ ( ( ( ( 𝑊 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑊 ‘ 1 ) ∈ 𝑉 ∧ ( 𝑊 ‘ 2 ) ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ∧ ( 𝑊 ‘ 1 ) = ( 𝑊 ‘ 1 ) ∧ ( 𝑊 ‘ 2 ) = ( 𝑊 ‘ 2 ) ) ) ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) |
37 |
8 14 20 26 36
|
syl31anc |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) |
38 |
|
df-3an |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ↔ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑐 ∈ 𝑉 ) ) |
39 |
|
eqwrds3 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ) → ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) |
40 |
39
|
ex |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) ) |
41 |
38 40
|
syl5bir |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑐 ∈ 𝑉 ) → ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) ) |
42 |
41
|
expd |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑐 ∈ 𝑉 → ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑐 ∈ 𝑉 → ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) ) ) |
44 |
43
|
imp31 |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ 𝑐 ∈ 𝑉 ) → ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) |
45 |
44
|
rexbidva |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ∃ 𝑐 ∈ 𝑉 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ∃ 𝑐 ∈ 𝑉 ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) |
46 |
45
|
2rexbidva |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ( ( ♯ ‘ 𝑊 ) = 3 ∧ ( ( 𝑊 ‘ 0 ) = 𝑎 ∧ ( 𝑊 ‘ 1 ) = 𝑏 ∧ ( 𝑊 ‘ 2 ) = 𝑐 ) ) ) ) |
47 |
37 46
|
mpbird |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ) |
48 |
|
s3cl |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → 〈“ 𝑎 𝑏 𝑐 ”〉 ∈ Word 𝑉 ) |
49 |
48
|
ad4ant123 |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑐 ∈ 𝑉 ) ∧ 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ) → 〈“ 𝑎 𝑏 𝑐 ”〉 ∈ Word 𝑉 ) |
50 |
|
s3len |
⊢ ( ♯ ‘ 〈“ 𝑎 𝑏 𝑐 ”〉 ) = 3 |
51 |
49 50
|
jctir |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑐 ∈ 𝑉 ) ∧ 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ) → ( 〈“ 𝑎 𝑏 𝑐 ”〉 ∈ Word 𝑉 ∧ ( ♯ ‘ 〈“ 𝑎 𝑏 𝑐 ”〉 ) = 3 ) ) |
52 |
|
eleq1 |
⊢ ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 → ( 𝑊 ∈ Word 𝑉 ↔ 〈“ 𝑎 𝑏 𝑐 ”〉 ∈ Word 𝑉 ) ) |
53 |
|
fveqeq2 |
⊢ ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 → ( ( ♯ ‘ 𝑊 ) = 3 ↔ ( ♯ ‘ 〈“ 𝑎 𝑏 𝑐 ”〉 ) = 3 ) ) |
54 |
52 53
|
anbi12d |
⊢ ( 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 → ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ↔ ( 〈“ 𝑎 𝑏 𝑐 ”〉 ∈ Word 𝑉 ∧ ( ♯ ‘ 〈“ 𝑎 𝑏 𝑐 ”〉 ) = 3 ) ) ) |
55 |
54
|
adantl |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑐 ∈ 𝑉 ) ∧ 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ↔ ( 〈“ 𝑎 𝑏 𝑐 ”〉 ∈ Word 𝑉 ∧ ( ♯ ‘ 〈“ 𝑎 𝑏 𝑐 ”〉 ) = 3 ) ) ) |
56 |
51 55
|
mpbird |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑐 ∈ 𝑉 ) ∧ 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ) → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ) |
57 |
56
|
rexlimdva2 |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ∃ 𝑐 ∈ 𝑉 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ) ) |
58 |
57
|
rexlimivv |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ) |
59 |
47 58
|
impbii |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 3 ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 𝑊 = 〈“ 𝑎 𝑏 𝑐 ”〉 ) |