Step |
Hyp |
Ref |
Expression |
1 |
|
psgnunilem4.g |
⊢ 𝐺 = ( SymGrp ‘ 𝐷 ) |
2 |
|
psgnunilem4.t |
⊢ 𝑇 = ran ( pmTrsp ‘ 𝐷 ) |
3 |
|
psgnunilem4.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑉 ) |
4 |
|
psgnunilem4.w1 |
⊢ ( 𝜑 → 𝑊 ∈ Word 𝑇 ) |
5 |
|
psgnunilem4.w2 |
⊢ ( 𝜑 → ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) ) |
6 |
|
wrdfin |
⊢ ( 𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin ) |
7 |
|
hashcl |
⊢ ( 𝑊 ∈ Fin → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
8 |
4 6 7
|
3syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
9 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
10 |
8 9
|
eleqtrdi |
⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 0 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑤 = ∅ → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ ∅ ) ) |
12 |
|
hash0 |
⊢ ( ♯ ‘ ∅ ) = 0 |
13 |
11 12
|
eqtrdi |
⊢ ( 𝑤 = ∅ → ( ♯ ‘ 𝑤 ) = 0 ) |
14 |
13
|
oveq2d |
⊢ ( 𝑤 = ∅ → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = ( - 1 ↑ 0 ) ) |
15 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
16 |
|
exp0 |
⊢ ( - 1 ∈ ℂ → ( - 1 ↑ 0 ) = 1 ) |
17 |
15 16
|
ax-mp |
⊢ ( - 1 ↑ 0 ) = 1 |
18 |
14 17
|
eqtrdi |
⊢ ( 𝑤 = ∅ → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) |
19 |
18
|
2a1d |
⊢ ( 𝑤 = ∅ → ( ( 𝜑 ∧ ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) |
20 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → 𝜑 ) |
21 |
20 3
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → 𝐷 ∈ 𝑉 ) |
22 |
|
simpl3l |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → 𝑤 ∈ Word 𝑇 ) |
23 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑤 ) ) |
24 |
|
wrdfin |
⊢ ( 𝑤 ∈ Word 𝑇 → 𝑤 ∈ Fin ) |
25 |
22 24
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → 𝑤 ∈ Fin ) |
26 |
|
simpl2 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → 𝑤 ≠ ∅ ) |
27 |
|
hashnncl |
⊢ ( 𝑤 ∈ Fin → ( ( ♯ ‘ 𝑤 ) ∈ ℕ ↔ 𝑤 ≠ ∅ ) ) |
28 |
27
|
biimpar |
⊢ ( ( 𝑤 ∈ Fin ∧ 𝑤 ≠ ∅ ) → ( ♯ ‘ 𝑤 ) ∈ ℕ ) |
29 |
25 26 28
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( ♯ ‘ 𝑤 ) ∈ ℕ ) |
30 |
|
simpl3r |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) |
31 |
|
fveqeq2 |
⊢ ( 𝑥 = 𝑦 → ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ↔ ( ♯ ‘ 𝑦 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ) ) |
32 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐺 Σg 𝑥 ) = ( 𝐺 Σg 𝑦 ) ) |
33 |
32
|
eqeq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ↔ ( 𝐺 Σg 𝑦 ) = ( I ↾ 𝐷 ) ) ) |
34 |
31 33
|
anbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ↔ ( ( ♯ ‘ 𝑦 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑦 ) = ( I ↾ 𝐷 ) ) ) ) |
35 |
34
|
cbvrexvw |
⊢ ( ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ↔ ∃ 𝑦 ∈ Word 𝑇 ( ( ♯ ‘ 𝑦 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑦 ) = ( I ↾ 𝐷 ) ) ) |
36 |
35
|
notbii |
⊢ ( ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ↔ ¬ ∃ 𝑦 ∈ Word 𝑇 ( ( ♯ ‘ 𝑦 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑦 ) = ( I ↾ 𝐷 ) ) ) |
37 |
36
|
biimpi |
⊢ ( ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ¬ ∃ 𝑦 ∈ Word 𝑇 ( ( ♯ ‘ 𝑦 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑦 ) = ( I ↾ 𝐷 ) ) ) |
38 |
37
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ¬ ∃ 𝑦 ∈ Word 𝑇 ( ( ♯ ‘ 𝑦 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑦 ) = ( I ↾ 𝐷 ) ) ) |
39 |
1 2 21 22 23 29 30 38
|
psgnunilem3 |
⊢ ¬ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) |
40 |
|
iman |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ↔ ¬ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ¬ ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) |
41 |
39 40
|
mpbir |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) |
42 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ Word 𝑇 ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) |
43 |
41 42
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ∃ 𝑥 ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) |
44 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → 𝑥 ∈ Word 𝑇 ) |
45 |
|
simprrr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) |
46 |
44 45
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) |
47 |
|
wrdfin |
⊢ ( 𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin ) |
48 |
|
hashcl |
⊢ ( 𝑥 ∈ Fin → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
49 |
44 47 48
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
50 |
|
simp3l |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → 𝑤 ∈ Word 𝑇 ) |
51 |
50 24
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → 𝑤 ∈ Fin ) |
52 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → 𝑤 ≠ ∅ ) |
53 |
51 52 28
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ( ♯ ‘ 𝑤 ) ∈ ℕ ) |
54 |
53
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑤 ) ∈ ℕ ) |
55 |
|
simprrl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ) |
56 |
54
|
nnred |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑤 ) ∈ ℝ ) |
57 |
|
2rp |
⊢ 2 ∈ ℝ+ |
58 |
|
ltsubrp |
⊢ ( ( ( ♯ ‘ 𝑤 ) ∈ ℝ ∧ 2 ∈ ℝ+ ) → ( ( ♯ ‘ 𝑤 ) − 2 ) < ( ♯ ‘ 𝑤 ) ) |
59 |
56 57 58
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ( ♯ ‘ 𝑤 ) − 2 ) < ( ♯ ‘ 𝑤 ) ) |
60 |
55 59
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑥 ) < ( ♯ ‘ 𝑤 ) ) |
61 |
|
elfzo0 |
⊢ ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) ↔ ( ( ♯ ‘ 𝑥 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑤 ) ∈ ℕ ∧ ( ♯ ‘ 𝑥 ) < ( ♯ ‘ 𝑤 ) ) ) |
62 |
49 54 60 61
|
syl3anbrc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) ) |
63 |
|
id |
⊢ ( ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) |
64 |
63
|
com13 |
⊢ ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) |
65 |
46 62 64
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) |
66 |
55
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = ( - 1 ↑ ( ( ♯ ‘ 𝑤 ) − 2 ) ) ) |
67 |
15
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → - 1 ∈ ℂ ) |
68 |
|
neg1ne0 |
⊢ - 1 ≠ 0 |
69 |
68
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → - 1 ≠ 0 ) |
70 |
|
2z |
⊢ 2 ∈ ℤ |
71 |
70
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → 2 ∈ ℤ ) |
72 |
54
|
nnzd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ♯ ‘ 𝑤 ) ∈ ℤ ) |
73 |
67 69 71 72
|
expsubd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( - 1 ↑ ( ( ♯ ‘ 𝑤 ) − 2 ) ) = ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) / ( - 1 ↑ 2 ) ) ) |
74 |
|
neg1sqe1 |
⊢ ( - 1 ↑ 2 ) = 1 |
75 |
74
|
oveq2i |
⊢ ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) / ( - 1 ↑ 2 ) ) = ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) / 1 ) |
76 |
|
m1expcl |
⊢ ( ( ♯ ‘ 𝑤 ) ∈ ℤ → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ∈ ℤ ) |
77 |
76
|
zcnd |
⊢ ( ( ♯ ‘ 𝑤 ) ∈ ℤ → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ∈ ℂ ) |
78 |
72 77
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ∈ ℂ ) |
79 |
78
|
div1d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) / 1 ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) |
80 |
75 79
|
eqtrid |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) / ( - 1 ↑ 2 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) |
81 |
66 73 80
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) |
82 |
81
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ↔ ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) |
83 |
65 82
|
sylibd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) ∧ ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) → ( ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) |
84 |
83
|
ex |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) |
85 |
84
|
com23 |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ( ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) |
86 |
85
|
alimdv |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ( ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ∀ 𝑥 ( ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) |
87 |
|
19.23v |
⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) |
88 |
86 87
|
syl6ib |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ( ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( ∃ 𝑥 ( 𝑥 ∈ Word 𝑇 ∧ ( ( ♯ ‘ 𝑥 ) = ( ( ♯ ‘ 𝑤 ) − 2 ) ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) |
89 |
43 88
|
mpid |
⊢ ( ( 𝜑 ∧ 𝑤 ≠ ∅ ∧ ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ) → ( ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) |
90 |
89
|
3exp |
⊢ ( 𝜑 → ( 𝑤 ≠ ∅ → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) ) |
91 |
90
|
com34 |
⊢ ( 𝜑 → ( 𝑤 ≠ ∅ → ( ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) ) |
92 |
91
|
com12 |
⊢ ( 𝑤 ≠ ∅ → ( 𝜑 → ( ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) ) |
93 |
92
|
impd |
⊢ ( 𝑤 ≠ ∅ → ( ( 𝜑 ∧ ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) ) |
94 |
19 93
|
pm2.61ine |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) |
95 |
94
|
3adant2 |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑤 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ∧ ∀ 𝑥 ( ( ♯ ‘ 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑤 ) ) → ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ) |
96 |
|
eleq1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇 ) ) |
97 |
|
oveq2 |
⊢ ( 𝑤 = 𝑥 → ( 𝐺 Σg 𝑤 ) = ( 𝐺 Σg 𝑥 ) ) |
98 |
97
|
eqeq1d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ↔ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) |
99 |
96 98
|
anbi12d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ↔ ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) ) ) |
100 |
|
fveq2 |
⊢ ( 𝑤 = 𝑥 → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑥 ) ) |
101 |
100
|
oveq2d |
⊢ ( 𝑤 = 𝑥 → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) ) |
102 |
101
|
eqeq1d |
⊢ ( 𝑤 = 𝑥 → ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ↔ ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) |
103 |
99 102
|
imbi12d |
⊢ ( 𝑤 = 𝑥 → ( ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ↔ ( ( 𝑥 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑥 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = 1 ) ) ) |
104 |
|
eleq1 |
⊢ ( 𝑤 = 𝑊 → ( 𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇 ) ) |
105 |
|
oveq2 |
⊢ ( 𝑤 = 𝑊 → ( 𝐺 Σg 𝑤 ) = ( 𝐺 Σg 𝑊 ) ) |
106 |
105
|
eqeq1d |
⊢ ( 𝑤 = 𝑊 → ( ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ↔ ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) ) ) |
107 |
104 106
|
anbi12d |
⊢ ( 𝑤 = 𝑊 → ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) ↔ ( 𝑊 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) ) ) ) |
108 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑊 ) ) |
109 |
108
|
oveq2d |
⊢ ( 𝑤 = 𝑊 → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) |
110 |
109
|
eqeq1d |
⊢ ( 𝑤 = 𝑊 → ( ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ↔ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = 1 ) ) |
111 |
107 110
|
imbi12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( 𝑤 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑤 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = 1 ) ↔ ( ( 𝑊 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = 1 ) ) ) |
112 |
4 10 95 103 111 100 108
|
uzindi |
⊢ ( 𝜑 → ( ( 𝑊 ∈ Word 𝑇 ∧ ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) ) → ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = 1 ) ) |
113 |
4 5 112
|
mp2and |
⊢ ( 𝜑 → ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = 1 ) |