Metamath Proof Explorer


Theorem psgnunilem5

Description: Lemma for psgnuni . It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving A in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015) (Revised by Mario Carneiro, 28-Feb-2016) (Proof shortened by AV, 12-Oct-2022)

Ref Expression
Hypotheses psgnunilem2.g 𝐺 = ( SymGrp ‘ 𝐷 )
psgnunilem2.t 𝑇 = ran ( pmTrsp ‘ 𝐷 )
psgnunilem2.d ( 𝜑𝐷𝑉 )
psgnunilem2.w ( 𝜑𝑊 ∈ Word 𝑇 )
psgnunilem2.id ( 𝜑 → ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) )
psgnunilem2.l ( 𝜑 → ( ♯ ‘ 𝑊 ) = 𝐿 )
psgnunilem2.ix ( 𝜑𝐼 ∈ ( 0 ..^ 𝐿 ) )
psgnunilem2.a ( 𝜑𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) )
psgnunilem2.al ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊𝑘 ) ∖ I ) )
Assertion psgnunilem5 ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) )

Proof

Step Hyp Ref Expression
1 psgnunilem2.g 𝐺 = ( SymGrp ‘ 𝐷 )
2 psgnunilem2.t 𝑇 = ran ( pmTrsp ‘ 𝐷 )
3 psgnunilem2.d ( 𝜑𝐷𝑉 )
4 psgnunilem2.w ( 𝜑𝑊 ∈ Word 𝑇 )
5 psgnunilem2.id ( 𝜑 → ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) )
6 psgnunilem2.l ( 𝜑 → ( ♯ ‘ 𝑊 ) = 𝐿 )
7 psgnunilem2.ix ( 𝜑𝐼 ∈ ( 0 ..^ 𝐿 ) )
8 psgnunilem2.a ( 𝜑𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) )
9 psgnunilem2.al ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊𝑘 ) ∖ I ) )
10 noel ¬ 𝐴 ∈ ∅
11 5 difeq1d ( 𝜑 → ( ( 𝐺 Σg 𝑊 ) ∖ I ) = ( ( I ↾ 𝐷 ) ∖ I ) )
12 11 dmeqd ( 𝜑 → dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) = dom ( ( I ↾ 𝐷 ) ∖ I ) )
13 resss ( I ↾ 𝐷 ) ⊆ I
14 ssdif0 ( ( I ↾ 𝐷 ) ⊆ I ↔ ( ( I ↾ 𝐷 ) ∖ I ) = ∅ )
15 13 14 mpbi ( ( I ↾ 𝐷 ) ∖ I ) = ∅
16 15 dmeqi dom ( ( I ↾ 𝐷 ) ∖ I ) = dom ∅
17 dm0 dom ∅ = ∅
18 16 17 eqtri dom ( ( I ↾ 𝐷 ) ∖ I ) = ∅
19 12 18 eqtrdi ( 𝜑 → dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) = ∅ )
20 19 eleq2d ( 𝜑 → ( 𝐴 ∈ dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) ↔ 𝐴 ∈ ∅ ) )
21 10 20 mtbiri ( 𝜑 → ¬ 𝐴 ∈ dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) )
22 1 symggrp ( 𝐷𝑉𝐺 ∈ Grp )
23 grpmnd ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
24 3 22 23 3syl ( 𝜑𝐺 ∈ Mnd )
25 eqid ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
26 2 1 25 symgtrf 𝑇 ⊆ ( Base ‘ 𝐺 )
27 sswrd ( 𝑇 ⊆ ( Base ‘ 𝐺 ) → Word 𝑇 ⊆ Word ( Base ‘ 𝐺 ) )
28 26 27 mp1i ( 𝜑 → Word 𝑇 ⊆ Word ( Base ‘ 𝐺 ) )
29 28 4 sseldd ( 𝜑𝑊 ∈ Word ( Base ‘ 𝐺 ) )
30 pfxcl ( 𝑊 ∈ Word ( Base ‘ 𝐺 ) → ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) )
31 29 30 syl ( 𝜑 → ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) )
32 25 gsumwcl ( ( 𝐺 ∈ Mnd ∧ ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) )
33 24 31 32 syl2anc ( 𝜑 → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) )
34 1 25 symgbasf1o ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷1-1-onto𝐷 )
35 33 34 syl ( 𝜑 → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷1-1-onto𝐷 )
36 35 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷1-1-onto𝐷 )
37 wrdf ( 𝑊 ∈ Word 𝑇𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑇 )
38 4 37 syl ( 𝜑𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑇 )
39 6 oveq2d ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 𝐿 ) )
40 7 39 eleqtrrd ( 𝜑𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
41 38 40 ffvelrnd ( 𝜑 → ( 𝑊𝐼 ) ∈ 𝑇 )
42 26 41 sselid ( 𝜑 → ( 𝑊𝐼 ) ∈ ( Base ‘ 𝐺 ) )
43 1 25 symgbasf1o ( ( 𝑊𝐼 ) ∈ ( Base ‘ 𝐺 ) → ( 𝑊𝐼 ) : 𝐷1-1-onto𝐷 )
44 42 43 syl ( 𝜑 → ( 𝑊𝐼 ) : 𝐷1-1-onto𝐷 )
45 44 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝑊𝐼 ) : 𝐷1-1-onto𝐷 )
46 1 25 symgsssg ( 𝐷𝑉 → { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubGrp ‘ 𝐺 ) )
47 subgsubm ( { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubGrp ‘ 𝐺 ) → { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubMnd ‘ 𝐺 ) )
48 3 46 47 3syl ( 𝜑 → { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubMnd ‘ 𝐺 ) )
49 fzossfz ( 0 ..^ 𝐿 ) ⊆ ( 0 ... 𝐿 )
50 49 7 sselid ( 𝜑𝐼 ∈ ( 0 ... 𝐿 ) )
51 6 oveq2d ( 𝜑 → ( 0 ... ( ♯ ‘ 𝑊 ) ) = ( 0 ... 𝐿 ) )
52 50 51 eleqtrrd ( 𝜑𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
53 pfxmpt ( ( 𝑊 ∈ Word 𝑇𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 prefix 𝐼 ) = ( 𝑠 ∈ ( 0 ..^ 𝐼 ) ↦ ( 𝑊𝑠 ) ) )
54 4 52 53 syl2anc ( 𝜑 → ( 𝑊 prefix 𝐼 ) = ( 𝑠 ∈ ( 0 ..^ 𝐼 ) ↦ ( 𝑊𝑠 ) ) )
55 difeq1 ( 𝑗 = ( 𝑊𝑠 ) → ( 𝑗 ∖ I ) = ( ( 𝑊𝑠 ) ∖ I ) )
56 55 dmeqd ( 𝑗 = ( 𝑊𝑠 ) → dom ( 𝑗 ∖ I ) = dom ( ( 𝑊𝑠 ) ∖ I ) )
57 56 sseq1d ( 𝑗 = ( 𝑊𝑠 ) → ( dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ dom ( ( 𝑊𝑠 ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) )
58 disj2 ( ( dom ( ( 𝑊𝑠 ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ dom ( ( 𝑊𝑠 ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) )
59 disjsn ( ( dom ( ( 𝑊𝑠 ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) )
60 58 59 bitr3i ( dom ( ( 𝑊𝑠 ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) )
61 57 60 bitrdi ( 𝑗 = ( 𝑊𝑠 ) → ( dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) ) )
62 elfzuz3 ( 𝐼 ∈ ( 0 ... 𝐿 ) → 𝐿 ∈ ( ℤ𝐼 ) )
63 50 62 syl ( 𝜑𝐿 ∈ ( ℤ𝐼 ) )
64 6 63 eqeltrd ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ( ℤ𝐼 ) )
65 fzoss2 ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ𝐼 ) → ( 0 ..^ 𝐼 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
66 64 65 syl ( 𝜑 → ( 0 ..^ 𝐼 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
67 66 sselda ( ( 𝜑𝑠 ∈ ( 0 ..^ 𝐼 ) ) → 𝑠 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
68 38 ffvelrnda ( ( 𝜑𝑠 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊𝑠 ) ∈ 𝑇 )
69 26 68 sselid ( ( 𝜑𝑠 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊𝑠 ) ∈ ( Base ‘ 𝐺 ) )
70 67 69 syldan ( ( 𝜑𝑠 ∈ ( 0 ..^ 𝐼 ) ) → ( 𝑊𝑠 ) ∈ ( Base ‘ 𝐺 ) )
71 fveq2 ( 𝑘 = 𝑠 → ( 𝑊𝑘 ) = ( 𝑊𝑠 ) )
72 71 difeq1d ( 𝑘 = 𝑠 → ( ( 𝑊𝑘 ) ∖ I ) = ( ( 𝑊𝑠 ) ∖ I ) )
73 72 dmeqd ( 𝑘 = 𝑠 → dom ( ( 𝑊𝑘 ) ∖ I ) = dom ( ( 𝑊𝑠 ) ∖ I ) )
74 73 eleq2d ( 𝑘 = 𝑠 → ( 𝐴 ∈ dom ( ( 𝑊𝑘 ) ∖ I ) ↔ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) ) )
75 74 notbid ( 𝑘 = 𝑠 → ( ¬ 𝐴 ∈ dom ( ( 𝑊𝑘 ) ∖ I ) ↔ ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) ) )
76 75 cbvralvw ( ∀ 𝑘 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊𝑘 ) ∖ I ) ↔ ∀ 𝑠 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) )
77 9 76 sylib ( 𝜑 → ∀ 𝑠 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) )
78 77 r19.21bi ( ( 𝜑𝑠 ∈ ( 0 ..^ 𝐼 ) ) → ¬ 𝐴 ∈ dom ( ( 𝑊𝑠 ) ∖ I ) )
79 61 70 78 elrabd ( ( 𝜑𝑠 ∈ ( 0 ..^ 𝐼 ) ) → ( 𝑊𝑠 ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
80 54 79 fmpt3d ( 𝜑 → ( 𝑊 prefix 𝐼 ) : ( 0 ..^ 𝐼 ) ⟶ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
81 80 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝑊 prefix 𝐼 ) : ( 0 ..^ 𝐼 ) ⟶ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
82 iswrdi ( ( 𝑊 prefix 𝐼 ) : ( 0 ..^ 𝐼 ) ⟶ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } → ( 𝑊 prefix 𝐼 ) ∈ Word { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
83 81 82 syl ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝑊 prefix 𝐼 ) ∈ Word { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
84 gsumwsubmcl ( ( { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubMnd ‘ 𝐺 ) ∧ ( 𝑊 prefix 𝐼 ) ∈ Word { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
85 48 83 84 syl2an2r ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } )
86 difeq1 ( 𝑗 = ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) → ( 𝑗 ∖ I ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) )
87 86 dmeqd ( 𝑗 = ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) → dom ( 𝑗 ∖ I ) = dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) )
88 87 sseq1d ( 𝑗 = ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) → ( dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) )
89 88 elrab ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ↔ ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) ∧ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) )
90 89 simprbi ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } → dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) )
91 disj2 ( ( dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) )
92 disjsn ( ( dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) )
93 91 92 bitr3i ( dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) )
94 90 93 sylib ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } → ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) )
95 85 94 syl ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) )
96 8 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) )
97 95 96 jca ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∧ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) )
98 97 olcd ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∧ ¬ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) ∨ ( ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∧ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) ) )
99 excxor ( ( 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊻ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) ↔ ( ( 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∧ ¬ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) ∨ ( ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∧ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) ) )
100 98 99 sylibr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊻ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) )
101 f1omvdco3 ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷1-1-onto𝐷 ∧ ( 𝑊𝐼 ) : 𝐷1-1-onto𝐷 ∧ ( 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊻ 𝐴 ∈ dom ( ( 𝑊𝐼 ) ∖ I ) ) ) → 𝐴 ∈ dom ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) ∖ I ) )
102 36 45 100 101 syl3anc ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝐴 ∈ dom ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) ∖ I ) )
103 elfzo0 ( 𝐼 ∈ ( 0 ..^ 𝐿 ) ↔ ( 𝐼 ∈ ℕ0𝐿 ∈ ℕ ∧ 𝐼 < 𝐿 ) )
104 103 simp2bi ( 𝐼 ∈ ( 0 ..^ 𝐿 ) → 𝐿 ∈ ℕ )
105 7 104 syl ( 𝜑𝐿 ∈ ℕ )
106 6 105 eqeltrd ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ℕ )
107 wrdfin ( 𝑊 ∈ Word 𝑇𝑊 ∈ Fin )
108 hashnncl ( 𝑊 ∈ Fin → ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ 𝑊 ≠ ∅ ) )
109 4 107 108 3syl ( 𝜑 → ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ 𝑊 ≠ ∅ ) )
110 106 109 mpbid ( 𝜑𝑊 ≠ ∅ )
111 110 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝑊 ≠ ∅ )
112 pfxlswccat ( ( 𝑊 ∈ Word 𝑇𝑊 ≠ ∅ ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ ⟨“ ( lastS ‘ 𝑊 ) ”⟩ ) = 𝑊 )
113 112 eqcomd ( ( 𝑊 ∈ Word 𝑇𝑊 ≠ ∅ ) → 𝑊 = ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ ⟨“ ( lastS ‘ 𝑊 ) ”⟩ ) )
114 4 111 113 syl2an2r ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝑊 = ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ ⟨“ ( lastS ‘ 𝑊 ) ”⟩ ) )
115 6 oveq1d ( 𝜑 → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( 𝐿 − 1 ) )
116 115 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( 𝐿 − 1 ) )
117 105 nncnd ( 𝜑𝐿 ∈ ℂ )
118 1cnd ( 𝜑 → 1 ∈ ℂ )
119 elfzoelz ( 𝐼 ∈ ( 0 ..^ 𝐿 ) → 𝐼 ∈ ℤ )
120 7 119 syl ( 𝜑𝐼 ∈ ℤ )
121 120 zcnd ( 𝜑𝐼 ∈ ℂ )
122 117 118 121 subadd2d ( 𝜑 → ( ( 𝐿 − 1 ) = 𝐼 ↔ ( 𝐼 + 1 ) = 𝐿 ) )
123 122 biimpar ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐿 − 1 ) = 𝐼 )
124 116 123 eqtrd ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 )
125 oveq2 ( ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 → ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 prefix 𝐼 ) )
126 125 adantl ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 prefix 𝐼 ) )
127 lsw ( 𝑊 ∈ Word 𝑇 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
128 4 127 syl ( 𝜑 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
129 fveq2 ( ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊𝐼 ) )
130 128 129 sylan9eq ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ( lastS ‘ 𝑊 ) = ( 𝑊𝐼 ) )
131 130 s1eqd ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ⟨“ ( lastS ‘ 𝑊 ) ”⟩ = ⟨“ ( 𝑊𝐼 ) ”⟩ )
132 126 131 oveq12d ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ ⟨“ ( lastS ‘ 𝑊 ) ”⟩ ) = ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) )
133 124 132 syldan ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ ⟨“ ( lastS ‘ 𝑊 ) ”⟩ ) = ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) )
134 114 133 eqtrd ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝑊 = ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) )
135 134 oveq2d ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) )
136 42 s1cld ( 𝜑 → ⟨“ ( 𝑊𝐼 ) ”⟩ ∈ Word ( Base ‘ 𝐺 ) )
137 eqid ( +g𝐺 ) = ( +g𝐺 )
138 25 137 gsumccat ( ( 𝐺 ∈ Mnd ∧ ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) ∧ ⟨“ ( 𝑊𝐼 ) ”⟩ ∈ Word ( Base ‘ 𝐺 ) ) → ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) )
139 24 31 136 138 syl3anc ( 𝜑 → ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) )
140 139 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) )
141 25 gsumws1 ( ( 𝑊𝐼 ) ∈ ( Base ‘ 𝐺 ) → ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) = ( 𝑊𝐼 ) )
142 42 141 syl ( 𝜑 → ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) = ( 𝑊𝐼 ) )
143 142 oveq2d ( 𝜑 → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝑊𝐼 ) ) )
144 1 25 137 symgov ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑊𝐼 ) ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝑊𝐼 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) )
145 33 42 144 syl2anc ( 𝜑 → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝑊𝐼 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) )
146 143 145 eqtrd ( 𝜑 → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) )
147 146 adantr ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g𝐺 ) ( 𝐺 Σg ⟨“ ( 𝑊𝐼 ) ”⟩ ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) )
148 135 140 147 3eqtrd ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg 𝑊 ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) )
149 148 difeq1d ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( 𝐺 Σg 𝑊 ) ∖ I ) = ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) ∖ I ) )
150 149 dmeqd ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) = dom ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊𝐼 ) ) ∖ I ) )
151 102 150 eleqtrrd ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝐴 ∈ dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) )
152 21 151 mtand ( 𝜑 → ¬ ( 𝐼 + 1 ) = 𝐿 )
153 fzostep1 ( 𝐼 ∈ ( 0 ..^ 𝐿 ) → ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) ∨ ( 𝐼 + 1 ) = 𝐿 ) )
154 7 153 syl ( 𝜑 → ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) ∨ ( 𝐼 + 1 ) = 𝐿 ) )
155 154 ord ( 𝜑 → ( ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) → ( 𝐼 + 1 ) = 𝐿 ) )
156 152 155 mt3d ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) )