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Theorem cycpmco2 30969
Description: The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.)
Hypotheses
Ref Expression
cycpmco2.c 𝑀 = (toCyc‘𝐷)
cycpmco2.s 𝑆 = (SymGrp‘𝐷)
cycpmco2.d (𝜑𝐷𝑉)
cycpmco2.w (𝜑𝑊 ∈ dom 𝑀)
cycpmco2.i (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))
cycpmco2.j (𝜑𝐽 ∈ ran 𝑊)
cycpmco2.e 𝐸 = ((𝑊𝐽) + 1)
cycpmco2.1 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
Assertion
Ref Expression
cycpmco2 (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))

Proof of Theorem cycpmco2
Dummy variables 𝑖 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cycpmco2.d . . . . . . 7 (𝜑𝐷𝑉)
2 cycpmco2.c . . . . . . . 8 𝑀 = (toCyc‘𝐷)
3 cycpmco2.s . . . . . . . 8 𝑆 = (SymGrp‘𝐷)
4 eqid 2738 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
52, 3, 4tocycf 30953 . . . . . . 7 (𝐷𝑉𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
61, 5syl 17 . . . . . 6 (𝜑𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
7 cycpmco2.w . . . . . . 7 (𝜑𝑊 ∈ dom 𝑀)
86fdmd 6509 . . . . . . 7 (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
97, 8eleqtrd 2835 . . . . . 6 (𝜑𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
106, 9ffvelrnd 6856 . . . . 5 (𝜑 → (𝑀𝑊) ∈ (Base‘𝑆))
113, 4symgbasf 18615 . . . . 5 ((𝑀𝑊) ∈ (Base‘𝑆) → (𝑀𝑊):𝐷𝐷)
1210, 11syl 17 . . . 4 (𝜑 → (𝑀𝑊):𝐷𝐷)
1312ffnd 6499 . . 3 (𝜑 → (𝑀𝑊) Fn 𝐷)
14 cycpmco2.i . . . . . . 7 (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))
1514eldifad 3853 . . . . . 6 (𝜑𝐼𝐷)
16 ssrab2 3967 . . . . . . . . . . 11 {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ⊆ Word 𝐷
1716, 9sseldi 3873 . . . . . . . . . 10 (𝜑𝑊 ∈ Word 𝐷)
18 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑊𝑤 = 𝑊)
19 dmeq 5740 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
20 eqidd 2739 . . . . . . . . . . . . 13 (𝑤 = 𝑊𝐷 = 𝐷)
2118, 19, 20f1eq123d 6604 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑤:dom 𝑤1-1𝐷𝑊:dom 𝑊1-1𝐷))
2221elrab3 3586 . . . . . . . . . . 11 (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ↔ 𝑊:dom 𝑊1-1𝐷))
2322biimpa 480 . . . . . . . . . 10 ((𝑊 ∈ Word 𝐷𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}) → 𝑊:dom 𝑊1-1𝐷)
2417, 9, 23syl2anc 587 . . . . . . . . 9 (𝜑𝑊:dom 𝑊1-1𝐷)
25 f1f 6568 . . . . . . . . 9 (𝑊:dom 𝑊1-1𝐷𝑊:dom 𝑊𝐷)
2624, 25syl 17 . . . . . . . 8 (𝜑𝑊:dom 𝑊𝐷)
2726frnd 6506 . . . . . . 7 (𝜑 → ran 𝑊𝐷)
28 cycpmco2.j . . . . . . 7 (𝜑𝐽 ∈ ran 𝑊)
2927, 28sseldd 3876 . . . . . 6 (𝜑𝐽𝐷)
3014eldifbd 3854 . . . . . . . 8 (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
31 nelne2 3031 . . . . . . . 8 ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽𝐼)
3228, 30, 31syl2anc 587 . . . . . . 7 (𝜑𝐽𝐼)
3332necomd 2989 . . . . . 6 (𝜑𝐼𝐽)
342, 1, 15, 29, 33, 3cycpm2cl 30956 . . . . 5 (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
353, 4symgbasf 18615 . . . . 5 ((𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝑀‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3634, 35syl 17 . . . 4 (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3736ffnd 6499 . . 3 (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷)
3836frnd 6506 . . 3 (𝜑 → ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷)
39 fnco 6447 . . 3 (((𝑀𝑊) Fn 𝐷 ∧ (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷 ∧ ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷) → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
4013, 37, 38, 39syl3anc 1372 . 2 (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
41 cycpmco2.1 . . . . . 6 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
4215s1cld 14039 . . . . . . 7 (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
43 splcl 14196 . . . . . . 7 ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
4417, 42, 43syl2anc 587 . . . . . 6 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
4541, 44eqeltrid 2837 . . . . 5 (𝜑𝑈 ∈ Word 𝐷)
46 cycpmco2.e . . . . . 6 𝐸 = ((𝑊𝐽) + 1)
472, 3, 1, 7, 14, 28, 46, 41cycpmco2f1 30960 . . . . 5 (𝜑𝑈:dom 𝑈1-1𝐷)
482, 1, 45, 47, 3cycpmcl 30952 . . . 4 (𝜑 → (𝑀𝑈) ∈ (Base‘𝑆))
493, 4symgbasf 18615 . . . 4 ((𝑀𝑈) ∈ (Base‘𝑆) → (𝑀𝑈):𝐷𝐷)
5048, 49syl 17 . . 3 (𝜑 → (𝑀𝑈):𝐷𝐷)
5150ffnd 6499 . 2 (𝜑 → (𝑀𝑈) Fn 𝐷)
52 fvco3 6761 . . . 4 (((𝑀‘⟨“𝐼𝐽”⟩):𝐷𝐷𝑖𝐷) → (((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
5336, 52sylan 583 . . 3 ((𝜑𝑖𝐷) → (((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
542, 1, 15, 29, 33, 3cyc2fv2 30958 . . . . . . . . . 10 (𝜑 → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
5554fveq2d 6672 . . . . . . . . 9 (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀𝑊)‘𝐼))
562, 3, 1, 7, 14, 28, 46, 41cycpmco2lem2 30963 . . . . . . . . . 10 (𝜑 → (𝑈𝐸) = 𝐼)
57 f1cnv 6635 . . . . . . . . . . . . . . . 16 (𝑊:dom 𝑊1-1𝐷𝑊:ran 𝑊1-1-onto→dom 𝑊)
58 f1of 6612 . . . . . . . . . . . . . . . 16 (𝑊:ran 𝑊1-1-onto→dom 𝑊𝑊:ran 𝑊⟶dom 𝑊)
5924, 57, 583syl 18 . . . . . . . . . . . . . . 15 (𝜑𝑊:ran 𝑊⟶dom 𝑊)
6059, 28ffvelrnd 6856 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝐽) ∈ dom 𝑊)
61 wrddm 13955 . . . . . . . . . . . . . . 15 (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
6217, 61syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
6360, 62eleqtrd 2835 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘𝑊)))
64 lencl 13967 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
6517, 64syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (♯‘𝑊) ∈ ℕ0)
6665nn0cnd 12031 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑊) ∈ ℂ)
67 1cnd 10707 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℂ)
68 ovexd 7199 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑊𝐽) + 1) ∈ V)
6946, 68eqeltrid 2837 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐸 ∈ V)
70 splval 14195 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
717, 69, 69, 42, 70syl13anc 1373 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
7241, 71syl5eq 2785 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
7372fveq2d 6672 . . . . . . . . . . . . . . . . . 18 (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
74 pfxcl 14121 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
7517, 74syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
76 ccatcl 14008 . . . . . . . . . . . . . . . . . . . 20 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
7775, 42, 76syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
78 swrdcl 14089 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
7917, 78syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
80 ccatlen 14009 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷) → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
8177, 79, 80syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
82 ccatws1len 14056 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
8317, 74, 823syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
84 fzofzp1 13218 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
8563, 84syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
8646, 85eqeltrid 2837 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐸 ∈ (0...(♯‘𝑊)))
87 pfxlen 14127 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
8817, 86, 87syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
8988oveq1d 7179 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
9083, 89eqtrd 2773 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
91 nn0fz0 13089 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
9265, 91sylib 221 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
93 swrdlen 14091 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
9417, 86, 92, 93syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
9590, 94oveq12d 7182 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
9673, 81, 953eqtrd 2777 . . . . . . . . . . . . . . . . 17 (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
97 fz0ssnn0 13086 . . . . . . . . . . . . . . . . . . . . . 22 (0...(♯‘𝑊)) ⊆ ℕ0
9897, 86sseldi 3873 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐸 ∈ ℕ0)
9998nn0zd 12159 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐸 ∈ ℤ)
10099peano2zd 12164 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐸 + 1) ∈ ℤ)
101100zcnd 12162 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸 + 1) ∈ ℂ)
10298nn0cnd 12031 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ ℂ)
103101, 66, 102addsubassd 11088 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
104102, 67, 66addassd 10734 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
105104oveq1d 7179 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
10696, 103, 1053eqtr2d 2779 . . . . . . . . . . . . . . . 16 (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
10767, 66addcld 10731 . . . . . . . . . . . . . . . . 17 (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
108102, 107pncan2d 11070 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
10967, 66addcomd 10913 . . . . . . . . . . . . . . . 16 (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
110106, 108, 1093eqtrd 2777 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
11166, 67, 110mvrraddd 11123 . . . . . . . . . . . . . 14 (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
112111oveq2d 7180 . . . . . . . . . . . . 13 (𝜑 → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
11363, 112eleqtrrd 2836 . . . . . . . . . . . 12 (𝜑 → (𝑊𝐽) ∈ (0..^((♯‘𝑈) − 1)))
1142, 1, 45, 47, 113cycpmfv1 30949 . . . . . . . . . . 11 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = (𝑈‘((𝑊𝐽) + 1)))
11546fveq2i 6671 . . . . . . . . . . 11 (𝑈𝐸) = (𝑈‘((𝑊𝐽) + 1))
116114, 115eqtr4di 2791 . . . . . . . . . 10 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = (𝑈𝐸))
1172, 1, 17, 24, 15, 30cycpmfv3 30951 . . . . . . . . . 10 (𝜑 → ((𝑀𝑊)‘𝐼) = 𝐼)
11856, 116, 1173eqtr4d 2783 . . . . . . . . 9 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = ((𝑀𝑊)‘𝐼))
11972fveq1d 6670 . . . . . . . . . . . 12 (𝜑 → (𝑈‘(𝑊𝐽)) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝑊𝐽)))
120 fzossfzop1 13199 . . . . . . . . . . . . . . . 16 (𝐸 ∈ ℕ0 → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
12198, 120syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
122 elfzonn0 13166 . . . . . . . . . . . . . . . . 17 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → (𝑊𝐽) ∈ ℕ0)
123 fzonn0p1 13198 . . . . . . . . . . . . . . . . 17 ((𝑊𝐽) ∈ ℕ0 → (𝑊𝐽) ∈ (0..^((𝑊𝐽) + 1)))
12463, 122, 1233syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑊𝐽) ∈ (0..^((𝑊𝐽) + 1)))
12546oveq2i 7175 . . . . . . . . . . . . . . . 16 (0..^𝐸) = (0..^((𝑊𝐽) + 1))
126124, 125eleqtrrdi 2844 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊𝐽) ∈ (0..^𝐸))
127121, 126sseldd 3876 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝐽) ∈ (0..^(𝐸 + 1)))
12890oveq2d 7180 . . . . . . . . . . . . . 14 (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
129127, 128eleqtrrd 2836 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
130 ccatval1 14012 . . . . . . . . . . . . 13 ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷 ∧ (𝑊𝐽) ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝑊𝐽)) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(𝑊𝐽)))
13177, 79, 129, 130syl3anc 1372 . . . . . . . . . . . 12 (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝑊𝐽)) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(𝑊𝐽)))
13288oveq2d 7180 . . . . . . . . . . . . . 14 (𝜑 → (0..^(♯‘(𝑊 prefix 𝐸))) = (0..^𝐸))
133126, 132eleqtrrd 2836 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘(𝑊 prefix 𝐸))))
134 ccatval1 14012 . . . . . . . . . . . . 13 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷 ∧ (𝑊𝐽) ∈ (0..^(♯‘(𝑊 prefix 𝐸)))) → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(𝑊𝐽)) = ((𝑊 prefix 𝐸)‘(𝑊𝐽)))
13575, 42, 133, 134syl3anc 1372 . . . . . . . . . . . 12 (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(𝑊𝐽)) = ((𝑊 prefix 𝐸)‘(𝑊𝐽)))
136119, 131, 1353eqtrd 2777 . . . . . . . . . . 11 (𝜑 → (𝑈‘(𝑊𝐽)) = ((𝑊 prefix 𝐸)‘(𝑊𝐽)))
137 pfxfv 14126 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (𝑊𝐽) ∈ (0..^𝐸)) → ((𝑊 prefix 𝐸)‘(𝑊𝐽)) = (𝑊‘(𝑊𝐽)))
13817, 86, 126, 137syl3anc 1372 . . . . . . . . . . 11 (𝜑 → ((𝑊 prefix 𝐸)‘(𝑊𝐽)) = (𝑊‘(𝑊𝐽)))
139 f1f1orn 6623 . . . . . . . . . . . . 13 (𝑊:dom 𝑊1-1𝐷𝑊:dom 𝑊1-1-onto→ran 𝑊)
14024, 139syl 17 . . . . . . . . . . . 12 (𝜑𝑊:dom 𝑊1-1-onto→ran 𝑊)
141 f1ocnvfv2 7039 . . . . . . . . . . . 12 ((𝑊:dom 𝑊1-1-onto→ran 𝑊𝐽 ∈ ran 𝑊) → (𝑊‘(𝑊𝐽)) = 𝐽)
142140, 28, 141syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝑊‘(𝑊𝐽)) = 𝐽)
143136, 138, 1423eqtrd 2777 . . . . . . . . . 10 (𝜑 → (𝑈‘(𝑊𝐽)) = 𝐽)
144143fveq2d 6672 . . . . . . . . 9 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = ((𝑀𝑈)‘𝐽))
14555, 118, 1443eqtr2d 2779 . . . . . . . 8 (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀𝑈)‘𝐽))
146145ad2antrr 726 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀𝑈)‘𝐽))
147 simpr 488 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → 𝑖 = 𝐽)
148147fveq2d 6672 . . . . . . . 8 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽))
149148fveq2d 6672 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)))
150147fveq2d 6672 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑈)‘𝐽))
151146, 149, 1503eqtr4d 2783 . . . . . 6 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
1521ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝐷𝑉)
15315, 29s2cld 14315 . . . . . . . . . 10 (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
154153ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
15515, 29, 33s2f1 30786 . . . . . . . . . 10 (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
156155ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
15727sselda 3875 . . . . . . . . . 10 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖𝐷)
158157adantr 484 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝑖𝐷)
159 simpr 488 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑊)
16030adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → ¬ 𝐼 ∈ ran 𝑊)
161 nelne2 3031 . . . . . . . . . . . . 13 ((𝑖 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝑖𝐼)
162159, 160, 161syl2anc 587 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖𝐼)
163162adantr 484 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝑖𝐼)
164 simpr 488 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝑖𝐽)
165163, 164nelprd 4544 . . . . . . . . . 10 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ¬ 𝑖 ∈ {𝐼, 𝐽})
16615, 29s2rn 30785 . . . . . . . . . . . . 13 (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
167166eleq2d 2818 . . . . . . . . . . . 12 (𝜑 → (𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ 𝑖 ∈ {𝐼, 𝐽}))
168167notbid 321 . . . . . . . . . . 11 (𝜑 → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
169168ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
170165, 169mpbird 260 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
1712, 152, 154, 156, 158, 170cycpmfv3 30951 . . . . . . . 8 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
172171fveq2d 6672 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘𝑖))
1731ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝐷𝑉)
1747ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝑊 ∈ dom 𝑀)
17514ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
17628ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝐽 ∈ ran 𝑊)
177 simpllr 776 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝑖 ∈ ran 𝑊)
178 simplr 769 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝑖𝐽)
179 simpr 488 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → (𝑈𝑖) ∈ (0..^𝐸))
1802, 3, 173, 174, 175, 176, 46, 41, 177, 178, 179cycpmco2lem7 30968 . . . . . . . 8 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
1811ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐷𝑉)
1827ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑊 ∈ dom 𝑀)
18314ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
18428ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐽 ∈ ran 𝑊)
185 simpllr 776 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖 ∈ ran 𝑊)
186162ad2antrr 726 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖𝐼)
187 simpr 488 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))
1882, 3, 181, 182, 183, 184, 46, 41, 185, 186, 187cycpmco2lem6 30967 . . . . . . . 8 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
1891ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝐷𝑉)
1907ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝑊 ∈ dom 𝑀)
19114ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
19228ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝐽 ∈ ran 𝑊)
193 simpllr 776 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝑖 ∈ ran 𝑊)
194 simpr 488 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → (𝑈𝑖) = ((♯‘𝑈) − 1))
1952, 3, 189, 190, 191, 192, 46, 41, 193, 194cycpmco2lem5 30966 . . . . . . . 8 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
196 f1f1orn 6623 . . . . . . . . . . . . . . . 16 (𝑈:dom 𝑈1-1𝐷𝑈:dom 𝑈1-1-onto→ran 𝑈)
19747, 196syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑈:dom 𝑈1-1-onto→ran 𝑈)
198 ssun1 4060 . . . . . . . . . . . . . . . . 17 ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
1992, 3, 1, 7, 14, 28, 46, 41cycpmco2rn 30961 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
200198, 199sseqtrrid 3928 . . . . . . . . . . . . . . . 16 (𝜑 → ran 𝑊 ⊆ ran 𝑈)
201200sselda 3875 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑈)
202 f1ocnvdm 7046 . . . . . . . . . . . . . . 15 ((𝑈:dom 𝑈1-1-onto→ran 𝑈𝑖 ∈ ran 𝑈) → (𝑈𝑖) ∈ dom 𝑈)
203197, 201, 202syl2an2r 685 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ran 𝑊) → (𝑈𝑖) ∈ dom 𝑈)
204 wrddm 13955 . . . . . . . . . . . . . . . 16 (𝑈 ∈ Word 𝐷 → dom 𝑈 = (0..^(♯‘𝑈)))
20545, 204syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
206205adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ran 𝑊) → dom 𝑈 = (0..^(♯‘𝑈)))
207203, 206eleqtrd 2835 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → (𝑈𝑖) ∈ (0..^(♯‘𝑈)))
20865nn0zd 12159 . . . . . . . . . . . . . . . . 17 (𝜑 → (♯‘𝑊) ∈ ℤ)
209208peano2zd 12164 . . . . . . . . . . . . . . . 16 (𝜑 → ((♯‘𝑊) + 1) ∈ ℤ)
210110, 209eqeltrd 2833 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑈) ∈ ℤ)
211 fzoval 13123 . . . . . . . . . . . . . . 15 ((♯‘𝑈) ∈ ℤ → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
212210, 211syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
213212adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
214207, 213eleqtrd 2835 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ran 𝑊) → (𝑈𝑖) ∈ (0...((♯‘𝑈) − 1)))
215 elfzr 13234 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ (0...((♯‘𝑈) − 1)) → ((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
216214, 215syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
217 simpr 488 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ ran 𝑊) ∧ (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1))) → (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)))
21899ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ ran 𝑊) ∧ (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1))) → 𝐸 ∈ ℤ)
219 fzospliti 13153 . . . . . . . . . . . . . 14 (((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∧ 𝐸 ∈ ℤ) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
220217, 218, 219syl2anc 587 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ ran 𝑊) ∧ (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1))) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
221220ex 416 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))))
222221orim1d 965 . . . . . . . . . . 11 ((𝜑𝑖 ∈ ran 𝑊) → (((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)) → (((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1))))
223216, 222mpd 15 . . . . . . . . . 10 ((𝜑𝑖 ∈ ran 𝑊) → (((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
224 df-3or 1089 . . . . . . . . . 10 (((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)) ↔ (((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
225223, 224sylibr 237 . . . . . . . . 9 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
226225adantr 484 . . . . . . . 8 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
227180, 188, 195, 226mpjao3dan 1432 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
228172, 227eqtr4d 2776 . . . . . 6 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
229151, 228pm2.61dane 3021 . . . . 5 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
230229adantlr 715 . . . 4 (((𝜑𝑖𝐷) ∧ 𝑖 ∈ ran 𝑊) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
2312, 3, 1, 7, 14, 28, 46, 41cycpmco2lem4 30965 . . . . . . . 8 (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
232231ad2antrr 726 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
233 simpr 488 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
234233fveq2d 6672 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼))
235234fveq2d 6672 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)))
236233fveq2d 6672 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑈)‘𝐼))
237232, 235, 2363eqtr4d 2783 . . . . . 6 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
2381ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝐷𝑉)
23917ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑊 ∈ Word 𝐷)
24024ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑊:dom 𝑊1-1𝐷)
241 simplr 769 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖 ∈ (𝐷 ∖ ran 𝑊))
242241eldifad 3853 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖𝐷)
243241eldifbd 3854 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ ran 𝑊)
2442, 238, 239, 240, 242, 243cycpmfv3 30951 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑊)‘𝑖) = 𝑖)
245153ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
246155ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
247 simpr 488 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖𝐼)
248 eldifn 4016 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝐷 ∖ ran 𝑊) → ¬ 𝑖 ∈ ran 𝑊)
249 nelne2 3031 . . . . . . . . . . . . . 14 ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ ran 𝑊) → 𝐽𝑖)
25028, 248, 249syl2an 599 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝐽𝑖)
251250necomd 2989 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝑖𝐽)
252251adantr 484 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖𝐽)
253247, 252nelprd 4544 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ {𝐼, 𝐽})
254168ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
255253, 254mpbird 260 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
2562, 238, 245, 246, 242, 255cycpmfv3 30951 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
257256fveq2d 6672 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘𝑖))
25845ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑈 ∈ Word 𝐷)
25947ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑈:dom 𝑈1-1𝐷)
260199ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
261 nelsn 4553 . . . . . . . . . 10 (𝑖𝐼 → ¬ 𝑖 ∈ {𝐼})
262261adantl 485 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ {𝐼})
263 nelun 30425 . . . . . . . . . 10 (ran 𝑈 = (ran 𝑊 ∪ {𝐼}) → (¬ 𝑖 ∈ ran 𝑈 ↔ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})))
264263biimpar 481 . . . . . . . . 9 ((ran 𝑈 = (ran 𝑊 ∪ {𝐼}) ∧ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})) → ¬ 𝑖 ∈ ran 𝑈)
265260, 243, 262, 264syl12anc 836 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ ran 𝑈)
2662, 238, 258, 259, 242, 265cycpmfv3 30951 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑈)‘𝑖) = 𝑖)
267244, 257, 2663eqtr4d 2783 . . . . . 6 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
268237, 267pm2.61dane 3021 . . . . 5 ((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
269268adantlr 715 . . . 4 (((𝜑𝑖𝐷) ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
270 undif 4368 . . . . . . . 8 (ran 𝑊𝐷 ↔ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
27127, 270sylib 221 . . . . . . 7 (𝜑 → (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
272271eleq2d 2818 . . . . . 6 (𝜑 → (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ 𝑖𝐷))
273 elun 4037 . . . . . 6 (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ (𝑖 ∈ ran 𝑊𝑖 ∈ (𝐷 ∖ ran 𝑊)))
274272, 273bitr3di 289 . . . . 5 (𝜑 → (𝑖𝐷 ↔ (𝑖 ∈ ran 𝑊𝑖 ∈ (𝐷 ∖ ran 𝑊))))
275274biimpa 480 . . . 4 ((𝜑𝑖𝐷) → (𝑖 ∈ ran 𝑊𝑖 ∈ (𝐷 ∖ ran 𝑊)))
276230, 269, 275mpjaodan 958 . . 3 ((𝜑𝑖𝐷) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
27753, 276eqtrd 2773 . 2 ((𝜑𝑖𝐷) → (((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀𝑈)‘𝑖))
27840, 51, 277eqfnfvd 6806 1 (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3o 1087   = wceq 1542  wcel 2113  wne 2934  {crab 3057  Vcvv 3397  cdif 3838  cun 3839  wss 3841  {csn 4513  {cpr 4515  cop 4519  cotp 4521  ccnv 5518  dom cdm 5519  ran crn 5520  ccom 5523   Fn wfn 6328  wf 6329  1-1wf1 6330  1-1-ontowf1o 6332  cfv 6333  (class class class)co 7164  0cc0 10608  1c1 10609   + caddc 10611  cmin 10941  0cn0 11969  cz 12055  ...cfz 12974  ..^cfzo 13117  chash 13775  Word cword 13948   ++ cconcat 14004  ⟨“cs1 14031   substr csubstr 14084   prefix cpfx 14114   splice csplice 14193  ⟨“cs2 14285  Basecbs 16579  SymGrpcsymg 18606  toCycctocyc 30942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685  ax-pre-sup 10686
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-ot 4522  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-oadd 8128  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-sup 8972  df-inf 8973  df-card 9434  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-div 11369  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-7 11777  df-8 11778  df-9 11779  df-n0 11970  df-xnn0 12042  df-z 12056  df-uz 12318  df-rp 12466  df-fz 12975  df-fzo 13118  df-fl 13246  df-mod 13322  df-hash 13776  df-word 13949  df-concat 14005  df-s1 14032  df-substr 14085  df-pfx 14115  df-splice 14194  df-csh 14233  df-s2 14292  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-plusg 16674  df-tset 16680  df-efmnd 18143  df-symg 18607  df-tocyc 30943
This theorem is referenced by: (None)
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