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Theorem cycpmco2 30777
 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 2823 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
52, 3, 4tocycf 30761 . . . . . . 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 6525 . . . . . . 7 (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
97, 8eleqtrd 2917 . . . . . 6 (𝜑𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
106, 9ffvelrnd 6854 . . . . 5 (𝜑 → (𝑀𝑊) ∈ (Base‘𝑆))
113, 4symgbasf 18506 . . . . 5 ((𝑀𝑊) ∈ (Base‘𝑆) → (𝑀𝑊):𝐷𝐷)
1210, 11syl 17 . . . 4 (𝜑 → (𝑀𝑊):𝐷𝐷)
1312ffnd 6517 . . 3 (𝜑 → (𝑀𝑊) Fn 𝐷)
14 cycpmco2.i . . . . . . 7 (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))
1514eldifad 3950 . . . . . 6 (𝜑𝐼𝐷)
16 ssrab2 4058 . . . . . . . . . . 11 {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ⊆ Word 𝐷
1716, 9sseldi 3967 . . . . . . . . . 10 (𝜑𝑊 ∈ Word 𝐷)
18 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑊𝑤 = 𝑊)
19 dmeq 5774 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
20 eqidd 2824 . . . . . . . . . . . . 13 (𝑤 = 𝑊𝐷 = 𝐷)
2118, 19, 20f1eq123d 6610 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑤:dom 𝑤1-1𝐷𝑊:dom 𝑊1-1𝐷))
2221elrab3 3683 . . . . . . . . . . 11 (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ↔ 𝑊:dom 𝑊1-1𝐷))
2322biimpa 479 . . . . . . . . . 10 ((𝑊 ∈ Word 𝐷𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}) → 𝑊:dom 𝑊1-1𝐷)
2417, 9, 23syl2anc 586 . . . . . . . . 9 (𝜑𝑊:dom 𝑊1-1𝐷)
25 f1f 6577 . . . . . . . . 9 (𝑊:dom 𝑊1-1𝐷𝑊:dom 𝑊𝐷)
2624, 25syl 17 . . . . . . . 8 (𝜑𝑊:dom 𝑊𝐷)
2726frnd 6523 . . . . . . 7 (𝜑 → ran 𝑊𝐷)
28 cycpmco2.j . . . . . . 7 (𝜑𝐽 ∈ ran 𝑊)
2927, 28sseldd 3970 . . . . . 6 (𝜑𝐽𝐷)
3014eldifbd 3951 . . . . . . . 8 (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
31 nelne2 3117 . . . . . . . 8 ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽𝐼)
3228, 30, 31syl2anc 586 . . . . . . 7 (𝜑𝐽𝐼)
3332necomd 3073 . . . . . 6 (𝜑𝐼𝐽)
342, 1, 15, 29, 33, 3cycpm2cl 30764 . . . . 5 (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
353, 4symgbasf 18506 . . . . 5 ((𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝑀‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3634, 35syl 17 . . . 4 (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3736ffnd 6517 . . 3 (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷)
3836frnd 6523 . . 3 (𝜑 → ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷)
39 fnco 6467 . . 3 (((𝑀𝑊) Fn 𝐷 ∧ (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷 ∧ ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷) → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
4013, 37, 38, 39syl3anc 1367 . 2 (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
41 cycpmco2.1 . . . . . 6 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
4215s1cld 13959 . . . . . . 7 (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
43 splcl 14116 . . . . . . 7 ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
4417, 42, 43syl2anc 586 . . . . . 6 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
4541, 44eqeltrid 2919 . . . . 5 (𝜑𝑈 ∈ Word 𝐷)
46 cycpmco2.e . . . . . 6 𝐸 = ((𝑊𝐽) + 1)
472, 3, 1, 7, 14, 28, 46, 41cycpmco2f1 30768 . . . . 5 (𝜑𝑈:dom 𝑈1-1𝐷)
482, 1, 45, 47, 3cycpmcl 30760 . . . 4 (𝜑 → (𝑀𝑈) ∈ (Base‘𝑆))
493, 4symgbasf 18506 . . . 4 ((𝑀𝑈) ∈ (Base‘𝑆) → (𝑀𝑈):𝐷𝐷)
5048, 49syl 17 . . 3 (𝜑 → (𝑀𝑈):𝐷𝐷)
5150ffnd 6517 . 2 (𝜑 → (𝑀𝑈) Fn 𝐷)
52 fvco3 6762 . . . 4 (((𝑀‘⟨“𝐼𝐽”⟩):𝐷𝐷𝑖𝐷) → (((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
5336, 52sylan 582 . . 3 ((𝜑𝑖𝐷) → (((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
542, 1, 15, 29, 33, 3cyc2fv2 30766 . . . . . . . . . 10 (𝜑 → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
5554fveq2d 6676 . . . . . . . . 9 (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀𝑊)‘𝐼))
562, 3, 1, 7, 14, 28, 46, 41cycpmco2lem2 30771 . . . . . . . . . 10 (𝜑 → (𝑈𝐸) = 𝐼)
57 f1cnv 6640 . . . . . . . . . . . . . . . 16 (𝑊:dom 𝑊1-1𝐷𝑊:ran 𝑊1-1-onto→dom 𝑊)
58 f1of 6617 . . . . . . . . . . . . . . . 16 (𝑊:ran 𝑊1-1-onto→dom 𝑊𝑊:ran 𝑊⟶dom 𝑊)
5924, 57, 583syl 18 . . . . . . . . . . . . . . 15 (𝜑𝑊:ran 𝑊⟶dom 𝑊)
6059, 28ffvelrnd 6854 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝐽) ∈ dom 𝑊)
61 wrddm 13871 . . . . . . . . . . . . . . 15 (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
6217, 61syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
6360, 62eleqtrd 2917 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘𝑊)))
64 lencl 13885 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
6517, 64syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (♯‘𝑊) ∈ ℕ0)
6665nn0cnd 11960 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑊) ∈ ℂ)
67 1cnd 10638 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℂ)
68 ovexd 7193 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑊𝐽) + 1) ∈ V)
6946, 68eqeltrid 2919 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐸 ∈ V)
70 splval 14115 . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
7241, 71syl5eq 2870 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
7372fveq2d 6676 . . . . . . . . . . . . . . . . . 18 (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
74 pfxcl 14041 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
7517, 74syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
76 ccatcl 13928 . . . . . . . . . . . . . . . . . . . 20 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
7775, 42, 76syl2anc 586 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
78 swrdcl 14009 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
7917, 78syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
80 ccatlen 13929 . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . 18 (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
82 ccatws1len 13976 . . . . . . . . . . . . . . . . . . . . 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 13137 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
8563, 84syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
8646, 85eqeltrid 2919 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐸 ∈ (0...(♯‘𝑊)))
87 pfxlen 14047 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
8817, 86, 87syl2anc 586 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
8988oveq1d 7173 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
9083, 89eqtrd 2858 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
91 nn0fz0 13008 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
9265, 91sylib 220 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
93 swrdlen 14011 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
9417, 86, 92, 93syl3anc 1367 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
9590, 94oveq12d 7176 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
9673, 81, 953eqtrd 2862 . . . . . . . . . . . . . . . . 17 (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
97 fz0ssnn0 13005 . . . . . . . . . . . . . . . . . . . . . 22 (0...(♯‘𝑊)) ⊆ ℕ0
9897, 86sseldi 3967 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐸 ∈ ℕ0)
9998nn0zd 12088 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐸 ∈ ℤ)
10099peano2zd 12093 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐸 + 1) ∈ ℤ)
101100zcnd 12091 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸 + 1) ∈ ℂ)
10298nn0cnd 11960 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ ℂ)
103101, 66, 102addsubassd 11019 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
104102, 67, 66addassd 10665 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
105104oveq1d 7173 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
10696, 103, 1053eqtr2d 2864 . . . . . . . . . . . . . . . 16 (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
10767, 66addcld 10662 . . . . . . . . . . . . . . . . 17 (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
108102, 107pncan2d 11001 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
10967, 66addcomd 10844 . . . . . . . . . . . . . . . 16 (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
110106, 108, 1093eqtrd 2862 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
11166, 67, 110mvrraddd 11054 . . . . . . . . . . . . . 14 (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
112111oveq2d 7174 . . . . . . . . . . . . 13 (𝜑 → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
11363, 112eleqtrrd 2918 . . . . . . . . . . . 12 (𝜑 → (𝑊𝐽) ∈ (0..^((♯‘𝑈) − 1)))
1142, 1, 45, 47, 113cycpmfv1 30757 . . . . . . . . . . 11 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = (𝑈‘((𝑊𝐽) + 1)))
11546fveq2i 6675 . . . . . . . . . . 11 (𝑈𝐸) = (𝑈‘((𝑊𝐽) + 1))
116114, 115syl6eqr 2876 . . . . . . . . . 10 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = (𝑈𝐸))
1172, 1, 17, 24, 15, 30cycpmfv3 30759 . . . . . . . . . 10 (𝜑 → ((𝑀𝑊)‘𝐼) = 𝐼)
11856, 116, 1173eqtr4d 2868 . . . . . . . . 9 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = ((𝑀𝑊)‘𝐼))
11972fveq1d 6674 . . . . . . . . . . . 12 (𝜑 → (𝑈‘(𝑊𝐽)) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝑊𝐽)))
120 fzossfzop1 13118 . . . . . . . . . . . . . . . 16 (𝐸 ∈ ℕ0 → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
12198, 120syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
122 elfzonn0 13085 . . . . . . . . . . . . . . . . 17 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → (𝑊𝐽) ∈ ℕ0)
123 fzonn0p1 13117 . . . . . . . . . . . . . . . . 17 ((𝑊𝐽) ∈ ℕ0 → (𝑊𝐽) ∈ (0..^((𝑊𝐽) + 1)))
12463, 122, 1233syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑊𝐽) ∈ (0..^((𝑊𝐽) + 1)))
12546oveq2i 7169 . . . . . . . . . . . . . . . 16 (0..^𝐸) = (0..^((𝑊𝐽) + 1))
126124, 125eleqtrrdi 2926 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊𝐽) ∈ (0..^𝐸))
127121, 126sseldd 3970 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝐽) ∈ (0..^(𝐸 + 1)))
12890oveq2d 7174 . . . . . . . . . . . . . 14 (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
129127, 128eleqtrrd 2918 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
130 ccatval1 13932 . . . . . . . . . . . . 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 1367 . . . . . . . . . . . 12 (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝑊𝐽)) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(𝑊𝐽)))
13288oveq2d 7174 . . . . . . . . . . . . . 14 (𝜑 → (0..^(♯‘(𝑊 prefix 𝐸))) = (0..^𝐸))
133126, 132eleqtrrd 2918 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘(𝑊 prefix 𝐸))))
134 ccatval1 13932 . . . . . . . . . . . . 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 1367 . . . . . . . . . . . 12 (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(𝑊𝐽)) = ((𝑊 prefix 𝐸)‘(𝑊𝐽)))
136119, 131, 1353eqtrd 2862 . . . . . . . . . . 11 (𝜑 → (𝑈‘(𝑊𝐽)) = ((𝑊 prefix 𝐸)‘(𝑊𝐽)))
137 pfxfv 14046 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (𝑊𝐽) ∈ (0..^𝐸)) → ((𝑊 prefix 𝐸)‘(𝑊𝐽)) = (𝑊‘(𝑊𝐽)))
13817, 86, 126, 137syl3anc 1367 . . . . . . . . . . 11 (𝜑 → ((𝑊 prefix 𝐸)‘(𝑊𝐽)) = (𝑊‘(𝑊𝐽)))
139 f1f1orn 6628 . . . . . . . . . . . . 13 (𝑊:dom 𝑊1-1𝐷𝑊:dom 𝑊1-1-onto→ran 𝑊)
14024, 139syl 17 . . . . . . . . . . . 12 (𝜑𝑊:dom 𝑊1-1-onto→ran 𝑊)
141 f1ocnvfv2 7036 . . . . . . . . . . . 12 ((𝑊:dom 𝑊1-1-onto→ran 𝑊𝐽 ∈ ran 𝑊) → (𝑊‘(𝑊𝐽)) = 𝐽)
142140, 28, 141syl2anc 586 . . . . . . . . . . 11 (𝜑 → (𝑊‘(𝑊𝐽)) = 𝐽)
143136, 138, 1423eqtrd 2862 . . . . . . . . . 10 (𝜑 → (𝑈‘(𝑊𝐽)) = 𝐽)
144143fveq2d 6676 . . . . . . . . 9 (𝜑 → ((𝑀𝑈)‘(𝑈‘(𝑊𝐽))) = ((𝑀𝑈)‘𝐽))
14555, 118, 1443eqtr2d 2864 . . . . . . . 8 (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀𝑈)‘𝐽))
146145ad2antrr 724 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀𝑈)‘𝐽))
147 simpr 487 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → 𝑖 = 𝐽)
148147fveq2d 6676 . . . . . . . 8 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽))
149148fveq2d 6676 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)))
150147fveq2d 6676 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑈)‘𝐽))
151146, 149, 1503eqtr4d 2868 . . . . . 6 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
1521ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝐷𝑉)
15315, 29s2cld 14235 . . . . . . . . . 10 (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
154153ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
15515, 29, 33s2f1 30623 . . . . . . . . . 10 (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
156155ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
15727sselda 3969 . . . . . . . . . 10 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖𝐷)
158157adantr 483 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝑖𝐷)
159 simpr 487 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑊)
16030adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → ¬ 𝐼 ∈ ran 𝑊)
161 nelne2 3117 . . . . . . . . . . . . 13 ((𝑖 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝑖𝐼)
162159, 160, 161syl2anc 586 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖𝐼)
163162adantr 483 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝑖𝐼)
164 simpr 487 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → 𝑖𝐽)
165163, 164nelprd 4598 . . . . . . . . . 10 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ¬ 𝑖 ∈ {𝐼, 𝐽})
16615, 29s2rn 30622 . . . . . . . . . . . . 13 (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
167166eleq2d 2900 . . . . . . . . . . . 12 (𝜑 → (𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ 𝑖 ∈ {𝐼, 𝐽}))
168167notbid 320 . . . . . . . . . . 11 (𝜑 → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
169168ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
170165, 169mpbird 259 . . . . . . . . 9 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
1712, 152, 154, 156, 158, 170cycpmfv3 30759 . . . . . . . 8 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
172171fveq2d 6676 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘𝑖))
1731ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝐷𝑉)
1747ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝑊 ∈ dom 𝑀)
17514ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
17628ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝐽 ∈ ran 𝑊)
177 simpllr 774 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝑖 ∈ ran 𝑊)
178 simplr 767 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → 𝑖𝐽)
179 simpr 487 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → (𝑈𝑖) ∈ (0..^𝐸))
1802, 3, 173, 174, 175, 176, 46, 41, 177, 178, 179cycpmco2lem7 30776 . . . . . . . 8 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (0..^𝐸)) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
1811ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐷𝑉)
1827ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑊 ∈ dom 𝑀)
18314ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
18428ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐽 ∈ ran 𝑊)
185 simpllr 774 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖 ∈ ran 𝑊)
186162ad2antrr 724 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖𝐼)
187 simpr 487 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))
1882, 3, 181, 182, 183, 184, 46, 41, 185, 186, 187cycpmco2lem6 30775 . . . . . . . 8 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
1891ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝐷𝑉)
1907ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝑊 ∈ dom 𝑀)
19114ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
19228ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝐽 ∈ ran 𝑊)
193 simpllr 774 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → 𝑖 ∈ ran 𝑊)
194 simpr 487 . . . . . . . . 9 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → (𝑈𝑖) = ((♯‘𝑈) − 1))
1952, 3, 189, 190, 191, 192, 46, 41, 193, 194cycpmco2lem5 30774 . . . . . . . 8 ((((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) ∧ (𝑈𝑖) = ((♯‘𝑈) − 1)) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
196 f1f1orn 6628 . . . . . . . . . . . . . . . 16 (𝑈:dom 𝑈1-1𝐷𝑈:dom 𝑈1-1-onto→ran 𝑈)
19747, 196syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑈:dom 𝑈1-1-onto→ran 𝑈)
198 ssun1 4150 . . . . . . . . . . . . . . . . 17 ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
1992, 3, 1, 7, 14, 28, 46, 41cycpmco2rn 30769 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
200198, 199sseqtrrid 4022 . . . . . . . . . . . . . . . 16 (𝜑 → ran 𝑊 ⊆ ran 𝑈)
201200sselda 3969 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑈)
202 f1ocnvdm 7043 . . . . . . . . . . . . . . 15 ((𝑈:dom 𝑈1-1-onto→ran 𝑈𝑖 ∈ ran 𝑈) → (𝑈𝑖) ∈ dom 𝑈)
203197, 201, 202syl2an2r 683 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ran 𝑊) → (𝑈𝑖) ∈ dom 𝑈)
204 wrddm 13871 . . . . . . . . . . . . . . . 16 (𝑈 ∈ Word 𝐷 → dom 𝑈 = (0..^(♯‘𝑈)))
20545, 204syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
206205adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ran 𝑊) → dom 𝑈 = (0..^(♯‘𝑈)))
207203, 206eleqtrd 2917 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → (𝑈𝑖) ∈ (0..^(♯‘𝑈)))
20865nn0zd 12088 . . . . . . . . . . . . . . . . 17 (𝜑 → (♯‘𝑊) ∈ ℤ)
209208peano2zd 12093 . . . . . . . . . . . . . . . 16 (𝜑 → ((♯‘𝑊) + 1) ∈ ℤ)
210110, 209eqeltrd 2915 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑈) ∈ ℤ)
211 fzoval 13042 . . . . . . . . . . . . . . 15 ((♯‘𝑈) ∈ ℤ → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
212210, 211syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
213212adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ran 𝑊) → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
214207, 213eleqtrd 2917 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ran 𝑊) → (𝑈𝑖) ∈ (0...((♯‘𝑈) − 1)))
215 elfzr 13153 . . . . . . . . . . . 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 487 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ ran 𝑊) ∧ (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1))) → (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)))
21899ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ ran 𝑊) ∧ (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1))) → 𝐸 ∈ ℤ)
219 fzospliti 13072 . . . . . . . . . . . . . 14 (((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∧ 𝐸 ∈ ℤ) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
220217, 218, 219syl2anc 586 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ ran 𝑊) ∧ (𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1))) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
221220ex 415 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑈𝑖) ∈ (0..^((♯‘𝑈) − 1)) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))))
222221orim1d 962 . . . . . . . . . . 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 1084 . . . . . . . . . 10 (((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)) ↔ (((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
225223, 224sylibr 236 . . . . . . . . 9 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
226225adantr 483 . . . . . . . 8 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑈𝑖) ∈ (0..^𝐸) ∨ (𝑈𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (𝑈𝑖) = ((♯‘𝑈) − 1)))
227180, 188, 195, 226mpjao3dan 1427 . . . . . . 7 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑊)‘𝑖))
228172, 227eqtr4d 2861 . . . . . 6 (((𝜑𝑖 ∈ ran 𝑊) ∧ 𝑖𝐽) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
229151, 228pm2.61dane 3106 . . . . 5 ((𝜑𝑖 ∈ ran 𝑊) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
230229adantlr 713 . . . 4 (((𝜑𝑖𝐷) ∧ 𝑖 ∈ ran 𝑊) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
2312, 3, 1, 7, 14, 28, 46, 41cycpmco2lem4 30773 . . . . . . . 8 (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
232231ad2antrr 724 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
233 simpr 487 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
234233fveq2d 6676 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼))
235234fveq2d 6676 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)))
236233fveq2d 6676 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑈)‘𝑖) = ((𝑀𝑈)‘𝐼))
237232, 235, 2363eqtr4d 2868 . . . . . 6 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
2381ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝐷𝑉)
23917ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑊 ∈ Word 𝐷)
24024ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑊:dom 𝑊1-1𝐷)
241 simplr 767 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖 ∈ (𝐷 ∖ ran 𝑊))
242241eldifad 3950 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖𝐷)
243241eldifbd 3951 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ ran 𝑊)
2442, 238, 239, 240, 242, 243cycpmfv3 30759 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑊)‘𝑖) = 𝑖)
245153ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
246155ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
247 simpr 487 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖𝐼)
248 eldifn 4106 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝐷 ∖ ran 𝑊) → ¬ 𝑖 ∈ ran 𝑊)
249 nelne2 3117 . . . . . . . . . . . . . 14 ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ ran 𝑊) → 𝐽𝑖)
25028, 248, 249syl2an 597 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝐽𝑖)
251250necomd 3073 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝑖𝐽)
252251adantr 483 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑖𝐽)
253247, 252nelprd 4598 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ {𝐼, 𝐽})
254168ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
255253, 254mpbird 259 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
2562, 238, 245, 246, 242, 255cycpmfv3 30759 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
257256fveq2d 6676 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑊)‘𝑖))
25845ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑈 ∈ Word 𝐷)
25947ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → 𝑈:dom 𝑈1-1𝐷)
260199ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
261 nelsn 4607 . . . . . . . . . 10 (𝑖𝐼 → ¬ 𝑖 ∈ {𝐼})
262261adantl 484 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ {𝐼})
263 nelun 30276 . . . . . . . . . 10 (ran 𝑈 = (ran 𝑊 ∪ {𝐼}) → (¬ 𝑖 ∈ ran 𝑈 ↔ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})))
264263biimpar 480 . . . . . . . . 9 ((ran 𝑈 = (ran 𝑊 ∪ {𝐼}) ∧ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})) → ¬ 𝑖 ∈ ran 𝑈)
265260, 243, 262, 264syl12anc 834 . . . . . . . 8 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ¬ 𝑖 ∈ ran 𝑈)
2662, 238, 258, 259, 242, 265cycpmfv3 30759 . . . . . . 7 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑈)‘𝑖) = 𝑖)
267244, 257, 2663eqtr4d 2868 . . . . . 6 (((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖𝐼) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
268237, 267pm2.61dane 3106 . . . . 5 ((𝜑𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
269268adantlr 713 . . . 4 (((𝜑𝑖𝐷) ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
270 elun 4127 . . . . . 6 (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ (𝑖 ∈ ran 𝑊𝑖 ∈ (𝐷 ∖ ran 𝑊)))
271 undif 4432 . . . . . . . 8 (ran 𝑊𝐷 ↔ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
27227, 271sylib 220 . . . . . . 7 (𝜑 → (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
273272eleq2d 2900 . . . . . 6 (𝜑 → (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ 𝑖𝐷))
274270, 273syl5rbbr 288 . . . . 5 (𝜑 → (𝑖𝐷 ↔ (𝑖 ∈ ran 𝑊𝑖 ∈ (𝐷 ∖ ran 𝑊))))
275274biimpa 479 . . . 4 ((𝜑𝑖𝐷) → (𝑖 ∈ ran 𝑊𝑖 ∈ (𝐷 ∖ ran 𝑊)))
276230, 269, 275mpjaodan 955 . . 3 ((𝜑𝑖𝐷) → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀𝑈)‘𝑖))
27753, 276eqtrd 2858 . 2 ((𝜑𝑖𝐷) → (((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀𝑈)‘𝑖))
27840, 51, 277eqfnfvd 6807 1 (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  {crab 3144  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  {csn 4569  {cpr 4571  ⟨cop 4575  ⟨cotp 4577  ◡ccnv 5556  dom cdm 5557  ran crn 5558   ∘ ccom 5561   Fn wfn 6352  ⟶wf 6353  –1-1→wf1 6354  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  0cc0 10539  1c1 10540   + caddc 10542   − cmin 10872  ℕ0cn0 11900  ℤcz 11984  ...cfz 12895  ..^cfzo 13036  ♯chash 13693  Word cword 13864   ++ cconcat 13924  ⟨“cs1 13951   substr csubstr 14004   prefix cpfx 14034   splice csplice 14113  ⟨“cs2 14205  Basecbs 16485  SymGrpcsymg 18497  toCycctocyc 30750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-hash 13694  df-word 13865  df-concat 13925  df-s1 13952  df-substr 14005  df-pfx 14035  df-splice 14114  df-csh 14153  df-s2 14212  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-tset 16586  df-efmnd 18036  df-symg 18498  df-tocyc 30751 This theorem is referenced by: (None)
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