| Step | Hyp | Ref
| Expression |
| 1 | | un23 4174 |
. 2
⊢ (((𝑊 “ (0..^𝐸)) ∪ {𝐼}) ∪ (𝑊 “ (𝐸..^(♯‘𝑊)))) = (((𝑊 “ (0..^𝐸)) ∪ (𝑊 “ (𝐸..^(♯‘𝑊)))) ∪ {𝐼}) |
| 2 | | cycpmco2.1 |
. . . . 5
⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) |
| 3 | | cycpmco2.w |
. . . . . 6
⊢ (𝜑 → 𝑊 ∈ dom 𝑀) |
| 4 | | cycpmco2.e |
. . . . . . 7
⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) |
| 5 | | ovexd 7466 |
. . . . . . 7
⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ V) |
| 6 | 4, 5 | eqeltrid 2845 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ V) |
| 7 | | cycpmco2.i |
. . . . . . . 8
⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) |
| 8 | 7 | eldifad 3963 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ 𝐷) |
| 9 | 8 | s1cld 14641 |
. . . . . 6
⊢ (𝜑 → 〈“𝐼”〉 ∈ Word 𝐷) |
| 10 | | splval 14789 |
. . . . . 6
⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ 〈“𝐼”〉 ∈ Word 𝐷)) → (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) = (((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ++ (𝑊 substr 〈𝐸, (♯‘𝑊)〉))) |
| 11 | 3, 6, 6, 9, 10 | syl13anc 1374 |
. . . . 5
⊢ (𝜑 → (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) = (((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ++ (𝑊 substr 〈𝐸, (♯‘𝑊)〉))) |
| 12 | 2, 11 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ++ (𝑊 substr 〈𝐸, (♯‘𝑊)〉))) |
| 13 | 12 | rneqd 5949 |
. . 3
⊢ (𝜑 → ran 𝑈 = ran (((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ++ (𝑊 substr 〈𝐸, (♯‘𝑊)〉))) |
| 14 | | ssrab2 4080 |
. . . . . . 7
⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷 |
| 15 | | cycpmco2.d |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| 16 | | cycpmco2.c |
. . . . . . . . . . 11
⊢ 𝑀 = (toCyc‘𝐷) |
| 17 | | cycpmco2.s |
. . . . . . . . . . 11
⊢ 𝑆 = (SymGrp‘𝐷) |
| 18 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝑆) =
(Base‘𝑆) |
| 19 | 16, 17, 18 | tocycf 33137 |
. . . . . . . . . 10
⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
| 20 | 15, 19 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
| 21 | 20 | fdmd 6746 |
. . . . . . . 8
⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
| 22 | 3, 21 | eleqtrd 2843 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
| 23 | 14, 22 | sselid 3981 |
. . . . . 6
⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
| 24 | | pfxcl 14715 |
. . . . . 6
⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷) |
| 25 | 23, 24 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷) |
| 26 | | ccatcl 14612 |
. . . . 5
⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ 〈“𝐼”〉 ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ∈ Word 𝐷) |
| 27 | 25, 9, 26 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ∈ Word 𝐷) |
| 28 | | swrdcl 14683 |
. . . . 5
⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr 〈𝐸, (♯‘𝑊)〉) ∈ Word 𝐷) |
| 29 | 23, 28 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑊 substr 〈𝐸, (♯‘𝑊)〉) ∈ Word 𝐷) |
| 30 | | ccatrn 14627 |
. . . 4
⊢ ((((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ∈ Word 𝐷 ∧ (𝑊 substr 〈𝐸, (♯‘𝑊)〉) ∈ Word 𝐷) → ran (((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ++ (𝑊 substr 〈𝐸, (♯‘𝑊)〉)) = (ran ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ∪ ran (𝑊 substr 〈𝐸, (♯‘𝑊)〉))) |
| 31 | 27, 29, 30 | syl2anc 584 |
. . 3
⊢ (𝜑 → ran (((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ++ (𝑊 substr 〈𝐸, (♯‘𝑊)〉)) = (ran ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ∪ ran (𝑊 substr 〈𝐸, (♯‘𝑊)〉))) |
| 32 | | ccatrn 14627 |
. . . . . 6
⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ 〈“𝐼”〉 ∈ Word 𝐷) → ran ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) = (ran (𝑊 prefix 𝐸) ∪ ran 〈“𝐼”〉)) |
| 33 | 25, 9, 32 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ran ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) = (ran (𝑊 prefix 𝐸) ∪ ran 〈“𝐼”〉)) |
| 34 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) |
| 35 | | dmeq 5914 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊) |
| 36 | | eqidd 2738 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷) |
| 37 | 34, 35, 36 | f1eq123d 6840 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
| 38 | 37 | elrab 3692 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷)) |
| 39 | 22, 38 | sylib 218 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷)) |
| 40 | 39 | simprd 495 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) |
| 41 | | f1cnv 6872 |
. . . . . . . . . . . 12
⊢ (𝑊:dom 𝑊–1-1→𝐷 → ◡𝑊:ran 𝑊–1-1-onto→dom
𝑊) |
| 42 | | f1of 6848 |
. . . . . . . . . . . 12
⊢ (◡𝑊:ran 𝑊–1-1-onto→dom
𝑊 → ◡𝑊:ran 𝑊⟶dom 𝑊) |
| 43 | 40, 41, 42 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ◡𝑊:ran 𝑊⟶dom 𝑊) |
| 44 | | cycpmco2.j |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ ran 𝑊) |
| 45 | 43, 44 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → (◡𝑊‘𝐽) ∈ dom 𝑊) |
| 46 | | wrddm 14559 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊))) |
| 47 | 23, 46 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
| 48 | 45, 47 | eleqtrd 2843 |
. . . . . . . . 9
⊢ (𝜑 → (◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊))) |
| 49 | | fzofzp1 13803 |
. . . . . . . . 9
⊢ ((◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊))) |
| 50 | 48, 49 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊))) |
| 51 | 4, 50 | eqeltrid 2845 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊))) |
| 52 | | pfxrn3 32925 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐸) = (𝑊 “ (0..^𝐸))) |
| 53 | 23, 51, 52 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ran (𝑊 prefix 𝐸) = (𝑊 “ (0..^𝐸))) |
| 54 | | s1rn 14637 |
. . . . . . 7
⊢ (𝐼 ∈ 𝐷 → ran 〈“𝐼”〉 = {𝐼}) |
| 55 | 8, 54 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 〈“𝐼”〉 = {𝐼}) |
| 56 | 53, 55 | uneq12d 4169 |
. . . . 5
⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∪ ran 〈“𝐼”〉) = ((𝑊 “ (0..^𝐸)) ∪ {𝐼})) |
| 57 | 33, 56 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → ran ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) = ((𝑊 “ (0..^𝐸)) ∪ {𝐼})) |
| 58 | | lencl 14571 |
. . . . . 6
⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈
ℕ0) |
| 59 | | nn0fz0 13665 |
. . . . . . 7
⊢
((♯‘𝑊)
∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊))) |
| 60 | 59 | biimpi 216 |
. . . . . 6
⊢
((♯‘𝑊)
∈ ℕ0 → (♯‘𝑊) ∈ (0...(♯‘𝑊))) |
| 61 | 23, 58, 60 | 3syl 18 |
. . . . 5
⊢ (𝜑 → (♯‘𝑊) ∈
(0...(♯‘𝑊))) |
| 62 | | swrdrn3 32940 |
. . . . 5
⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈
(0...(♯‘𝑊)))
→ ran (𝑊 substr
〈𝐸,
(♯‘𝑊)〉) =
(𝑊 “ (𝐸..^(♯‘𝑊)))) |
| 63 | 23, 51, 61, 62 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → ran (𝑊 substr 〈𝐸, (♯‘𝑊)〉) = (𝑊 “ (𝐸..^(♯‘𝑊)))) |
| 64 | 57, 63 | uneq12d 4169 |
. . 3
⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ 〈“𝐼”〉) ∪ ran (𝑊 substr 〈𝐸, (♯‘𝑊)〉)) = (((𝑊 “ (0..^𝐸)) ∪ {𝐼}) ∪ (𝑊 “ (𝐸..^(♯‘𝑊))))) |
| 65 | 13, 31, 64 | 3eqtrd 2781 |
. 2
⊢ (𝜑 → ran 𝑈 = (((𝑊 “ (0..^𝐸)) ∪ {𝐼}) ∪ (𝑊 “ (𝐸..^(♯‘𝑊))))) |
| 66 | | fzosplit 13732 |
. . . . . 6
⊢ (𝐸 ∈
(0...(♯‘𝑊))
→ (0..^(♯‘𝑊)) = ((0..^𝐸) ∪ (𝐸..^(♯‘𝑊)))) |
| 67 | 51, 66 | syl 17 |
. . . . 5
⊢ (𝜑 → (0..^(♯‘𝑊)) = ((0..^𝐸) ∪ (𝐸..^(♯‘𝑊)))) |
| 68 | 67 | imaeq2d 6078 |
. . . 4
⊢ (𝜑 → (𝑊 “ (0..^(♯‘𝑊))) = (𝑊 “ ((0..^𝐸) ∪ (𝐸..^(♯‘𝑊))))) |
| 69 | | wrdf 14557 |
. . . . . . 7
⊢ (𝑊 ∈ Word 𝐷 → 𝑊:(0..^(♯‘𝑊))⟶𝐷) |
| 70 | 23, 69 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝐷) |
| 71 | 70 | ffnd 6737 |
. . . . 5
⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) |
| 72 | | fnima 6698 |
. . . . 5
⊢ (𝑊 Fn (0..^(♯‘𝑊)) → (𝑊 “ (0..^(♯‘𝑊))) = ran 𝑊) |
| 73 | 71, 72 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑊 “ (0..^(♯‘𝑊))) = ran 𝑊) |
| 74 | | elfzuz3 13561 |
. . . . . 6
⊢ (𝐸 ∈
(0...(♯‘𝑊))
→ (♯‘𝑊)
∈ (ℤ≥‘𝐸)) |
| 75 | | fzoss2 13727 |
. . . . . 6
⊢
((♯‘𝑊)
∈ (ℤ≥‘𝐸) → (0..^𝐸) ⊆ (0..^(♯‘𝑊))) |
| 76 | 51, 74, 75 | 3syl 18 |
. . . . 5
⊢ (𝜑 → (0..^𝐸) ⊆ (0..^(♯‘𝑊))) |
| 77 | | fz0ssnn0 13662 |
. . . . . . . 8
⊢
(0...(♯‘𝑊)) ⊆
ℕ0 |
| 78 | 77, 51 | sselid 3981 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈
ℕ0) |
| 79 | | nn0uz 12920 |
. . . . . . 7
⊢
ℕ0 = (ℤ≥‘0) |
| 80 | 78, 79 | eleqtrdi 2851 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈
(ℤ≥‘0)) |
| 81 | | fzoss1 13726 |
. . . . . 6
⊢ (𝐸 ∈
(ℤ≥‘0) → (𝐸..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))) |
| 82 | 80, 81 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐸..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))) |
| 83 | | unima 6984 |
. . . . 5
⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ (0..^𝐸) ⊆ (0..^(♯‘𝑊)) ∧ (𝐸..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))) → (𝑊 “ ((0..^𝐸) ∪ (𝐸..^(♯‘𝑊)))) = ((𝑊 “ (0..^𝐸)) ∪ (𝑊 “ (𝐸..^(♯‘𝑊))))) |
| 84 | 71, 76, 82, 83 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → (𝑊 “ ((0..^𝐸) ∪ (𝐸..^(♯‘𝑊)))) = ((𝑊 “ (0..^𝐸)) ∪ (𝑊 “ (𝐸..^(♯‘𝑊))))) |
| 85 | 68, 73, 84 | 3eqtr3d 2785 |
. . 3
⊢ (𝜑 → ran 𝑊 = ((𝑊 “ (0..^𝐸)) ∪ (𝑊 “ (𝐸..^(♯‘𝑊))))) |
| 86 | 85 | uneq1d 4167 |
. 2
⊢ (𝜑 → (ran 𝑊 ∪ {𝐼}) = (((𝑊 “ (0..^𝐸)) ∪ (𝑊 “ (𝐸..^(♯‘𝑊)))) ∪ {𝐼})) |
| 87 | 1, 65, 86 | 3eqtr4a 2803 |
1
⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼})) |