Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fzto1st Structured version   Visualization version   GIF version

Theorem fzto1st 32001
Description: The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
Hypotheses
Ref Expression
psgnfzto1st.d 𝐷 = (1...𝑁)
psgnfzto1st.p 𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
psgnfzto1st.g 𝐺 = (SymGrp‘𝐷)
psgnfzto1st.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
fzto1st (𝐼𝐷𝑃𝐵)
Distinct variable groups:   𝐷,𝑖   𝑖,𝐼   𝑖,𝑁   𝐵,𝑖
Allowed substitution hints:   𝑃(𝑖)   𝐺(𝑖)

Proof of Theorem fzto1st
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfz1b 13516 . . . . 5 (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
21biimpi 215 . . . 4 (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
3 psgnfzto1st.d . . . 4 𝐷 = (1...𝑁)
42, 3eleq2s 2852 . . 3 (𝐼𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
5 3ancoma 1099 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
64, 5sylibr 233 . 2 (𝐼𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁))
7 df-3an 1090 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁))
8 breq1 5109 . . . . . 6 (𝑚 = 1 → (𝑚𝑁 ↔ 1 ≤ 𝑁))
9 simpl 484 . . . . . . . . . 10 ((𝑚 = 1 ∧ 𝑖𝐷) → 𝑚 = 1)
109breq2d 5118 . . . . . . . . . . 11 ((𝑚 = 1 ∧ 𝑖𝐷) → (𝑖𝑚𝑖 ≤ 1))
1110ifbid 4510 . . . . . . . . . 10 ((𝑚 = 1 ∧ 𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
129, 11ifeq12d 4508 . . . . . . . . 9 ((𝑚 = 1 ∧ 𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
1312mpteq2dva 5206 . . . . . . . 8 (𝑚 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))))
14 eqid 2733 . . . . . . . . 9 1 = 1
15 eqid 2733 . . . . . . . . . 10 (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
163, 15fzto1st1 32000 . . . . . . . . 9 (1 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
1714, 16ax-mp 5 . . . . . . . 8 (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷)
1813, 17eqtrdi 2789 . . . . . . 7 (𝑚 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
1918eleq1d 2819 . . . . . 6 (𝑚 = 1 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ ( I ↾ 𝐷) ∈ 𝐵))
208, 19imbi12d 345 . . . . 5 (𝑚 = 1 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵)))
21 breq1 5109 . . . . . 6 (𝑚 = 𝑛 → (𝑚𝑁𝑛𝑁))
22 simpl 484 . . . . . . . . 9 ((𝑚 = 𝑛𝑖𝐷) → 𝑚 = 𝑛)
2322breq2d 5118 . . . . . . . . . 10 ((𝑚 = 𝑛𝑖𝐷) → (𝑖𝑚𝑖𝑛))
2423ifbid 4510 . . . . . . . . 9 ((𝑚 = 𝑛𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝑛, (𝑖 − 1), 𝑖))
2522, 24ifeq12d 4508 . . . . . . . 8 ((𝑚 = 𝑛𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))
2625mpteq2dva 5206 . . . . . . 7 (𝑚 = 𝑛 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))
2726eleq1d 2819 . . . . . 6 (𝑚 = 𝑛 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵))
2821, 27imbi12d 345 . . . . 5 (𝑚 = 𝑛 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)))
29 breq1 5109 . . . . . 6 (𝑚 = (𝑛 + 1) → (𝑚𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
30 simpl 484 . . . . . . . . 9 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → 𝑚 = (𝑛 + 1))
3130breq2d 5118 . . . . . . . . . 10 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → (𝑖𝑚𝑖 ≤ (𝑛 + 1)))
3231ifbid 4510 . . . . . . . . 9 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
3330, 32ifeq12d 4508 . . . . . . . 8 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))
3433mpteq2dva 5206 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))))
3534eleq1d 2819 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵))
3629, 35imbi12d 345 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ ((𝑛 + 1) ≤ 𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵)))
37 breq1 5109 . . . . . 6 (𝑚 = 𝐼 → (𝑚𝑁𝐼𝑁))
38 simpl 484 . . . . . . . . . 10 ((𝑚 = 𝐼𝑖𝐷) → 𝑚 = 𝐼)
3938breq2d 5118 . . . . . . . . . . 11 ((𝑚 = 𝐼𝑖𝐷) → (𝑖𝑚𝑖𝐼))
4039ifbid 4510 . . . . . . . . . 10 ((𝑚 = 𝐼𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝐼, (𝑖 − 1), 𝑖))
4138, 40ifeq12d 4508 . . . . . . . . 9 ((𝑚 = 𝐼𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
4241mpteq2dva 5206 . . . . . . . 8 (𝑚 = 𝐼 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖))))
43 psgnfzto1st.p . . . . . . . 8 𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
4442, 43eqtr4di 2791 . . . . . . 7 (𝑚 = 𝐼 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
4544eleq1d 2819 . . . . . 6 (𝑚 = 𝐼 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵𝑃𝐵))
4637, 45imbi12d 345 . . . . 5 (𝑚 = 𝐼 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝐼𝑁𝑃𝐵)))
47 fzfi 13883 . . . . . . . . 9 (1...𝑁) ∈ Fin
483, 47eqeltri 2830 . . . . . . . 8 𝐷 ∈ Fin
49 psgnfzto1st.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐷)
5049idresperm 19172 . . . . . . . 8 (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (Base‘𝐺))
5148, 50ax-mp 5 . . . . . . 7 ( I ↾ 𝐷) ∈ (Base‘𝐺)
52 psgnfzto1st.b . . . . . . 7 𝐵 = (Base‘𝐺)
5351, 52eleqtrri 2833 . . . . . 6 ( I ↾ 𝐷) ∈ 𝐵
54532a1i 12 . . . . 5 (𝑁 ∈ ℕ → (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵))
55 simplr 768 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
5655peano2nnd 12175 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
57 simpll 766 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
58 simpr 486 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
5956, 57, 583jca 1129 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
60 elfz1b 13516 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
6159, 60sylibr 233 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
6261, 3eleqtrrdi 2845 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
633psgnfzto1stlem 31998 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
6455, 62, 63syl2anc 585 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
6564adantlr 714 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
66 eqid 2733 . . . . . . . . . 10 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
6766, 49, 52symgtrf 19256 . . . . . . . . 9 ran (pmTrsp‘𝐷) ⊆ 𝐵
68 eqid 2733 . . . . . . . . . . . 12 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
693, 68pmtrto1cl 31997 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7055, 62, 69syl2anc 585 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7170adantlr 714 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7267, 71sselid 3943 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵)
7355nnred 12173 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
74 1red 11161 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
7573, 74readdcld 11189 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
7657nnred 12173 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
7773lep1d 12091 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
7873, 75, 76, 77, 58letrd 11317 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
7978adantlr 714 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
80 simplr 768 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵))
8179, 80mpd 15 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
82 eqid 2733 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
8349, 52, 82symgov 19170 . . . . . . . . 9 ((((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})(+g𝐺)(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
8449, 52, 82symgcl 19171 . . . . . . . . 9 ((((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})(+g𝐺)(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) ∈ 𝐵)
8583, 84eqeltrrd 2835 . . . . . . . 8 ((((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) ∈ 𝐵)
8672, 81, 85syl2anc 585 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) ∈ 𝐵)
8765, 86eqeltrd 2834 . . . . . 6 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵)
8887ex 414 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) → ((𝑛 + 1) ≤ 𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵))
8920, 28, 36, 46, 54, 88nnindd 12178 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼𝑁𝑃𝐵))
9089imp 408 . . 3 (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁) → 𝑃𝐵)
917, 90sylbi 216 . 2 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) → 𝑃𝐵)
926, 91syl 17 1 (𝐼𝐷𝑃𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  ifcif 4487  {cpr 4589   class class class wbr 5106  cmpt 5189   I cid 5531  ran crn 5635  cres 5636  ccom 5638  cfv 6497  (class class class)co 7358  Fincfn 8886  1c1 11057   + caddc 11059  cle 11195  cmin 11390  cn 12158  ...cfz 13430  Basecbs 17088  +gcplusg 17138  SymGrpcsymg 19153  pmTrspcpmtr 19228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-tset 17157  df-efmnd 18684  df-symg 19154  df-pmtr 19229
This theorem is referenced by:  fzto1stinvn  32002  psgnfzto1st  32003  madjusmdetlem2  32466  madjusmdetlem3  32467  madjusmdetlem4  32468
  Copyright terms: Public domain W3C validator