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Theorem fzto1st 31952
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 13510 . . . . 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 2856 . . 3 (𝐼𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
5 3ancoma 1098 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
64, 5sylibr 233 . 2 (𝐼𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁))
7 df-3an 1089 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁))
8 breq1 5108 . . . . . 6 (𝑚 = 1 → (𝑚𝑁 ↔ 1 ≤ 𝑁))
9 simpl 483 . . . . . . . . . 10 ((𝑚 = 1 ∧ 𝑖𝐷) → 𝑚 = 1)
109breq2d 5117 . . . . . . . . . . 11 ((𝑚 = 1 ∧ 𝑖𝐷) → (𝑖𝑚𝑖 ≤ 1))
1110ifbid 4509 . . . . . . . . . 10 ((𝑚 = 1 ∧ 𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
129, 11ifeq12d 4507 . . . . . . . . 9 ((𝑚 = 1 ∧ 𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
1312mpteq2dva 5205 . . . . . . . 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 2736 . . . . . . . . 9 1 = 1
15 eqid 2736 . . . . . . . . . 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 31951 . . . . . . . . 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 2792 . . . . . . 7 (𝑚 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
1918eleq1d 2822 . . . . . 6 (𝑚 = 1 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ ( I ↾ 𝐷) ∈ 𝐵))
208, 19imbi12d 344 . . . . 5 (𝑚 = 1 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵)))
21 breq1 5108 . . . . . 6 (𝑚 = 𝑛 → (𝑚𝑁𝑛𝑁))
22 simpl 483 . . . . . . . . 9 ((𝑚 = 𝑛𝑖𝐷) → 𝑚 = 𝑛)
2322breq2d 5117 . . . . . . . . . 10 ((𝑚 = 𝑛𝑖𝐷) → (𝑖𝑚𝑖𝑛))
2423ifbid 4509 . . . . . . . . 9 ((𝑚 = 𝑛𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝑛, (𝑖 − 1), 𝑖))
2522, 24ifeq12d 4507 . . . . . . . 8 ((𝑚 = 𝑛𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))
2625mpteq2dva 5205 . . . . . . 7 (𝑚 = 𝑛 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))
2726eleq1d 2822 . . . . . 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 344 . . . . 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 5108 . . . . . 6 (𝑚 = (𝑛 + 1) → (𝑚𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
30 simpl 483 . . . . . . . . 9 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → 𝑚 = (𝑛 + 1))
3130breq2d 5117 . . . . . . . . . 10 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → (𝑖𝑚𝑖 ≤ (𝑛 + 1)))
3231ifbid 4509 . . . . . . . . 9 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
3330, 32ifeq12d 4507 . . . . . . . 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 5205 . . . . . . 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 2822 . . . . . 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 344 . . . . 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 5108 . . . . . 6 (𝑚 = 𝐼 → (𝑚𝑁𝐼𝑁))
38 simpl 483 . . . . . . . . . 10 ((𝑚 = 𝐼𝑖𝐷) → 𝑚 = 𝐼)
3938breq2d 5117 . . . . . . . . . . 11 ((𝑚 = 𝐼𝑖𝐷) → (𝑖𝑚𝑖𝐼))
4039ifbid 4509 . . . . . . . . . 10 ((𝑚 = 𝐼𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝐼, (𝑖 − 1), 𝑖))
4138, 40ifeq12d 4507 . . . . . . . . 9 ((𝑚 = 𝐼𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
4241mpteq2dva 5205 . . . . . . . 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 2794 . . . . . . 7 (𝑚 = 𝐼 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
4544eleq1d 2822 . . . . . 6 (𝑚 = 𝐼 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵𝑃𝐵))
4637, 45imbi12d 344 . . . . 5 (𝑚 = 𝐼 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝐼𝑁𝑃𝐵)))
47 fzfi 13877 . . . . . . . . 9 (1...𝑁) ∈ Fin
483, 47eqeltri 2834 . . . . . . . 8 𝐷 ∈ Fin
49 psgnfzto1st.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐷)
5049idresperm 19167 . . . . . . . 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 2837 . . . . . 6 ( I ↾ 𝐷) ∈ 𝐵
54532a1i 12 . . . . 5 (𝑁 ∈ ℕ → (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵))
55 simplr 767 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
5655peano2nnd 12170 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
57 simpll 765 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
58 simpr 485 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
5956, 57, 583jca 1128 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
60 elfz1b 13510 . . . . . . . . . . 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 2849 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
633psgnfzto1stlem 31949 . . . . . . . . 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 584 . . . . . . . 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 713 . . . . . . 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 2736 . . . . . . . . . 10 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
6766, 49, 52symgtrf 19251 . . . . . . . . 9 ran (pmTrsp‘𝐷) ⊆ 𝐵
68 eqid 2736 . . . . . . . . . . . 12 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
693, 68pmtrto1cl 31948 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7055, 62, 69syl2anc 584 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7170adantlr 713 . . . . . . . . 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 3942 . . . . . . . 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 12168 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
74 1red 11156 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
7573, 74readdcld 11184 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
7657nnred 12168 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
7773lep1d 12086 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
7873, 75, 76, 77, 58letrd 11312 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
7978adantlr 713 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
80 simplr 767 . . . . . . . . 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 2736 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
8349, 52, 82symgov 19165 . . . . . . . . 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 19166 . . . . . . . . 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 2839 . . . . . . . 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 584 . . . . . . 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 2838 . . . . . 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 413 . . . . 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 12173 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼𝑁𝑃𝐵))
9089imp 407 . . 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 396  w3a 1087   = wceq 1541  wcel 2106  ifcif 4486  {cpr 4588   class class class wbr 5105  cmpt 5188   I cid 5530  ran crn 5634  cres 5635  ccom 5637  cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052   + caddc 11054  cle 11190  cmin 11385  cn 12153  ...cfz 13424  Basecbs 17083  +gcplusg 17133  SymGrpcsymg 19148  pmTrspcpmtr 19223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-tset 17152  df-efmnd 18679  df-symg 19149  df-pmtr 19224
This theorem is referenced by:  fzto1stinvn  31953  psgnfzto1st  31954  madjusmdetlem2  32409  madjusmdetlem3  32410  madjusmdetlem4  32411
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