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Theorem fzto1st 30162
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 12602 . . . . 5 (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
21biimpi 206 . . . 4 (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
3 psgnfzto1st.d . . . 4 𝐷 = (1...𝑁)
42, 3eleq2s 2857 . . 3 (𝐼𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
5 3ancoma 1084 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
64, 5sylibr 224 . 2 (𝐼𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁))
7 df-3an 1074 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁))
8 breq1 4807 . . . . . 6 (𝑚 = 1 → (𝑚𝑁 ↔ 1 ≤ 𝑁))
9 simpl 474 . . . . . . . . . 10 ((𝑚 = 1 ∧ 𝑖𝐷) → 𝑚 = 1)
109breq2d 4816 . . . . . . . . . . 11 ((𝑚 = 1 ∧ 𝑖𝐷) → (𝑖𝑚𝑖 ≤ 1))
1110ifbid 4252 . . . . . . . . . 10 ((𝑚 = 1 ∧ 𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
129, 11ifeq12d 4250 . . . . . . . . 9 ((𝑚 = 1 ∧ 𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
1312mpteq2dva 4896 . . . . . . . 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 2760 . . . . . . . . 9 1 = 1
15 eqid 2760 . . . . . . . . . 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 30161 . . . . . . . . 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, 17syl6eq 2810 . . . . . . 7 (𝑚 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
1918eleq1d 2824 . . . . . 6 (𝑚 = 1 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ ( I ↾ 𝐷) ∈ 𝐵))
208, 19imbi12d 333 . . . . 5 (𝑚 = 1 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵)))
21 breq1 4807 . . . . . 6 (𝑚 = 𝑛 → (𝑚𝑁𝑛𝑁))
22 simpl 474 . . . . . . . . 9 ((𝑚 = 𝑛𝑖𝐷) → 𝑚 = 𝑛)
2322breq2d 4816 . . . . . . . . . 10 ((𝑚 = 𝑛𝑖𝐷) → (𝑖𝑚𝑖𝑛))
2423ifbid 4252 . . . . . . . . 9 ((𝑚 = 𝑛𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝑛, (𝑖 − 1), 𝑖))
2522, 24ifeq12d 4250 . . . . . . . 8 ((𝑚 = 𝑛𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))
2625mpteq2dva 4896 . . . . . . 7 (𝑚 = 𝑛 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))
2726eleq1d 2824 . . . . . 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 333 . . . . 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 4807 . . . . . 6 (𝑚 = (𝑛 + 1) → (𝑚𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
30 simpl 474 . . . . . . . . 9 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → 𝑚 = (𝑛 + 1))
3130breq2d 4816 . . . . . . . . . 10 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → (𝑖𝑚𝑖 ≤ (𝑛 + 1)))
3231ifbid 4252 . . . . . . . . 9 ((𝑚 = (𝑛 + 1) ∧ 𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
3330, 32ifeq12d 4250 . . . . . . . 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 4896 . . . . . . 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 2824 . . . . . 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 333 . . . . 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 4807 . . . . . 6 (𝑚 = 𝐼 → (𝑚𝑁𝐼𝑁))
38 simpl 474 . . . . . . . . . 10 ((𝑚 = 𝐼𝑖𝐷) → 𝑚 = 𝐼)
3938breq2d 4816 . . . . . . . . . . 11 ((𝑚 = 𝐼𝑖𝐷) → (𝑖𝑚𝑖𝐼))
4039ifbid 4252 . . . . . . . . . 10 ((𝑚 = 𝐼𝑖𝐷) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝐼, (𝑖 − 1), 𝑖))
4138, 40ifeq12d 4250 . . . . . . . . 9 ((𝑚 = 𝐼𝑖𝐷) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
4241mpteq2dva 4896 . . . . . . . 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, 43syl6eqr 2812 . . . . . . 7 (𝑚 = 𝐼 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
4544eleq1d 2824 . . . . . 6 (𝑚 = 𝐼 → ((𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵𝑃𝐵))
4637, 45imbi12d 333 . . . . 5 (𝑚 = 𝐼 → ((𝑚𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝐼𝑁𝑃𝐵)))
47 fzfi 12965 . . . . . . . . 9 (1...𝑁) ∈ Fin
483, 47eqeltri 2835 . . . . . . . 8 𝐷 ∈ Fin
49 psgnfzto1st.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐷)
5049idresperm 18029 . . . . . . . 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 2838 . . . . . 6 ( I ↾ 𝐷) ∈ 𝐵
54532a1i 12 . . . . 5 (𝑁 ∈ ℕ → (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵))
55 simplr 809 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
5655peano2nnd 11229 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
57 simpll 807 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
58 simpr 479 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
5956, 57, 583jca 1123 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
60 elfz1b 12602 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
6159, 60sylibr 224 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
6261, 3syl6eleqr 2850 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
633psgnfzto1stlem 30159 . . . . . . . . 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 696 . . . . . . . 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 753 . . . . . . 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 2760 . . . . . . . . . 10 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
6766, 49, 52symgtrf 18089 . . . . . . . . 9 ran (pmTrsp‘𝐷) ⊆ 𝐵
68 eqid 2760 . . . . . . . . . . . 12 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
693, 68pmtrto1cl 30158 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7055, 62, 69syl2anc 696 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
7170adantlr 753 . . . . . . . . 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, 71sseldi 3742 . . . . . . . 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 11227 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
74 1red 10247 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
7573, 74readdcld 10261 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
7657nnred 11227 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
7773lep1d 11147 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
7873, 75, 76, 77, 58letrd 10386 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
7978adantlr 753 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
80 simplr 809 . . . . . . . . 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 2760 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
8349, 52, 82symgov 18010 . . . . . . . . 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 18011 . . . . . . . . 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 2840 . . . . . . . 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 696 . . . . . . 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 2839 . . . . . 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 449 . . . . 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 29875 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼𝑁𝑃𝐵))
9089imp 444 . . 3 (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁) → 𝑃𝐵)
917, 90sylbi 207 . 2 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) → 𝑃𝐵)
926, 91syl 17 1 (𝐼𝐷𝑃𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  ifcif 4230  {cpr 4323   class class class wbr 4804  cmpt 4881   I cid 5173  ran crn 5267  cres 5268  ccom 5270  cfv 6049  (class class class)co 6813  Fincfn 8121  1c1 10129   + caddc 10131  cle 10267  cmin 10458  cn 11212  ...cfz 12519  Basecbs 16059  +gcplusg 16143  SymGrpcsymg 17997  pmTrspcpmtr 18061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-plusg 16156  df-tset 16162  df-symg 17998  df-pmtr 18062
This theorem is referenced by:  fzto1stinvn  30163  psgnfzto1st  30164  madjusmdetlem2  30203  madjusmdetlem3  30204  madjusmdetlem4  30205
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