Theorem fzto1st 30745

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 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
psgnfzto1st.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnfzto1st.b	⊢ 𝐵 = (Base‘𝐺)

Assertion

Ref	Expression
fzto1st	⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵)

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.

Step	Hyp	Ref	Expression
1		elfz1b 12975	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
2	1	biimpi 218	. . . 4 ⊢ (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
3		psgnfzto1st.d	. . . 4 ⊢ 𝐷 = (1...𝑁)
4	2, 3	eleq2s 2931	. . 3 ⊢ (𝐼 ∈ 𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
5		3ancoma 1094	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
6	4, 5	sylibr 236	. 2 ⊢ (𝐼 ∈ 𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
7		df-3an 1085	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁))
8		breq1 5068	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 ≤ 𝑁 ↔ 1 ≤ 𝑁))
9		simpl 485	. . . . . . . . . 10 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → 𝑚 = 1)
10	9	breq2d 5077	. . . . . . . . . . 11 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 1))
11	10	ifbid 4488	. . . . . . . . . 10 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
12	9, 11	ifeq12d 4486	. . . . . . . . 9 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
13	12	mpteq2dva 5160	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))))
14		eqid 2821	. . . . . . . . 9 ⊢ 1 = 1
15		eqid 2821	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
16	3, 15	fzto1st1 30744	. . . . . . . . 9 ⊢ (1 = 1 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
17	14, 16	ax-mp 5	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷)
18	13, 17	syl6eq 2872	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
19	18	eleq1d 2897	. . . . . 6 ⊢ (𝑚 = 1 → ((𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ ( I ↾ 𝐷) ∈ 𝐵))
20	8, 19	imbi12d 347	. . . . 5 ⊢ (𝑚 = 1 → ((𝑚 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵)))
21		breq1 5068	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
22		simpl 485	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → 𝑚 = 𝑛)
23	22	breq2d 5077	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝑛))
24	23	ifbid 4488	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))
25	22, 24	ifeq12d 4486	. . . . . . . 8 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))
26	25	mpteq2dva 5160	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))
27	26	eleq1d 2897	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵))
28	21, 27	imbi12d 347	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)))
29		breq1 5068	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ 𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
30		simpl 485	. . . . . . . . 9 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → 𝑚 = (𝑛 + 1))
31	30	breq2d 5077	. . . . . . . . . 10 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ (𝑛 + 1)))
32	31	ifbid 4488	. . . . . . . . 9 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
33	30, 32	ifeq12d 4486	. . . . . . . 8 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))
34	33	mpteq2dva 5160	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))))
35	34	eleq1d 2897	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵))
36	29, 35	imbi12d 347	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ ((𝑛 + 1) ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵)))
37		breq1 5068	. . . . . 6 ⊢ (𝑚 = 𝐼 → (𝑚 ≤ 𝑁 ↔ 𝐼 ≤ 𝑁))
38		simpl 485	. . . . . . . . . 10 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → 𝑚 = 𝐼)
39	38	breq2d 5077	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝐼))
40	39	ifbid 4488	. . . . . . . . . 10 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))
41	38, 40	ifeq12d 4486	. . . . . . . . 9 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
42	41	mpteq2dva 5160	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))))
43		psgnfzto1st.p	. . . . . . . 8 ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
44	42, 43	syl6eqr 2874	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
45	44	eleq1d 2897	. . . . . 6 ⊢ (𝑚 = 𝐼 → ((𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ 𝑃 ∈ 𝐵))
46	37, 45	imbi12d 347	. . . . 5 ⊢ (𝑚 = 𝐼 → ((𝑚 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝐼 ≤ 𝑁 → 𝑃 ∈ 𝐵)))
47		fzfi 13339	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
48	3, 47	eqeltri 2909	. . . . . . . 8 ⊢ 𝐷 ∈ Fin
49		psgnfzto1st.g	. . . . . . . . 9 ⊢ 𝐺 = (SymGrp‘𝐷)
50	49	idresperm 18513	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (Base‘𝐺))
51	48, 50	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ 𝐷) ∈ (Base‘𝐺)
52		psgnfzto1st.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
53	51, 52	eleqtrri 2912	. . . . . 6 ⊢ ( I ↾ 𝐷) ∈ 𝐵
54	53	2a1i 12	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵))
55		simplr 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
56	55	peano2nnd 11654	. . . . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
59	56, 57, 58	3jca 1124	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
60		elfz1b 12975	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
61	59, 60	sylibr 236	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
62	61, 3	eleqtrrdi 2924	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
63	3	psgnfzto1stlem 30742	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))))
64	55, 62, 63	syl2anc 586	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))))
65	64	adantlr 713	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))))
66		eqid 2821	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
67	66, 49, 52	symgtrf 18596	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) ⊆ 𝐵
68		eqid 2821	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
69	3, 68	pmtrto1cl 30741	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
70	55, 62, 69	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
71	70	adantlr 713	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
72	67, 71	sseldi 3964	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵)
73	55	nnred 11652	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
74		1red 10641	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
75	73, 74	readdcld 10669	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
76	57	nnred 11652	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
77	73	lep1d 11570	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
78	73, 75, 76, 77, 58	letrd 10796	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
79	78	adantlr 713	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
80		simplr 767	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵))
81	79, 80	mpd 15	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
82		eqid 2821	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
83	49, 52, 82	symgov 18511	. . . . . . . . 9 ⊢ ((((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})(+g‘𝐺)(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))))
84	49, 52, 82	symgcl 18512	. . . . . . . . 9 ⊢ ((((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})(+g‘𝐺)(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) ∈ 𝐵)
85	83, 84	eqeltrrd 2914	. . . . . . . 8 ⊢ ((((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) ∈ 𝐵)
86	72, 81, 85	syl2anc 586	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) ∈ 𝐵)
87	65, 86	eqeltrd 2913	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵)
88	87	ex 415	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)) → ((𝑛 + 1) ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) ∈ 𝐵))
89	20, 28, 36, 46, 54, 88	nnindd 30535	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼 ≤ 𝑁 → 𝑃 ∈ 𝐵))
90	89	imp 409	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁) → 𝑃 ∈ 𝐵)
91	7, 90	sylbi 219	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) → 𝑃 ∈ 𝐵)
92	6, 91	syl 17	1 ⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ifcif 4466  {cpr 4568   class class class wbr 5065   ↦ cmpt 5145   I cid 5458  ran crn 5555   ↾ cres 5556   ∘ ccom 5558  ‘cfv 6354  (class class class)co 7155  Fincfn 8508  1c1 10537   + caddc 10539   ≤ cle 10675   − cmin 10869  ℕcn 11637  ...cfz 12891  Basecbs 16482  +_gcplusg 16564  SymGrpcsymg 18494  pmTrspcpmtr 18568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-tset 16583  df-efmnd 18033  df-symg 18495  df-pmtr 18569

This theorem is referenced by:  fzto1stinvn  30746  psgnfzto1st  30747  madjusmdetlem2  31093  madjusmdetlem3  31094  madjusmdetlem4  31095