Theorem fzto1st 33114

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 13610	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
2	1	biimpi 216	. . . 4 ⊢ (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
3		psgnfzto1st.d	. . . 4 ⊢ 𝐷 = (1...𝑁)
4	2, 3	eleq2s 2852	. . 3 ⊢ (𝐼 ∈ 𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
5		3ancoma 1097	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
6	4, 5	sylibr 234	. 2 ⊢ (𝐼 ∈ 𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
7		df-3an 1088	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁))
8		breq1 5122	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 ≤ 𝑁 ↔ 1 ≤ 𝑁))
9		simpl 482	. . . . . . . . . 10 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → 𝑚 = 1)
10	9	breq2d 5131	. . . . . . . . . . 11 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 1))
11	10	ifbid 4524	. . . . . . . . . 10 ⊢ ((𝑚 = 1 ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
12	9, 11	ifeq12d 4522	. . . . . . . . 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 5214	. . . . . . . 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 2735	. . . . . . . . 9 ⊢ 1 = 1
15		eqid 2735	. . . . . . . . . 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 33113	. . . . . . . . 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	eqtrdi 2786	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
19	18	eleq1d 2819	. . . . . 6 ⊢ (𝑚 = 1 → ((𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ ( I ↾ 𝐷) ∈ 𝐵))
20	8, 19	imbi12d 344	. . . . 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 5122	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
22		simpl 482	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → 𝑚 = 𝑛)
23	22	breq2d 5131	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝑛))
24	23	ifbid 4524	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))
25	22, 24	ifeq12d 4522	. . . . . . . 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 5214	. . . . . . 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 2819	. . . . . 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 344	. . . . 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 5122	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ 𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
30		simpl 482	. . . . . . . . 9 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → 𝑚 = (𝑛 + 1))
31	30	breq2d 5131	. . . . . . . . . 10 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ (𝑛 + 1)))
32	31	ifbid 4524	. . . . . . . . 9 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
33	30, 32	ifeq12d 4522	. . . . . . . 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 5214	. . . . . . 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 2819	. . . . . 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 344	. . . . 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 5122	. . . . . 6 ⊢ (𝑚 = 𝐼 → (𝑚 ≤ 𝑁 ↔ 𝐼 ≤ 𝑁))
38		simpl 482	. . . . . . . . . 10 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → 𝑚 = 𝐼)
39	38	breq2d 5131	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝐼))
40	39	ifbid 4524	. . . . . . . . . 10 ⊢ ((𝑚 = 𝐼 ∧ 𝑖 ∈ 𝐷) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))
41	38, 40	ifeq12d 4522	. . . . . . . . 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 5214	. . . . . . . 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	eqtr4di 2788	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
45	44	eleq1d 2819	. . . . . 6 ⊢ (𝑚 = 𝐼 → ((𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵 ↔ 𝑃 ∈ 𝐵))
46	37, 45	imbi12d 344	. . . . 5 ⊢ (𝑚 = 𝐼 → ((𝑚 ≤ 𝑁 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) ∈ 𝐵) ↔ (𝐼 ≤ 𝑁 → 𝑃 ∈ 𝐵)))
47		fzfi 13990	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
48	3, 47	eqeltri 2830	. . . . . . . 8 ⊢ 𝐷 ∈ Fin
49		psgnfzto1st.g	. . . . . . . . 9 ⊢ 𝐺 = (SymGrp‘𝐷)
50	49	idresperm 19367	. . . . . . . 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 2833	. . . . . 6 ⊢ ( I ↾ 𝐷) ∈ 𝐵
54	53	2a1i 12	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 → ( I ↾ 𝐷) ∈ 𝐵))
55		simplr 768	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
56	55	peano2nnd 12257	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
59	56, 57, 58	3jca 1128	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
60		elfz1b 13610	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
61	59, 60	sylibr 234	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
62	61, 3	eleqtrrdi 2845	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
63	3	psgnfzto1stlem 33111	. . . . . . . . 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 584	. . . . . . . 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 715	. . . . . . 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 2735	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
67	66, 49, 52	symgtrf 19450	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) ⊆ 𝐵
68		eqid 2735	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
69	3, 68	pmtrto1cl 33110	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
70	55, 62, 69	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
71	70	adantlr 715	. . . . . . . . 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	sselid 3956	. . . . . . . 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 12255	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
74		1red 11236	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
75	73, 74	readdcld 11264	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
76	57	nnred 12255	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
77	73	lep1d 12173	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
78	73, 75, 76, 77, 58	letrd 11392	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
79	78	adantlr 715	. . . . . . . . 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 768	. . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
83	49, 52, 82	symgov 19365	. . . . . . . . 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 19366	. . . . . . . . 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 2835	. . . . . . . 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 584	. . . . . . 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 2834	. . . . . 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 412	. . . . 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 12260	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼 ≤ 𝑁 → 𝑃 ∈ 𝐵))
90	89	imp 406	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁) → 𝑃 ∈ 𝐵)
91	7, 90	sylbi 217	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) → 𝑃 ∈ 𝐵)
92	6, 91	syl 17	1 ⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ifcif 4500  {cpr 4603   class class class wbr 5119   ↦ cmpt 5201   I cid 5547  ran crn 5655   ↾ cres 5656   ∘ ccom 5658  ‘cfv 6531  (class class class)co 7405  Fincfn 8959  1c1 11130   + caddc 11132   ≤ cle 11270   − cmin 11466  ℕcn 12240  ...cfz 13524  Basecbs 17228  +_gcplusg 17271  SymGrpcsymg 19350  pmTrspcpmtr 19422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-tset 17290  df-efmnd 18847  df-symg 19351  df-pmtr 19423

This theorem is referenced by:  fzto1stinvn  33115  psgnfzto1st  33116  madjusmdetlem2  33859  madjusmdetlem3  33860  madjusmdetlem4  33861