Theorem psgnfzto1st 33184

Description: The permutation sign for moving one element to the first position. (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)
psgnfzto1st.s	⊢ 𝑆 = (pmSgn‘𝐷)

Assertion

Ref	Expression
psgnfzto1st	⊢ (𝐼 ∈ 𝐷 → (𝑆‘𝑃) = (-1↑(𝐼 + 1)))

Distinct variable groups:   𝐷,𝑖   𝑖,𝐼   𝑖,𝑁   𝐵,𝑖

Allowed substitution hints:   𝑃(𝑖)   𝑆(𝑖)   𝐺(𝑖)

Proof of Theorem psgnfzto1st

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfz1b 13541	. . . . 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 2855	. . 3 ⊢ (𝐼 ∈ 𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
5		3ancoma 1098	. . 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 1089	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁))
8		breq1 5089	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 ≤ 𝑁 ↔ 1 ≤ 𝑁))
9		id 22	. . . . . . . . . 10 ⊢ (𝑚 = 1 → 𝑚 = 1)
10		breq2 5090	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 1))
11	10	ifbid 4491	. . . . . . . . . 10 ⊢ (𝑚 = 1 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
12	9, 11	ifeq12d 4489	. . . . . . . . 9 ⊢ (𝑚 = 1 → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
13	12	mpteq2dv 5180	. . . . . . . 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	13	fveq2d 6839	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))))
15		oveq1 7368	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝑚 + 1) = (1 + 1))
16	15	oveq2d 7377	. . . . . . 7 ⊢ (𝑚 = 1 → (-1↑(𝑚 + 1)) = (-1↑(1 + 1)))
17	14, 16	eqeq12d 2753	. . . . . 6 ⊢ (𝑚 = 1 → ((𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1))))
18	8, 17	imbi12d 344	. . . . 5 ⊢ (𝑚 = 1 → ((𝑚 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ (1 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1)))))
19		breq1 5089	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
20		id 22	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
21		breq2 5090	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝑛))
22	21	ifbid 4491	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))
23	20, 22	ifeq12d 4489	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))
24	23	mpteq2dv 5180	. . . . . . . 8 ⊢ (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))))
25	24	fveq2d 6839	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))))
26		oveq1 7368	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
27	26	oveq2d 7377	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (-1↑(𝑚 + 1)) = (-1↑(𝑛 + 1)))
28	25, 27	eqeq12d 2753	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1))))
29	19, 28	imbi12d 344	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))))
30		breq1 5089	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ 𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
31		id 22	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
32		breq2 5090	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ (𝑛 + 1)))
33	32	ifbid 4491	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
34	31, 33	ifeq12d 4489	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))
35	34	mpteq2dv 5180	. . . . . . . 8 ⊢ (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))))
36	35	fveq2d 6839	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))))
37		oveq1 7368	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
38	37	oveq2d 7377	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (-1↑(𝑚 + 1)) = (-1↑((𝑛 + 1) + 1)))
39	36, 38	eqeq12d 2753	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1))))
40	30, 39	imbi12d 344	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ ((𝑛 + 1) ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1)))))
41		breq1 5089	. . . . . 6 ⊢ (𝑚 = 𝐼 → (𝑚 ≤ 𝑁 ↔ 𝐼 ≤ 𝑁))
42		id 22	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → 𝑚 = 𝐼)
43		breq2 5090	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝐼 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝐼))
44	43	ifbid 4491	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))
45	42, 44	ifeq12d 4489	. . . . . . . . . 10 ⊢ (𝑚 = 𝐼 → if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
46	45	mpteq2dv 5180	. . . . . . . . 9 ⊢ (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))))
47		psgnfzto1st.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
48	46, 47	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
49	48	fveq2d 6839	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘𝑃))
50		oveq1 7368	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑚 + 1) = (𝐼 + 1))
51	50	oveq2d 7377	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (-1↑(𝑚 + 1)) = (-1↑(𝐼 + 1)))
52	49, 51	eqeq12d 2753	. . . . . 6 ⊢ (𝑚 = 𝐼 → ((𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘𝑃) = (-1↑(𝐼 + 1))))
53	41, 52	imbi12d 344	. . . . 5 ⊢ (𝑚 = 𝐼 → ((𝑚 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ (𝐼 ≤ 𝑁 → (𝑆‘𝑃) = (-1↑(𝐼 + 1)))))
54		fzfi 13928	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
55	3, 54	eqeltri 2833	. . . . . . . 8 ⊢ 𝐷 ∈ Fin
56		psgnfzto1st.s	. . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝐷)
57	56	psgnid 33176	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
58	55, 57	ax-mp 5	. . . . . . 7 ⊢ (𝑆‘( I ↾ 𝐷)) = 1
59		eqid 2737	. . . . . . . . 9 ⊢ 1 = 1
60		eqid 2737	. . . . . . . . . 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)))
61	3, 60	fzto1st1 33181	. . . . . . . . 9 ⊢ (1 = 1 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
62	59, 61	ax-mp 5	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷)
63	62	fveq2i 6838	. . . . . . 7 ⊢ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (𝑆‘( I ↾ 𝐷))
64		1p1e2 12295	. . . . . . . . 9 ⊢ (1 + 1) = 2
65	64	oveq2i 7372	. . . . . . . 8 ⊢ (-1↑(1 + 1)) = (-1↑2)
66		neg1sqe1 14152	. . . . . . . 8 ⊢ (-1↑2) = 1
67	65, 66	eqtri 2760	. . . . . . 7 ⊢ (-1↑(1 + 1)) = 1
68	58, 63, 67	3eqtr4i 2770	. . . . . 6 ⊢ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1))
69	68	2a1i 12	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1))))
70		simplr 769	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
71	70	peano2nnd 12185	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
72		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
73		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
74	71, 72, 73	3jca 1129	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
75		elfz1b 13541	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
76	74, 75	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
77	76, 3	eleqtrrdi 2848	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
78	3	psgnfzto1stlem 33179	. . . . . . . . . 10 ⊢ ((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)))))
79	70, 77, 78	syl2anc 585	. . . . . . . . 9 ⊢ (((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)))))
80	79	adantlr 716	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (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)))))
81	80	fveq2d 6839	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (𝑆‘(((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))))
82	55	a1i 11	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → 𝐷 ∈ Fin)
83		eqid 2737	. . . . . . . . . 10 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
84		psgnfzto1st.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
85		psgnfzto1st.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
86	83, 84, 85	symgtrf 19438	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) ⊆ 𝐵
87		eqid 2737	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 87	pmtrto1cl 33178	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
89	70, 77, 88	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
90	89	adantlr 716	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
91	86, 90	sselid 3920	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵)
92	70	nnred 12183	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
93		1red 11139	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
94	92, 93	readdcld 11168	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
95	72	nnred 12183	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
96	92	lep1d 12081	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
97	92, 94, 95, 96, 73	letrd 11297	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
98	70, 72, 97	3jca 1129	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛 ≤ 𝑁))
99		elfz1b 13541	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛 ≤ 𝑁))
100	98, 99	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ (1...𝑁))
101	100, 3	eleqtrrdi 2848	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ 𝐷)
102	101	adantlr 716	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ 𝐷)
103		eqid 2737	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))
104	3, 103, 84, 85	fzto1st 33182	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
105	102, 104	syl 17	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
106	84, 56, 85	psgnco 21576	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (𝑆‘(((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))) = ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))))
107	82, 91, 105, 106	syl3anc 1374	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))) = ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))))
108	84, 83, 56	psgnpmtr 19479	. . . . . . . . . . 11 ⊢ (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
109	89, 108	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
110	109	adantlr 716	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
111	97	adantlr 716	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
112		simplr 769	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1))))
113	111, 112	mpd 15	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))
114	110, 113	oveq12d 7379	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))) = (-1 · (-1↑(𝑛 + 1))))
115		neg1cn 12138	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
116		peano2nn 12180	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
117	116	nnnn0d 12492	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ0)
118		expp1 14024	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (𝑛 + 1) ∈ ℕ0) → (-1↑((𝑛 + 1) + 1)) = ((-1↑(𝑛 + 1)) · -1))
119	115, 117, 118	sylancr 588	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (-1↑((𝑛 + 1) + 1)) = ((-1↑(𝑛 + 1)) · -1))
120	115	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → -1 ∈ ℂ)
121	120, 117	expcld 14102	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (-1↑(𝑛 + 1)) ∈ ℂ)
122	121, 120	mulcomd 11160	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((-1↑(𝑛 + 1)) · -1) = (-1 · (-1↑(𝑛 + 1))))
123	119, 122	eqtr2d 2773	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (-1 · (-1↑(𝑛 + 1))) = (-1↑((𝑛 + 1) + 1)))
124	123	ad3antlr 732	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (-1 · (-1↑(𝑛 + 1))) = (-1↑((𝑛 + 1) + 1)))
125	114, 124	eqtrd 2772	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))))) = (-1↑((𝑛 + 1) + 1)))
126	81, 107, 125	3eqtrd 2776	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1)))
127	126	ex 412	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) → ((𝑛 + 1) ≤ 𝑁 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1))))
128	18, 29, 40, 53, 69, 127	nnindd 12188	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼 ≤ 𝑁 → (𝑆‘𝑃) = (-1↑(𝐼 + 1))))
129	128	imp 406	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁) → (𝑆‘𝑃) = (-1↑(𝐼 + 1)))
130	7, 129	sylbi 217	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) → (𝑆‘𝑃) = (-1↑(𝐼 + 1)))
131	6, 130	syl 17	1 ⊢ (𝐼 ∈ 𝐷 → (𝑆‘𝑃) = (-1↑(𝐼 + 1)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ifcif 4467  {cpr 4570   class class class wbr 5086   ↦ cmpt 5167   I cid 5519  ran crn 5626   ↾ cres 5627   ∘ ccom 5629  ‘cfv 6493  (class class class)co 7361  Fincfn 8887  ℂcc 11030  1c1 11033   + caddc 11035   · cmul 11037   ≤ cle 11174   − cmin 11371  -cneg 11372  ℕcn 12168  2c2 12230  ℕ₀cn0 12431  ...cfz 13455  ↑cexp 14017  Basecbs 17173  SymGrpcsymg 19338  pmTrspcpmtr 19410  pmSgncpsgn 19458

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-addf 11111  ax-mulf 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-xnn0 12505  df-z 12519  df-dec 12639  df-uz 12783  df-rp 12937  df-fz 13456  df-fzo 13603  df-seq 13958  df-exp 14018  df-hash 14287  df-word 14470  df-lsw 14519  df-concat 14527  df-s1 14553  df-substr 14598  df-pfx 14628  df-splice 14706  df-reverse 14715  df-s2 14804  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-0g 17398  df-gsum 17399  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-efmnd 18831  df-grp 18906  df-minusg 18907  df-subg 19093  df-ghm 19182  df-gim 19228  df-oppg 19315  df-symg 19339  df-pmtr 19411  df-psgn 19460  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-oppr 20311  df-dvdsr 20331  df-unit 20332  df-invr 20362  df-dvr 20375  df-drng 20702  df-cnfld 21348

This theorem is referenced by:  madjusmdetlem4  33993