Theorem psgnfzto1st 33166

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 13547	. . . . 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 2854	. . 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 5088	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 ≤ 𝑁 ↔ 1 ≤ 𝑁))
9		id 22	. . . . . . . . . 10 ⊢ (𝑚 = 1 → 𝑚 = 1)
10		breq2 5089	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 1))
11	10	ifbid 4490	. . . . . . . . . 10 ⊢ (𝑚 = 1 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
12	9, 11	ifeq12d 4488	. . . . . . . . 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 5179	. . . . . . . 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 6844	. . . . . . 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 7374	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝑚 + 1) = (1 + 1))
16	15	oveq2d 7383	. . . . . . 7 ⊢ (𝑚 = 1 → (-1↑(𝑚 + 1)) = (-1↑(1 + 1)))
17	14, 16	eqeq12d 2752	. . . . . 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 5088	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
20		id 22	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
21		breq2 5089	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝑛))
22	21	ifbid 4490	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))
23	20, 22	ifeq12d 4488	. . . . . . . . 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 5179	. . . . . . . 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 6844	. . . . . . 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 7374	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
27	26	oveq2d 7383	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (-1↑(𝑚 + 1)) = (-1↑(𝑛 + 1)))
28	25, 27	eqeq12d 2752	. . . . . 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 5088	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ 𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
31		id 22	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
32		breq2 5089	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ (𝑛 + 1)))
33	32	ifbid 4490	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
34	31, 33	ifeq12d 4488	. . . . . . . . 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 5179	. . . . . . . 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 6844	. . . . . . 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 7374	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
38	37	oveq2d 7383	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (-1↑(𝑚 + 1)) = (-1↑((𝑛 + 1) + 1)))
39	36, 38	eqeq12d 2752	. . . . . 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 5088	. . . . . 6 ⊢ (𝑚 = 𝐼 → (𝑚 ≤ 𝑁 ↔ 𝐼 ≤ 𝑁))
42		id 22	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → 𝑚 = 𝐼)
43		breq2 5089	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝐼 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝐼))
44	43	ifbid 4490	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))
45	42, 44	ifeq12d 4488	. . . . . . . . . 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 5179	. . . . . . . . 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 2789	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
49	48	fveq2d 6844	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘𝑃))
50		oveq1 7374	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑚 + 1) = (𝐼 + 1))
51	50	oveq2d 7383	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (-1↑(𝑚 + 1)) = (-1↑(𝐼 + 1)))
52	49, 51	eqeq12d 2752	. . . . . 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 13934	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
55	3, 54	eqeltri 2832	. . . . . . . 8 ⊢ 𝐷 ∈ Fin
56		psgnfzto1st.s	. . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝐷)
57	56	psgnid 33158	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
58	55, 57	ax-mp 5	. . . . . . 7 ⊢ (𝑆‘( I ↾ 𝐷)) = 1
59		eqid 2736	. . . . . . . . 9 ⊢ 1 = 1
60		eqid 2736	. . . . . . . . . 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 33163	. . . . . . . . 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 6843	. . . . . . 7 ⊢ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (𝑆‘( I ↾ 𝐷))
64		1p1e2 12301	. . . . . . . . 9 ⊢ (1 + 1) = 2
65	64	oveq2i 7378	. . . . . . . 8 ⊢ (-1↑(1 + 1)) = (-1↑2)
66		neg1sqe1 14158	. . . . . . . 8 ⊢ (-1↑2) = 1
67	65, 66	eqtri 2759	. . . . . . 7 ⊢ (-1↑(1 + 1)) = 1
68	58, 63, 67	3eqtr4i 2769	. . . . . 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 12191	. . . . . . . . . . . . 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 13547	. . . . . . . . . . . 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 2847	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
78	3	psgnfzto1stlem 33161	. . . . . . . . . 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 6844	. . . . . . 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 2736	. . . . . . . . . 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 19444	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) ⊆ 𝐵
87		eqid 2736	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 87	pmtrto1cl 33160	. . . . . . . . . . 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 3919	. . . . . . . 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 12189	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
93		1red 11145	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
94	92, 93	readdcld 11174	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
95	72	nnred 12189	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
96	92	lep1d 12087	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
97	92, 94, 95, 96, 73	letrd 11303	. . . . . . . . . . . . 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 13547	. . . . . . . . . . . 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 2847	. . . . . . . . . 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 2736	. . . . . . . . . 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 33164	. . . . . . . . 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 21563	. . . . . . . 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 19485	. . . . . . . . . . 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 7385	. . . . . . . 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 12144	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
116		peano2nn 12186	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
117	116	nnnn0d 12498	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ0)
118		expp1 14030	. . . . . . . . . . 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 14108	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (-1↑(𝑛 + 1)) ∈ ℂ)
122	121, 120	mulcomd 11166	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((-1↑(𝑛 + 1)) · -1) = (-1 · (-1↑(𝑛 + 1))))
123	119, 122	eqtr2d 2772	. . . . . . . . 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 2771	. . . . . . 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 2775	. . . . . 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 12194	. . . 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 4466  {cpr 4569   class class class wbr 5085   ↦ cmpt 5166   I cid 5525  ran crn 5632   ↾ cres 5633   ∘ ccom 5635  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  ℂcc 11036  1c1 11039   + caddc 11041   · cmul 11043   ≤ cle 11180   − cmin 11377  -cneg 11378  ℕcn 12174  2c2 12236  ℕ₀cn0 12437  ...cfz 13461  ↑cexp 14023  Basecbs 17179  SymGrpcsymg 19344  pmTrspcpmtr 19416  pmSgncpsgn 19464

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-addf 11117  ax-mulf 11118

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-splice 14712  df-reverse 14721  df-s2 14810  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-0g 17404  df-gsum 17405  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-efmnd 18837  df-grp 18912  df-minusg 18913  df-subg 19099  df-ghm 19188  df-gim 19234  df-oppg 19321  df-symg 19345  df-pmtr 19417  df-psgn 19466  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-dvr 20381  df-drng 20708  df-cnfld 21353

This theorem is referenced by:  madjusmdetlem4  33974