Theorem psgnfzto1st 31954

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 13510	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
2	1	biimpi 215	. . . 4 ⊢ (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
3		psgnfzto1st.d	. . . 4 ⊢ 𝐷 = (1...𝑁)
4	2, 3	eleq2s 2856	. . 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 233	. 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 5108	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 ≤ 𝑁 ↔ 1 ≤ 𝑁))
9		id 22	. . . . . . . . . 10 ⊢ (𝑚 = 1 → 𝑚 = 1)
10		breq2 5109	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 1))
11	10	ifbid 4509	. . . . . . . . . 10 ⊢ (𝑚 = 1 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
12	9, 11	ifeq12d 4507	. . . . . . . . 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 5207	. . . . . . . 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 6846	. . . . . . 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 7364	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝑚 + 1) = (1 + 1))
16	15	oveq2d 7373	. . . . . . 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 5108	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
20		id 22	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
21		breq2 5109	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝑛))
22	21	ifbid 4509	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))
23	20, 22	ifeq12d 4507	. . . . . . . . 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 5207	. . . . . . . 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 6846	. . . . . . 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 7364	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
27	26	oveq2d 7373	. . . . . . 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 5108	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ 𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
31		id 22	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
32		breq2 5109	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ (𝑛 + 1)))
33	32	ifbid 4509	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
34	31, 33	ifeq12d 4507	. . . . . . . . 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 5207	. . . . . . . 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 6846	. . . . . . 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 7364	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
38	37	oveq2d 7373	. . . . . . 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 5108	. . . . . 6 ⊢ (𝑚 = 𝐼 → (𝑚 ≤ 𝑁 ↔ 𝐼 ≤ 𝑁))
42		id 22	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → 𝑚 = 𝐼)
43		breq2 5109	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝐼 → (𝑖 ≤ 𝑚 ↔ 𝑖 ≤ 𝐼))
44	43	ifbid 4509	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))
45	42, 44	ifeq12d 4507	. . . . . . . . . 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 5207	. . . . . . . . 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 2794	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
49	48	fveq2d 6846	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘𝑃))
50		oveq1 7364	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑚 + 1) = (𝐼 + 1))
51	50	oveq2d 7373	. . . . . . 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 13877	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
55	3, 54	eqeltri 2834	. . . . . . . 8 ⊢ 𝐷 ∈ Fin
56		psgnfzto1st.s	. . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝐷)
57	56	psgnid 31946	. . . . . . . 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 31951	. . . . . . . . 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 6845	. . . . . . 7 ⊢ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (𝑆‘( I ↾ 𝐷))
64		1p1e2 12278	. . . . . . . . 9 ⊢ (1 + 1) = 2
65	64	oveq2i 7368	. . . . . . . 8 ⊢ (-1↑(1 + 1)) = (-1↑2)
66		neg1sqe1 14100	. . . . . . . 8 ⊢ (-1↑2) = 1
67	65, 66	eqtri 2764	. . . . . . 7 ⊢ (-1↑(1 + 1)) = 1
68	58, 63, 67	3eqtr4i 2774	. . . . . 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 767	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
71	70	peano2nnd 12170	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
72		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
73		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
74	71, 72, 73	3jca 1128	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
75		elfz1b 13510	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
76	74, 75	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
77	76, 3	eleqtrrdi 2849	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
78	3	psgnfzto1stlem 31949	. . . . . . . . . 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 584	. . . . . . . . 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 713	. . . . . . . 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 6846	. . . . . . 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 19251	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) ⊆ 𝐵
87		eqid 2736	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 87	pmtrto1cl 31948	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
89	70, 77, 88	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
90	89	adantlr 713	. . . . . . . . 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 3942	. . . . . . . 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 12168	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
93		1red 11156	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
94	92, 93	readdcld 11184	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
95	72	nnred 12168	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
96	92	lep1d 12086	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
97	92, 94, 95, 96, 73	letrd 11312	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
98	70, 72, 97	3jca 1128	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛 ≤ 𝑁))
99		elfz1b 13510	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛 ≤ 𝑁))
100	98, 99	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ (1...𝑁))
101	100, 3	eleqtrrdi 2849	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ 𝐷)
102	101	adantlr 713	. . . . . . . . 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 31952	. . . . . . . . 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 20987	. . . . . . . 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 1371	. . . . . . 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 19292	. . . . . . . . . . 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 713	. . . . . . . . 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 713	. . . . . . . . . 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 767	. . . . . . . . . 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 7375	. . . . . . . 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 12267	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
116		peano2nn 12165	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
117	116	nnnn0d 12473	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ0)
118		expp1 13974	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (𝑛 + 1) ∈ ℕ0) → (-1↑((𝑛 + 1) + 1)) = ((-1↑(𝑛 + 1)) · -1))
119	115, 117, 118	sylancr 587	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (-1↑((𝑛 + 1) + 1)) = ((-1↑(𝑛 + 1)) · -1))
120	115	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → -1 ∈ ℂ)
121	120, 117	expcld 14051	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (-1↑(𝑛 + 1)) ∈ ℂ)
122	121, 120	mulcomd 11176	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((-1↑(𝑛 + 1)) · -1) = (-1 · (-1↑(𝑛 + 1))))
123	119, 122	eqtr2d 2777	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (-1 · (-1↑(𝑛 + 1))) = (-1↑((𝑛 + 1) + 1)))
124	123	ad3antlr 729	. . . . . . . 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 2776	. . . . . . 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 2780	. . . . . 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 413	. . . . 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 12173	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼 ≤ 𝑁 → (𝑆‘𝑃) = (-1↑(𝐼 + 1))))
129	128	imp 407	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁) → (𝑆‘𝑃) = (-1↑(𝐼 + 1)))
130	7, 129	sylbi 216	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ifcif 4486  {cpr 4588   class class class wbr 5105   ↦ cmpt 5188   I cid 5530  ran crn 5634   ↾ cres 5635   ∘ ccom 5637  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  ℂcc 11049  1c1 11052   + caddc 11054   · cmul 11056   ≤ cle 11190   − cmin 11385  -cneg 11386  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ...cfz 13424  ↑cexp 13967  Basecbs 17083  SymGrpcsymg 19148  pmTrspcpmtr 19223  pmSgncpsgn 19271

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-reverse 14647  df-s2 14737  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-minusg 18752  df-subg 18925  df-ghm 19006  df-gim 19049  df-oppg 19124  df-symg 19149  df-pmtr 19224  df-psgn 19273  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-cnfld 20797

This theorem is referenced by:  madjusmdetlem4  32411