Theorem psgnfzto1st 33765

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 13767	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
2	1	biimpi 219	. . . 4 ⊢ (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
3		psgnfzto1st.d	. . . 4 ⊢ 𝐷 = (1...𝑁)
4	2, 3	eleq2s 2882	. . 3 ⊢ (𝐼 ∈ 𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
5		3ancoma 1115	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
6	4, 5	sylibr 237	. 2 ⊢ (𝐼 ∈ 𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁))
7		df-3an 1105	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁))
8		breq1 5111	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 ≤ 𝑁 ↔ 1 ≤ 𝑁))
9		id 23	. . . . . . . . . 10 ⊢ (𝑚 = 1 → 𝑚 = 1)
10		breq2 5112	. . . . . . . . . . 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 5205	. . . . . . . 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 6920	. . . . . . 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 7458	. . . . . . . 8 ⊢ (𝑚 = 1 → (𝑚 + 1) = (1 + 1))
16	15	oveq2d 7467	. . . . . . 7 ⊢ (𝑚 = 1 → ( -1↑(𝑚 + 1)) = ( -1↑(1 + 1)))
17	14, 16	eqeq12d 2779	. . . . . 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 347	. . . . 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 5111	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
20		id 23	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
21		breq2 5112	. . . . . . . . . . 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 5205	. . . . . . . 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 6920	. . . . . . 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 7458	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
27	26	oveq2d 7467	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ( -1↑(𝑚 + 1)) = ( -1↑(𝑛 + 1)))
28	25, 27	eqeq12d 2779	. . . . . 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 347	. . . . 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 5111	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ≤ 𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
31		id 23	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
32		breq2 5112	. . . . . . . . . . 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 5205	. . . . . . . 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 6920	. . . . . . 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 7458	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
38	37	oveq2d 7467	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → ( -1↑(𝑚 + 1)) = ( -1↑((𝑛 + 1) + 1)))
39	36, 38	eqeq12d 2779	. . . . . 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 347	. . . . 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 5111	. . . . . 6 ⊢ (𝑚 = 𝐼 → (𝑚 ≤ 𝑁 ↔ 𝐼 ≤ 𝑁))
42		id 23	. . . . . . . . . . 11 ⊢ (𝑚 = 𝐼 → 𝑚 = 𝐼)
43		breq2 5112	. . . . . . . . . . . 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 5205	. . . . . . . . 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 2816	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
49	48	fveq2d 6920	. . . . . . 7 ⊢ (𝑚 = 𝐼 → (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖 ≤ 𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘𝑃))
50		oveq1 7458	. . . . . . . 8 ⊢ (𝑚 = 𝐼 → (𝑚 + 1) = (𝐼 + 1))
51	50	oveq2d 7467	. . . . . . 7 ⊢ (𝑚 = 𝐼 → ( -1↑(𝑚 + 1)) = ( -1↑(𝐼 + 1)))
52	49, 51	eqeq12d 2779	. . . . . 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 347	. . . . 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 14155	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
55	3, 54	eqeltri 2860	. . . . . . . 8 ⊢ 𝐷 ∈ Fin
56		psgnfzto1st.s	. . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝐷)
57	56	psgnid 33757	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
58	55, 57	ax-mp 5	. . . . . . 7 ⊢ (𝑆‘( I ↾ 𝐷)) = 1
59		eqid 2763	. . . . . . . . 9 ⊢ 1 = 1
60		eqid 2763	. . . . . . . . . 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 33762	. . . . . . . . 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 6919	. . . . . . 7 ⊢ (𝑆‘(𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (𝑆‘( I ↾ 𝐷))
64		1p1e2 12506	. . . . . . . . 9 ⊢ (1 + 1) = 2
65	64	oveq2i 7462	. . . . . . . 8 ⊢ ( -1↑(1 + 1)) = ( -1↑2)
66		neg1sqe1 14379	. . . . . . . 8 ⊢ ( -1↑2) = 1
67	65, 66	eqtri 2786	. . . . . . 7 ⊢ ( -1↑(1 + 1)) = 1
68	58, 63, 67	3eqtr4i 2796	. . . . . 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 781	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
71	70	peano2nnd 12392	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
72		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
73		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
74	71, 72, 73	3jca 1146	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
75		elfz1b 13767	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
76	74, 75	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
77	76, 3	eleqtrrdi 2875	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
78	3	psgnfzto1stlem 33760	. . . . . . . . . 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 596	. . . . . . . . 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 728	. . . . . . . 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 6920	. . . . . . 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 2763	. . . . . . . . . 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 19734	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) ⊆ 𝐵
87		eqid 2763	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 87	pmtrto1cl 33759	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
89	70, 77, 88	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
90	89	adantlr 728	. . . . . . . . 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 3911	. . . . . . . 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 12390	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
93		1red 11348	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
94	92, 93	readdcld 11377	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
95	72	nnred 12390	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
96	92	lep1d 12288	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
97	92, 94, 95, 96, 73	letrd 11506	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ 𝑁)
98	70, 72, 97	3jca 1146	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛 ≤ 𝑁))
99		elfz1b 13767	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛 ≤ 𝑁))
100	98, 99	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ (1...𝑁))
101	100, 3	eleqtrrdi 2875	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ 𝐷)
102	101	adantlr 728	. . . . . . . . 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 2763	. . . . . . . . . 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 33763	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖 ≤ 𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
105	102, 104	syl 18	. . . . . . . 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 21944	. . . . . . . 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 1399	. . . . . . 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 19775	. . . . . . . . . . 11 ⊢ (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
109	89, 108	syl 18	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
110	109	adantlr 728	. . . . . . . . 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 728	. . . . . . . . . 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 781	. . . . . . . . . 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 16	. . . . . . . . 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 7469	. . . . . . . 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 12345	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
116		peano2nn 12387	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
117	116	nnnn0d 12707	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ0)
118		expp1 14251	. . . . . . . . . . 11 ⊢ (( -1 ∈ ℂ ∧ (𝑛 + 1) ∈ ℕ0) → ( -1↑((𝑛 + 1) + 1)) = (( -1↑(𝑛 + 1)) · -1))
119	115, 117, 118	sylancr 599	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ( -1↑((𝑛 + 1) + 1)) = (( -1↑(𝑛 + 1)) · -1))
120	115	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → -1 ∈ ℂ)
121	120, 117	expcld 14329	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ( -1↑(𝑛 + 1)) ∈ ℂ)
122	121, 120	mulcomd 11369	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (( -1↑(𝑛 + 1)) · -1) = ( -1 · ( -1↑(𝑛 + 1))))
123	119, 122	eqtr2d 2799	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ( -1 · ( -1↑(𝑛 + 1))) = ( -1↑((𝑛 + 1) + 1)))
124	123	ad3antlr 744	. . . . . . . 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 2798	. . . . . . 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 2802	. . . . . 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 418	. . . . 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 12395	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼 ≤ 𝑁 → (𝑆‘𝑃) = ( -1↑(𝐼 + 1))))
129	128	imp 412	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼 ≤ 𝑁) → (𝑆‘𝑃) = ( -1↑(𝐼 + 1)))
130	7, 129	sylbi 220	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝑁) → (𝑆‘𝑃) = ( -1↑(𝐼 + 1)))
131	6, 130	syl 18	1 ⊢ (𝐼 ∈ 𝐷 → (𝑆‘𝑃) = ( -1↑(𝐼 + 1)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ifcif 4485  {cpr 4590   class class class wbr 5108   ↦ cmpt 5192   I cid 5552  ran crn 5659   ↾ cres 5660   ∘ ccom 5662  ‘cfv 6566  (class class class)co 7451  Fincfn 9011  ℂcc 11237  1c1 11240   + caddc 11242   · cmul 11244   ≤ cle 11383   − cmin 11580   -cneg 11581  ℕcn 12375  2c2 12437  ℕ₀cn0 12646  ...cfz 13679  ↑cexp 14244  Basecbs 17431  SymGrpcsymg 19634  pmTrspcpmtr 19706  pmSgncpsgn 19754

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-addf 11318  ax-mulf 11319

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-xnn0 12720  df-z 12734  df-dec 12855  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-word 14700  df-lsw 14749  df-concat 14757  df-s1 14784  df-substr 14831  df-pfx 14863  df-splice 14941  df-reverse 14950  df-s2 15041  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-tset 17492  df-ple 17493  df-ds 17495  df-unif 17496  df-euf 17499  df-0g 17662  df-gsum 17663  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-efmnd 19115  df-grp 19197  df-minusg 19198  df-subg 19383  df-ghm 19478  df-gim 19523  df-oppg 19611  df-symg 19635  df-pmtr 19707  df-psgn 19756  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-unit 20639  df-invr 20669  df-dvr 20682  df-drng 21033  df-cnfld 21730

This theorem is used by:  madjusmdetlem4  34573