Theorem evpmodpmf1o 21636

Description: The function for performing an even permutation after a fixed odd permutation is one to one onto all odd permutations. (Contributed by SO, 9-Jul-2018.)

Hypotheses

Ref	Expression
evpmodpmf1o.s	⊢ 𝑆 = (SymGrp‘𝐷)
evpmodpmf1o.p	⊢ 𝑃 = (Base‘𝑆)

Assertion

Ref	Expression
evpmodpmf1o	⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)):(pmEven‘𝐷)–1-1-onto→(𝑃 ∖ (pmEven‘𝐷)))

Distinct variable groups:   𝑆,𝑓   𝐷,𝑓   𝑃,𝑓   𝑓,𝐹

Proof of Theorem evpmodpmf1o

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 776	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin)
2		evpmodpmf1o.s	. . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷)
3	2	symggrp 19431	. . . . . 6 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
4	3	ad2antrr 736	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp)
5		eldifi 4082	. . . . . 6 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹 ∈ 𝑃)
6	5	ad2antlr 737	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹 ∈ 𝑃)
7		evpmodpmf1o.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝑆)
8	2, 7	evpmss 21626	. . . . . . 7 ⊢ (pmEven‘𝐷) ⊆ 𝑃
9	8	sseli 3930	. . . . . 6 ⊢ (𝑓 ∈ (pmEven‘𝐷) → 𝑓 ∈ 𝑃)
10	9	adantl 485	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓 ∈ 𝑃)
11		eqid 2761	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
12	7, 11	grpcl 18974	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃 ∧ 𝑓 ∈ 𝑃) → (𝐹(+g‘𝑆)𝑓) ∈ 𝑃)
13	4, 6, 10, 12	syl3anc 1389	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g‘𝑆)𝑓) ∈ 𝑃)
14		eqid 2761	. . . . . . . 8 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
15		eqid 2761	. . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})
16	2, 14, 15	psgnghm2 21621	. . . . . . 7 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
17	16	ad2antrr 736	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
18		prex 5392	. . . . . . . 8 ⊢ {1, -1} ∈ V
19		eqid 2761	. . . . . . . . . 10 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20		cnfldmul 21420	. . . . . . . . . 10 ⊢ · = (.r‘ℂfld)
21	19, 20	mgpplusg 20181	. . . . . . . . 9 ⊢ · = (+g‘(mulGrp‘ℂfld))
22	15, 21	ressplusg 17311	. . . . . . . 8 ⊢ ({1, -1} ∈ V → · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1})))
23	18, 22	ax-mp 5	. . . . . . 7 ⊢ · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1}))
24	7, 11, 23	ghmlin 19252	. . . . . 6 ⊢ (((pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ 𝑃 ∧ 𝑓 ∈ 𝑃) → ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
25	17, 6, 10, 24	syl3anc 1389	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
26	2, 7, 14	psgnodpm 21628	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
27	26	adantr 484	. . . . . . 7 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝐹) = -1)
28	2, 7, 14	psgnevpm 21629	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
29	28	adantlr 725	. . . . . . 7 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
30	27, 29	oveq12d 7409	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1))
31		ax-1cn 11125	. . . . . . 7 ⊢ 1 ∈ ℂ
32	31	mulm1i 11626	. . . . . 6 ⊢ (-1 · 1) = -1
33	30, 32	eqtrdi 2812	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1)
34	25, 33	eqtrd 2796	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = -1)
35	2, 7, 14	psgnodpmr 21630	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ (𝐹(+g‘𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = -1) → (𝐹(+g‘𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
36	1, 13, 34, 35	syl3anc 1389	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g‘𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
37	36	fmpttd 7091	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷)))
38	3	ad2antrr 736	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp)
39		eqid 2761	. . . . . . . 8 ⊢ (invg‘𝑆) = (invg‘𝑆)
40	7, 39	grpinvcl 19020	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃) → ((invg‘𝑆)‘𝐹) ∈ 𝑃)
41	3, 5, 40	syl2an 605	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg‘𝑆)‘𝐹) ∈ 𝑃)
42	41	adantr 484	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg‘𝑆)‘𝐹) ∈ 𝑃)
43		eldifi 4082	. . . . . 6 ⊢ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔 ∈ 𝑃)
44	43	adantl 485	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔 ∈ 𝑃)
45	7, 11	grpcl 18974	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ ((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝑔 ∈ 𝑃) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃)
46	38, 42, 44, 45	syl3anc 1389	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃)
47	16	ad2antrr 736	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
48	7, 11, 23	ghmlin 19252	. . . . . 6 ⊢ (((pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) ∧ ((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝑔 ∈ 𝑃) → ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)))
49	47, 42, 44, 48	syl3anc 1389	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)))
50	2, 7, 39	symginv 19433	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑃 → ((invg‘𝑆)‘𝐹) = ◡𝐹)
51	5, 50	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg‘𝑆)‘𝐹) = ◡𝐹)
52	51	ad2antlr 737	. . . . . . 7 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg‘𝑆)‘𝐹) = ◡𝐹)
53	52	fveq2d 6866	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘◡𝐹))
54	2, 7, 14	psgnodpm 21628	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
55	54	adantlr 725	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
56	53, 55	oveq12d 7409	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘◡𝐹) · -1))
57		simpll 776	. . . . . . . . 9 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin)
58	5	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹 ∈ 𝑃)
59	2, 14, 7	psgninv 21622	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((pmSgn‘𝐷)‘◡𝐹) = ((pmSgn‘𝐷)‘𝐹))
60	57, 58, 59	syl2anc 593	. . . . . . . 8 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘◡𝐹) = ((pmSgn‘𝐷)‘𝐹))
61	26	adantr 484	. . . . . . . 8 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
62	60, 61	eqtrd 2796	. . . . . . 7 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘◡𝐹) = -1)
63	62	oveq1d 7406	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘◡𝐹) · -1) = (-1 · -1))
64		neg1mulneg1e1 12427	. . . . . 6 ⊢ (-1 · -1) = 1
65	63, 64	eqtrdi 2812	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘◡𝐹) · -1) = 1)
66	49, 56, 65	3eqtrd 2800	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 1)
67	2, 7, 14	psgnevpmb 21627	. . . . 5 ⊢ (𝐷 ∈ Fin → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 1)))
68	67	ad2antrr 736	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 1)))
69	46, 66, 68	mpbir2and 723	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ (pmEven‘𝐷))
70	69	fmpttd 7091	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷))
71		eqidd 2762	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)))
72		eqidd 2762	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)))
73		oveq2 7399	. . . . 5 ⊢ (𝑔 = (𝐹(+g‘𝑆)𝑓) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) = (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓)))
74	36, 71, 72, 73	fmptco 7106	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓))))
75		eqid 2761	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
76	7, 11, 75, 39	grplinv 19022	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹) = (0g‘𝑆))
77	4, 6, 76	syl2anc 593	. . . . . . 7 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹) = (0g‘𝑆))
78	77	oveq1d 7406	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹)(+g‘𝑆)𝑓) = ((0g‘𝑆)(+g‘𝑆)𝑓))
79	41	adantr 484	. . . . . . 7 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg‘𝑆)‘𝐹) ∈ 𝑃)
80	7, 11	grpass 18975	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ (((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝐹 ∈ 𝑃 ∧ 𝑓 ∈ 𝑃)) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹)(+g‘𝑆)𝑓) = (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓)))
81	4, 79, 6, 10, 80	syl13anc 1390	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹)(+g‘𝑆)𝑓) = (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓)))
82	7, 11, 75	grplid 19000	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ 𝑓 ∈ 𝑃) → ((0g‘𝑆)(+g‘𝑆)𝑓) = 𝑓)
83	4, 10, 82	syl2anc 593	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g‘𝑆)(+g‘𝑆)𝑓) = 𝑓)
84	78, 81, 83	3eqtr3d 2804	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓)) = 𝑓)
85	84	mpteq2dva 5190	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
86	74, 85	eqtrd 2796	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
87		mptresid 6036	. . 3 ⊢ ( I ↾ (pmEven‘𝐷)) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓)
88	86, 87	eqtr4di 2814	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) = ( I ↾ (pmEven‘𝐷)))
89		oveq2 7399	. . . . 5 ⊢ (𝑓 = (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) → (𝐹(+g‘𝑆)𝑓) = (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)))
90	69, 72, 71, 89	fmptco 7106	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))))
91	7, 11, 75, 39	grprinv 19023	. . . . . . . . 9 ⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃) → (𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹)) = (0g‘𝑆))
92	3, 5, 91	syl2an 605	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹)) = (0g‘𝑆))
93	92	oveq1d 7406	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = ((0g‘𝑆)(+g‘𝑆)𝑔))
94	93	adantr 484	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = ((0g‘𝑆)(+g‘𝑆)𝑔))
95	7, 11	grpass 18975	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ (𝐹 ∈ 𝑃 ∧ ((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝑔 ∈ 𝑃)) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)))
96	38, 58, 42, 44, 95	syl13anc 1390	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)))
97	7, 11, 75	grplid 19000	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ 𝑔 ∈ 𝑃) → ((0g‘𝑆)(+g‘𝑆)𝑔) = 𝑔)
98	38, 44, 97	syl2anc 593	. . . . . 6 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g‘𝑆)(+g‘𝑆)𝑔) = 𝑔)
99	94, 96, 98	3eqtr3d 2804	. . . . 5 ⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 𝑔)
100	99	mpteq2dva 5190	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
101	90, 100	eqtrd 2796	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
102		mptresid 6036	. . 3 ⊢ ( I ↾ (𝑃 ∖ (pmEven‘𝐷))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔)
103	101, 102	eqtr4di 2814	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷))))
104	37, 70, 88, 103	fcof1od 7273	1 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)):(pmEven‘𝐷)–1-1-onto→(𝑃 ∖ (pmEven‘𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3899  {cpr 4581   ↦ cmpt 5178   I cid 5537  ^◡ccnv 5642   ↾ cres 5645   ∘ ccom 5647  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391  Fincfn 8921  1c1 11068   · cmul 11072  -cneg 11409  Basecbs 17236   ↾_s cress 17257  +_gcplusg 17277  0_gc0g 17459  Grpcgrp 18966  inv_gcminusg 18967   GrpHom cghm 19244  SymGrpcsymg 19400  pmSgncpsgn 19520  pmEvencevpm 19521  mulGrpcmgp 20177  ℂ_fldccnfld 21412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-addf 11146  ax-mulf 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-xor 1531  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-tpos 8200  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-div 11839  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-xnn0 12549  df-z 12563  df-dec 12683  df-uz 12834  df-rp 12988  df-fz 13507  df-fzo 13654  df-seq 14009  df-exp 14069  df-hash 14338  df-word 14521  df-lsw 14570  df-concat 14578  df-s1 14604  df-substr 14649  df-pfx 14679  df-splice 14757  df-reverse 14766  df-s2 14855  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-mulr 17291  df-starv 17292  df-tset 17296  df-ple 17297  df-ds 17299  df-unif 17300  df-0g 17461  df-gsum 17462  df-mre 17605  df-mrc 17606  df-acs 17608  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-efmnd 18894  df-grp 18969  df-minusg 18970  df-subg 19156  df-ghm 19245  df-gim 19290  df-oppg 19377  df-symg 19401  df-pmtr 19473  df-psgn 19522  df-evpm 19523  df-cmn 19813  df-abl 19814  df-mgp 20178  df-rng 20190  df-ur 20219  df-ring 20272  df-cring 20273  df-oppr 20373  df-dvdsr 20393  df-unit 20394  df-invr 20424  df-dvr 20437  df-drng 20768  df-cnfld 21413

This theorem is referenced by:  mdetralt  22656