MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evpmodpmf1o Structured version   Visualization version   GIF version

Theorem evpmodpmf1o 21016
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(+g𝑆)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.
StepHypRef Expression
1 simpll 766 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin)
2 evpmodpmf1o.s . . . . . . 7 𝑆 = (SymGrp‘𝐷)
32symggrp 19187 . . . . . 6 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
43ad2antrr 725 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp)
5 eldifi 4087 . . . . . 6 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹𝑃)
65ad2antlr 726 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹𝑃)
7 evpmodpmf1o.p . . . . . . . 8 𝑃 = (Base‘𝑆)
82, 7evpmss 21006 . . . . . . 7 (pmEven‘𝐷) ⊆ 𝑃
98sseli 3941 . . . . . 6 (𝑓 ∈ (pmEven‘𝐷) → 𝑓𝑃)
109adantl 483 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓𝑃)
11 eqid 2733 . . . . . 6 (+g𝑆) = (+g𝑆)
127, 11grpcl 18761 . . . . 5 ((𝑆 ∈ Grp ∧ 𝐹𝑃𝑓𝑃) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
134, 6, 10, 12syl3anc 1372 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
14 eqid 2733 . . . . . . . 8 (pmSgn‘𝐷) = (pmSgn‘𝐷)
15 eqid 2733 . . . . . . . 8 ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})
162, 14, 15psgnghm2 21001 . . . . . . 7 (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
1716ad2antrr 725 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
18 prex 5390 . . . . . . . 8 {1, -1} ∈ V
19 eqid 2733 . . . . . . . . . 10 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20 cnfldmul 20818 . . . . . . . . . 10 · = (.r‘ℂfld)
2119, 20mgpplusg 19905 . . . . . . . . 9 · = (+g‘(mulGrp‘ℂfld))
2215, 21ressplusg 17176 . . . . . . . 8 ({1, -1} ∈ V → · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1})))
2318, 22ax-mp 5 . . . . . . 7 · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1}))
247, 11, 23ghmlin 19018 . . . . . 6 (((pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹𝑃𝑓𝑃) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
2517, 6, 10, 24syl3anc 1372 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
262, 7, 14psgnodpm 21008 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
2726adantr 482 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝐹) = -1)
282, 7, 14psgnevpm 21009 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
2928adantlr 714 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
3027, 29oveq12d 7376 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1))
31 ax-1cn 11114 . . . . . . 7 1 ∈ ℂ
3231mulm1i 11605 . . . . . 6 (-1 · 1) = -1
3330, 32eqtrdi 2789 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1)
3425, 33eqtrd 2773 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1)
352, 7, 14psgnodpmr 21010 . . . 4 ((𝐷 ∈ Fin ∧ (𝐹(+g𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
361, 13, 34, 35syl3anc 1372 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
3736fmpttd 7064 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷)))
383ad2antrr 725 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp)
39 eqid 2733 . . . . . . . 8 (invg𝑆) = (invg𝑆)
407, 39grpinvcl 18803 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → ((invg𝑆)‘𝐹) ∈ 𝑃)
413, 5, 40syl2an 597 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
4241adantr 482 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
43 eldifi 4087 . . . . . 6 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔𝑃)
4443adantl 483 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔𝑃)
457, 11grpcl 18761 . . . . 5 ((𝑆 ∈ Grp ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4638, 42, 44, 45syl3anc 1372 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4716ad2antrr 725 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
487, 11, 23ghmlin 19018 . . . . . 6 (((pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)))
4947, 42, 44, 48syl3anc 1372 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)))
502, 7, 39symginv 19189 . . . . . . . . 9 (𝐹𝑃 → ((invg𝑆)‘𝐹) = 𝐹)
515, 50syl 17 . . . . . . . 8 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) = 𝐹)
5251ad2antlr 726 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) = 𝐹)
5352fveq2d 6847 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘𝐹))
542, 7, 14psgnodpm 21008 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5554adantlr 714 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5653, 55oveq12d 7376 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘𝐹) · -1))
57 simpll 766 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin)
585ad2antlr 726 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹𝑃)
592, 14, 7psgninv 21002 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝐹𝑃) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6057, 58, 59syl2anc 585 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6126adantr 482 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6260, 61eqtrd 2773 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6362oveq1d 7373 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = (-1 · -1))
64 neg1mulneg1e1 12371 . . . . . 6 (-1 · -1) = 1
6563, 64eqtrdi 2789 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = 1)
6649, 56, 653eqtrd 2777 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)
672, 7, 14psgnevpmb 21007 . . . . 5 (𝐷 ∈ Fin → ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)))
6867ad2antrr 725 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)))
6946, 66, 68mpbir2and 712 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷))
7069fmpttd 7064 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷))
71 eqidd 2734 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)))
72 eqidd 2734 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
73 oveq2 7366 . . . . 5 (𝑔 = (𝐹(+g𝑆)𝑓) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
7436, 71, 72, 73fmptco 7076 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓))))
75 eqid 2733 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
767, 11, 75, 39grplinv 18805 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
774, 6, 76syl2anc 585 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
7877oveq1d 7373 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = ((0g𝑆)(+g𝑆)𝑓))
7941adantr 482 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) ∈ 𝑃)
807, 11grpass 18762 . . . . . . 7 ((𝑆 ∈ Grp ∧ (((invg𝑆)‘𝐹) ∈ 𝑃𝐹𝑃𝑓𝑃)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
814, 79, 6, 10, 80syl13anc 1373 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
827, 11, 75grplid 18785 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑓𝑃) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
834, 10, 82syl2anc 585 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
8478, 81, 833eqtr3d 2781 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)) = 𝑓)
8584mpteq2dva 5206 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
8674, 85eqtrd 2773 . . 3 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
87 mptresid 6005 . . 3 ( I ↾ (pmEven‘𝐷)) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓)
8886, 87eqtr4di 2791 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))) = ( I ↾ (pmEven‘𝐷)))
89 oveq2 7366 . . . . 5 (𝑓 = (((invg𝑆)‘𝐹)(+g𝑆)𝑔) → (𝐹(+g𝑆)𝑓) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9069, 72, 71, 89fmptco 7076 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔))))
917, 11, 75, 39grprinv 18806 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
923, 5, 91syl2an 597 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
9392oveq1d 7373 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
9493adantr 482 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
957, 11grpass 18762 . . . . . . 7 ((𝑆 ∈ Grp ∧ (𝐹𝑃 ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃)) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9638, 58, 42, 44, 95syl13anc 1373 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
977, 11, 75grplid 18785 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑔𝑃) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
9838, 44, 97syl2anc 585 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
9994, 96, 983eqtr3d 2781 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 𝑔)
10099mpteq2dva 5206 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
10190, 100eqtrd 2773 . . 3 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
102 mptresid 6005 . . 3 ( I ↾ (𝑃 ∖ (pmEven‘𝐷))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔)
103101, 102eqtr4di 2791 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷))))
10437, 70, 88, 103fcof1od 7241 1 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)):(pmEven‘𝐷)–1-1-onto→(𝑃 ∖ (pmEven‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  cdif 3908  {cpr 4589  cmpt 5189   I cid 5531  ccnv 5633  cres 5636  ccom 5638  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  Fincfn 8886  1c1 11057   · cmul 11061  -cneg 11391  Basecbs 17088  s cress 17117  +gcplusg 17138  0gc0g 17326  Grpcgrp 18753  invgcminusg 18754   GrpHom cghm 19010  SymGrpcsymg 19153  pmSgncpsgn 19276  pmEvencevpm 19277  mulGrpcmgp 19901  fldccnfld 20812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-addf 11135  ax-mulf 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-xor 1511  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-xnn0 12491  df-z 12505  df-dec 12624  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-word 14409  df-lsw 14457  df-concat 14465  df-s1 14490  df-substr 14535  df-pfx 14565  df-splice 14644  df-reverse 14653  df-s2 14743  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-0g 17328  df-gsum 17329  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-efmnd 18684  df-grp 18756  df-minusg 18757  df-subg 18930  df-ghm 19011  df-gim 19054  df-oppg 19129  df-symg 19154  df-pmtr 19229  df-psgn 19278  df-evpm 19279  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-cring 19972  df-oppr 20054  df-dvdsr 20075  df-unit 20076  df-invr 20106  df-dvr 20117  df-drng 20199  df-cnfld 20813
This theorem is referenced by:  mdetralt  21973
  Copyright terms: Public domain W3C validator