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Theorem evpmodpmf1o 20151
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 750 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin)
2 evpmodpmf1o.s . . . . . . 7 𝑆 = (SymGrp‘𝐷)
32symggrp 18020 . . . . . 6 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
43ad2antrr 705 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp)
5 eldifi 3883 . . . . . 6 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹𝑃)
65ad2antlr 706 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹𝑃)
7 evpmodpmf1o.p . . . . . . . 8 𝑃 = (Base‘𝑆)
82, 7evpmss 20140 . . . . . . 7 (pmEven‘𝐷) ⊆ 𝑃
98sseli 3748 . . . . . 6 (𝑓 ∈ (pmEven‘𝐷) → 𝑓𝑃)
109adantl 467 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓𝑃)
11 eqid 2771 . . . . . 6 (+g𝑆) = (+g𝑆)
127, 11grpcl 17631 . . . . 5 ((𝑆 ∈ Grp ∧ 𝐹𝑃𝑓𝑃) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
134, 6, 10, 12syl3anc 1476 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
14 eqid 2771 . . . . . . . 8 (pmSgn‘𝐷) = (pmSgn‘𝐷)
15 eqid 2771 . . . . . . . 8 ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})
162, 14, 15psgnghm2 20135 . . . . . . 7 (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
1716ad2antrr 705 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
18 prex 5037 . . . . . . . 8 {1, -1} ∈ V
19 eqid 2771 . . . . . . . . . 10 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20 cnfldmul 19960 . . . . . . . . . 10 · = (.r‘ℂfld)
2119, 20mgpplusg 18694 . . . . . . . . 9 · = (+g‘(mulGrp‘ℂfld))
2215, 21ressplusg 16194 . . . . . . . 8 ({1, -1} ∈ V → · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1})))
2318, 22ax-mp 5 . . . . . . 7 · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1}))
247, 11, 23ghmlin 17866 . . . . . 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 1476 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
262, 7, 14psgnodpm 20142 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
2726adantr 466 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝐹) = -1)
282, 7, 14psgnevpm 20143 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
2928adantlr 694 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
3027, 29oveq12d 6809 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1))
31 ax-1cn 10194 . . . . . . 7 1 ∈ ℂ
3231mulm1i 10675 . . . . . 6 (-1 · 1) = -1
3330, 32syl6eq 2821 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1)
3425, 33eqtrd 2805 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1)
352, 7, 14psgnodpmr 20144 . . . 4 ((𝐷 ∈ Fin ∧ (𝐹(+g𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
361, 13, 34, 35syl3anc 1476 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
37 eqid 2771 . . 3 (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))
3836, 37fmptd 6525 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷)))
393ad2antrr 705 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp)
40 eqid 2771 . . . . . . . 8 (invg𝑆) = (invg𝑆)
417, 40grpinvcl 17668 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → ((invg𝑆)‘𝐹) ∈ 𝑃)
423, 5, 41syl2an 583 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
4342adantr 466 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
44 eldifi 3883 . . . . . 6 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔𝑃)
4544adantl 467 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔𝑃)
467, 11grpcl 17631 . . . . 5 ((𝑆 ∈ Grp ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4739, 43, 45, 46syl3anc 1476 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4816ad2antrr 705 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
497, 11, 23ghmlin 17866 . . . . . 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)))
5048, 43, 45, 49syl3anc 1476 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)))
512, 7, 40symginv 18022 . . . . . . . . 9 (𝐹𝑃 → ((invg𝑆)‘𝐹) = 𝐹)
525, 51syl 17 . . . . . . . 8 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) = 𝐹)
5352ad2antlr 706 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) = 𝐹)
5453fveq2d 6334 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘𝐹))
552, 7, 14psgnodpm 20142 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5655adantlr 694 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5754, 56oveq12d 6809 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘𝐹) · -1))
58 simpll 750 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin)
595ad2antlr 706 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹𝑃)
602, 14, 7psgninv 20136 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝐹𝑃) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6158, 59, 60syl2anc 573 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6226adantr 466 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6361, 62eqtrd 2805 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6463oveq1d 6806 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = (-1 · -1))
65 neg1mulneg1e1 11445 . . . . . 6 (-1 · -1) = 1
6664, 65syl6eq 2821 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = 1)
6750, 57, 663eqtrd 2809 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)
682, 7, 14psgnevpmb 20141 . . . . 5 (𝐷 ∈ Fin → ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)))
6968ad2antrr 705 . . . 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)))
7047, 67, 69mpbir2and 692 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷))
71 eqid 2771 . . 3 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔))
7270, 71fmptd 6525 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷))
73 eqidd 2772 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)))
74 eqidd 2772 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
75 oveq2 6799 . . . . 5 (𝑔 = (𝐹(+g𝑆)𝑓) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
7636, 73, 74, 75fmptco 6537 . . . 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))))
77 eqid 2771 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
787, 11, 77, 40grplinv 17669 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
794, 6, 78syl2anc 573 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
8079oveq1d 6806 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = ((0g𝑆)(+g𝑆)𝑓))
8142adantr 466 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) ∈ 𝑃)
827, 11grpass 17632 . . . . . . 7 ((𝑆 ∈ Grp ∧ (((invg𝑆)‘𝐹) ∈ 𝑃𝐹𝑃𝑓𝑃)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
834, 81, 6, 10, 82syl13anc 1478 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
847, 11, 77grplid 17653 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑓𝑃) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
854, 10, 84syl2anc 573 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
8680, 83, 853eqtr3d 2813 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)) = 𝑓)
8786mpteq2dva 4878 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
8876, 87eqtrd 2805 . . 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))
89 mptresid 5595 . . 3 (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓) = ( I ↾ (pmEven‘𝐷))
9088, 89syl6eq 2821 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))) = ( I ↾ (pmEven‘𝐷)))
91 oveq2 6799 . . . . 5 (𝑓 = (((invg𝑆)‘𝐹)(+g𝑆)𝑔) → (𝐹(+g𝑆)𝑓) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9270, 74, 73, 91fmptco 6537 . . . 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))))
937, 11, 77, 40grprinv 17670 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
943, 5, 93syl2an 583 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
9594oveq1d 6806 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
9695adantr 466 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
977, 11grpass 17632 . . . . . . 7 ((𝑆 ∈ Grp ∧ (𝐹𝑃 ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃)) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9839, 59, 43, 45, 97syl13anc 1478 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
997, 11, 77grplid 17653 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑔𝑃) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
10039, 45, 99syl2anc 573 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
10196, 98, 1003eqtr3d 2813 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 𝑔)
102101mpteq2dva 4878 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
10392, 102eqtrd 2805 . . 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))
104 mptresid 5595 . . 3 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷)))
105103, 104syl6eq 2821 . 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))))
10638, 72, 90, 105fcof1od 6690 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 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  {cpr 4318  cmpt 4863   I cid 5156  ccnv 5248  cres 5251  ccom 5253  1-1-ontowf1o 6028  cfv 6029  (class class class)co 6791  Fincfn 8107  1c1 10137   · cmul 10141  -cneg 10467  Basecbs 16057  s cress 16058  +gcplusg 16142  0gc0g 16301  Grpcgrp 17623  invgcminusg 17624   GrpHom cghm 17858  SymGrpcsymg 17997  pmSgncpsgn 18109  pmEvencevpm 18110  mulGrpcmgp 18690  fldccnfld 19954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-addf 10215  ax-mulf 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-xor 1613  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-ot 4325  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-tpos 7502  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-7 11284  df-8 11285  df-9 11286  df-n0 11493  df-xnn0 11564  df-z 11578  df-dec 11694  df-uz 11887  df-rp 12029  df-fz 12527  df-fzo 12667  df-seq 13002  df-exp 13061  df-hash 13315  df-word 13488  df-lsw 13489  df-concat 13490  df-s1 13491  df-substr 13492  df-splice 13493  df-reverse 13494  df-s2 13795  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16155  df-mulr 16156  df-starv 16157  df-tset 16161  df-ple 16162  df-ds 16165  df-unif 16166  df-0g 16303  df-gsum 16304  df-mre 16447  df-mrc 16448  df-acs 16450  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-subg 17792  df-ghm 17859  df-gim 17902  df-oppg 17976  df-symg 17998  df-pmtr 18062  df-psgn 18111  df-evpm 18112  df-cmn 18395  df-abl 18396  df-mgp 18691  df-ur 18703  df-ring 18750  df-cring 18751  df-oppr 18824  df-dvdsr 18842  df-unit 18843  df-invr 18873  df-dvr 18884  df-drng 18952  df-cnfld 19955
This theorem is referenced by:  mdetralt  20625
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