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Theorem evpmodpmf1o 19761
 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 786 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin)
2 evpmodpmf1o.s . . . . . . 7 𝑆 = (SymGrp‘𝐷)
32symggrp 17643 . . . . . 6 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
43ad2antrr 758 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp)
5 eldifi 3694 . . . . . 6 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹𝑃)
65ad2antlr 759 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹𝑃)
7 evpmodpmf1o.p . . . . . . . 8 𝑃 = (Base‘𝑆)
82, 7evpmss 19751 . . . . . . 7 (pmEven‘𝐷) ⊆ 𝑃
98sseli 3564 . . . . . 6 (𝑓 ∈ (pmEven‘𝐷) → 𝑓𝑃)
109adantl 481 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓𝑃)
11 eqid 2610 . . . . . 6 (+g𝑆) = (+g𝑆)
127, 11grpcl 17253 . . . . 5 ((𝑆 ∈ Grp ∧ 𝐹𝑃𝑓𝑃) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
134, 6, 10, 12syl3anc 1318 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
14 eqid 2610 . . . . . . . 8 (pmSgn‘𝐷) = (pmSgn‘𝐷)
15 eqid 2610 . . . . . . . 8 ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})
162, 14, 15psgnghm2 19746 . . . . . . 7 (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
1716ad2antrr 758 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
18 prex 4836 . . . . . . . 8 {1, -1} ∈ V
19 eqid 2610 . . . . . . . . . 10 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20 cnfldmul 19573 . . . . . . . . . 10 · = (.r‘ℂfld)
2119, 20mgpplusg 18316 . . . . . . . . 9 · = (+g‘(mulGrp‘ℂfld))
2215, 21ressplusg 15818 . . . . . . . 8 ({1, -1} ∈ V → · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1})))
2318, 22ax-mp 5 . . . . . . 7 · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1}))
247, 11, 23ghmlin 17488 . . . . . 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 1318 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
262, 7, 14psgnodpm 19753 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
2726adantr 480 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝐹) = -1)
282, 7, 14psgnevpm 19754 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
2928adantlr 747 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
3027, 29oveq12d 6567 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1))
31 ax-1cn 9873 . . . . . . 7 1 ∈ ℂ
3231mulm1i 10354 . . . . . 6 (-1 · 1) = -1
3330, 32syl6eq 2660 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1)
3425, 33eqtrd 2644 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1)
352, 7, 14psgnodpmr 19755 . . . 4 ((𝐷 ∈ Fin ∧ (𝐹(+g𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
361, 13, 34, 35syl3anc 1318 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
37 eqid 2610 . . 3 (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))
3836, 37fmptd 6292 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷)))
393ad2antrr 758 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp)
40 eqid 2610 . . . . . . . 8 (invg𝑆) = (invg𝑆)
417, 40grpinvcl 17290 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → ((invg𝑆)‘𝐹) ∈ 𝑃)
423, 5, 41syl2an 493 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
4342adantr 480 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
44 eldifi 3694 . . . . . 6 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔𝑃)
4544adantl 481 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔𝑃)
467, 11grpcl 17253 . . . . 5 ((𝑆 ∈ Grp ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4739, 43, 45, 46syl3anc 1318 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4816ad2antrr 758 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
497, 11, 23ghmlin 17488 . . . . . 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 1318 . . . . 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 17645 . . . . . . . . 9 (𝐹𝑃 → ((invg𝑆)‘𝐹) = 𝐹)
525, 51syl 17 . . . . . . . 8 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) = 𝐹)
5352ad2antlr 759 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) = 𝐹)
5453fveq2d 6107 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘𝐹))
552, 7, 14psgnodpm 19753 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5655adantlr 747 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5754, 56oveq12d 6567 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘𝐹) · -1))
58 simpll 786 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin)
595ad2antlr 759 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹𝑃)
602, 14, 7psgninv 19747 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝐹𝑃) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6158, 59, 60syl2anc 691 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6226adantr 480 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6361, 62eqtrd 2644 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6463oveq1d 6564 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = (-1 · -1))
65 neg1mulneg1e1 11122 . . . . . 6 (-1 · -1) = 1
6664, 65syl6eq 2660 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = 1)
6750, 57, 663eqtrd 2648 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)
682, 7, 14psgnevpmb 19752 . . . . 5 (𝐷 ∈ Fin → ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)))
6968ad2antrr 758 . . . 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 959 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷))
71 eqid 2610 . . 3 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔))
7270, 71fmptd 6292 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷))
73 eqidd 2611 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)))
74 eqidd 2611 . . . . 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 6557 . . . . 5 (𝑔 = (𝐹(+g𝑆)𝑓) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
7636, 73, 74, 75fmptco 6303 . . . 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 2610 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
787, 11, 77, 40grplinv 17291 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
794, 6, 78syl2anc 691 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
8079oveq1d 6564 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = ((0g𝑆)(+g𝑆)𝑓))
8142adantr 480 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) ∈ 𝑃)
827, 11grpass 17254 . . . . . . 7 ((𝑆 ∈ Grp ∧ (((invg𝑆)‘𝐹) ∈ 𝑃𝐹𝑃𝑓𝑃)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
834, 81, 6, 10, 82syl13anc 1320 . . . . . 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 17275 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑓𝑃) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
854, 10, 84syl2anc 691 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
8680, 83, 853eqtr3d 2652 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)) = 𝑓)
8786mpteq2dva 4672 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
8876, 87eqtrd 2644 . . 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 5375 . . 3 (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓) = ( I ↾ (pmEven‘𝐷))
9088, 89syl6eq 2660 . 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 6557 . . . . 5 (𝑓 = (((invg𝑆)‘𝐹)(+g𝑆)𝑔) → (𝐹(+g𝑆)𝑓) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9270, 74, 73, 91fmptco 6303 . . . 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 17292 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
943, 5, 93syl2an 493 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
9594oveq1d 6564 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
9695adantr 480 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
977, 11grpass 17254 . . . . . . 7 ((𝑆 ∈ Grp ∧ (𝐹𝑃 ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃)) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9839, 59, 43, 45, 97syl13anc 1320 . . . . . 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 17275 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑔𝑃) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
10039, 45, 99syl2anc 691 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
10196, 98, 1003eqtr3d 2652 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 𝑔)
102101mpteq2dva 4672 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
10392, 102eqtrd 2644 . . 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 5375 . . 3 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷)))
105103, 104syl6eq 2660 . 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 6449 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  {cpr 4127   ↦ cmpt 4643   I cid 4948  ◡ccnv 5037   ↾ cres 5040   ∘ ccom 5042  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   · cmul 9820  -cneg 10146  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246   GrpHom cghm 17480  SymGrpcsymg 17620  pmSgncpsgn 17732  pmEvencevpm 17733  mulGrpcmgp 18312  ℂfldccnfld 19567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-ghm 17481  df-gim 17524  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734  df-evpm 17735  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-cnfld 19568 This theorem is referenced by:  mdetralt  20233
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