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Theorem evpmodpmf1o 21000
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 765 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin)
2 evpmodpmf1o.s . . . . . . 7 𝑆 = (SymGrp‘𝐷)
32symggrp 19182 . . . . . 6 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
43ad2antrr 724 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp)
5 eldifi 4086 . . . . . 6 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹𝑃)
65ad2antlr 725 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹𝑃)
7 evpmodpmf1o.p . . . . . . . 8 𝑃 = (Base‘𝑆)
82, 7evpmss 20990 . . . . . . 7 (pmEven‘𝐷) ⊆ 𝑃
98sseli 3940 . . . . . 6 (𝑓 ∈ (pmEven‘𝐷) → 𝑓𝑃)
109adantl 482 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓𝑃)
11 eqid 2736 . . . . . 6 (+g𝑆) = (+g𝑆)
127, 11grpcl 18756 . . . . 5 ((𝑆 ∈ Grp ∧ 𝐹𝑃𝑓𝑃) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
134, 6, 10, 12syl3anc 1371 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
14 eqid 2736 . . . . . . . 8 (pmSgn‘𝐷) = (pmSgn‘𝐷)
15 eqid 2736 . . . . . . . 8 ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})
162, 14, 15psgnghm2 20985 . . . . . . 7 (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
1716ad2antrr 724 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
18 prex 5389 . . . . . . . 8 {1, -1} ∈ V
19 eqid 2736 . . . . . . . . . 10 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20 cnfldmul 20802 . . . . . . . . . 10 · = (.r‘ℂfld)
2119, 20mgpplusg 19900 . . . . . . . . 9 · = (+g‘(mulGrp‘ℂfld))
2215, 21ressplusg 17171 . . . . . . . 8 ({1, -1} ∈ V → · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1})))
2318, 22ax-mp 5 . . . . . . 7 · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1}))
247, 11, 23ghmlin 19013 . . . . . 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 1371 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
262, 7, 14psgnodpm 20992 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
2726adantr 481 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝐹) = -1)
282, 7, 14psgnevpm 20993 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
2928adantlr 713 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
3027, 29oveq12d 7375 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1))
31 ax-1cn 11109 . . . . . . 7 1 ∈ ℂ
3231mulm1i 11600 . . . . . 6 (-1 · 1) = -1
3330, 32eqtrdi 2792 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1)
3425, 33eqtrd 2776 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1)
352, 7, 14psgnodpmr 20994 . . . 4 ((𝐷 ∈ Fin ∧ (𝐹(+g𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
361, 13, 34, 35syl3anc 1371 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
3736fmpttd 7063 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷)))
383ad2antrr 724 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp)
39 eqid 2736 . . . . . . . 8 (invg𝑆) = (invg𝑆)
407, 39grpinvcl 18798 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → ((invg𝑆)‘𝐹) ∈ 𝑃)
413, 5, 40syl2an 596 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
4241adantr 481 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
43 eldifi 4086 . . . . . 6 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔𝑃)
4443adantl 482 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔𝑃)
457, 11grpcl 18756 . . . . 5 ((𝑆 ∈ Grp ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4638, 42, 44, 45syl3anc 1371 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4716ad2antrr 724 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
487, 11, 23ghmlin 19013 . . . . . 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 1371 . . . . 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 19184 . . . . . . . . 9 (𝐹𝑃 → ((invg𝑆)‘𝐹) = 𝐹)
515, 50syl 17 . . . . . . . 8 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) = 𝐹)
5251ad2antlr 725 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) = 𝐹)
5352fveq2d 6846 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘𝐹))
542, 7, 14psgnodpm 20992 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5554adantlr 713 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5653, 55oveq12d 7375 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘𝐹) · -1))
57 simpll 765 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin)
585ad2antlr 725 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹𝑃)
592, 14, 7psgninv 20986 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝐹𝑃) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6057, 58, 59syl2anc 584 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6126adantr 481 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6260, 61eqtrd 2776 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6362oveq1d 7372 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = (-1 · -1))
64 neg1mulneg1e1 12366 . . . . . 6 (-1 · -1) = 1
6563, 64eqtrdi 2792 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = 1)
6649, 56, 653eqtrd 2780 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)
672, 7, 14psgnevpmb 20991 . . . . 5 (𝐷 ∈ Fin → ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)))
6867ad2antrr 724 . . . 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 711 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷))
7069fmpttd 7063 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷))
71 eqidd 2737 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)))
72 eqidd 2737 . . . . 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 7365 . . . . 5 (𝑔 = (𝐹(+g𝑆)𝑓) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
7436, 71, 72, 73fmptco 7075 . . . 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 2736 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
767, 11, 75, 39grplinv 18800 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
774, 6, 76syl2anc 584 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
7877oveq1d 7372 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = ((0g𝑆)(+g𝑆)𝑓))
7941adantr 481 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) ∈ 𝑃)
807, 11grpass 18757 . . . . . . 7 ((𝑆 ∈ Grp ∧ (((invg𝑆)‘𝐹) ∈ 𝑃𝐹𝑃𝑓𝑃)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
814, 79, 6, 10, 80syl13anc 1372 . . . . . 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 18780 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑓𝑃) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
834, 10, 82syl2anc 584 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
8478, 81, 833eqtr3d 2784 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)) = 𝑓)
8584mpteq2dva 5205 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
8674, 85eqtrd 2776 . . 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 6004 . . 3 ( I ↾ (pmEven‘𝐷)) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓)
8886, 87eqtr4di 2794 . 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 7365 . . . . 5 (𝑓 = (((invg𝑆)‘𝐹)(+g𝑆)𝑔) → (𝐹(+g𝑆)𝑓) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9069, 72, 71, 89fmptco 7075 . . . 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 18801 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
923, 5, 91syl2an 596 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
9392oveq1d 7372 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
9493adantr 481 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
957, 11grpass 18757 . . . . . . 7 ((𝑆 ∈ Grp ∧ (𝐹𝑃 ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃)) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9638, 58, 42, 44, 95syl13anc 1372 . . . . . 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 18780 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑔𝑃) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
9838, 44, 97syl2anc 584 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
9994, 96, 983eqtr3d 2784 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 𝑔)
10099mpteq2dva 5205 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
10190, 100eqtrd 2776 . . 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 6004 . . 3 ( I ↾ (𝑃 ∖ (pmEven‘𝐷))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔)
103101, 102eqtr4di 2794 . 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 7240 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 396   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  {cpr 4588  cmpt 5188   I cid 5530  ccnv 5632  cres 5635  ccom 5637  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052   · cmul 11056  -cneg 11386  Basecbs 17083  s cress 17112  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749   GrpHom cghm 19005  SymGrpcsymg 19148  pmSgncpsgn 19271  pmEvencevpm 19272  mulGrpcmgp 19896  fldccnfld 20796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-reverse 14647  df-s2 14737  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-minusg 18752  df-subg 18925  df-ghm 19006  df-gim 19049  df-oppg 19124  df-symg 19149  df-pmtr 19224  df-psgn 19273  df-evpm 19274  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-cnfld 20797
This theorem is referenced by:  mdetralt  21957
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