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Theorem evpmodpmf1o 19861
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 789 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin)
2 evpmodpmf1o.s . . . . . . 7 𝑆 = (SymGrp‘𝐷)
32symggrp 17741 . . . . . 6 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
43ad2antrr 761 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp)
5 eldifi 3710 . . . . . 6 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹𝑃)
65ad2antlr 762 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹𝑃)
7 evpmodpmf1o.p . . . . . . . 8 𝑃 = (Base‘𝑆)
82, 7evpmss 19851 . . . . . . 7 (pmEven‘𝐷) ⊆ 𝑃
98sseli 3579 . . . . . 6 (𝑓 ∈ (pmEven‘𝐷) → 𝑓𝑃)
109adantl 482 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓𝑃)
11 eqid 2621 . . . . . 6 (+g𝑆) = (+g𝑆)
127, 11grpcl 17351 . . . . 5 ((𝑆 ∈ Grp ∧ 𝐹𝑃𝑓𝑃) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
134, 6, 10, 12syl3anc 1323 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ 𝑃)
14 eqid 2621 . . . . . . . 8 (pmSgn‘𝐷) = (pmSgn‘𝐷)
15 eqid 2621 . . . . . . . 8 ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})
162, 14, 15psgnghm2 19846 . . . . . . 7 (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
1716ad2antrr 761 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
18 prex 4870 . . . . . . . 8 {1, -1} ∈ V
19 eqid 2621 . . . . . . . . . 10 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20 cnfldmul 19671 . . . . . . . . . 10 · = (.r‘ℂfld)
2119, 20mgpplusg 18414 . . . . . . . . 9 · = (+g‘(mulGrp‘ℂfld))
2215, 21ressplusg 15914 . . . . . . . 8 ({1, -1} ∈ V → · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1})))
2318, 22ax-mp 5 . . . . . . 7 · = (+g‘((mulGrp‘ℂfld) ↾s {1, -1}))
247, 11, 23ghmlin 17586 . . . . . 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 1323 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)))
262, 7, 14psgnodpm 19853 . . . . . . . 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 19854 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
2928adantlr 750 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1)
3027, 29oveq12d 6622 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1))
31 ax-1cn 9938 . . . . . . 7 1 ∈ ℂ
3231mulm1i 10419 . . . . . 6 (-1 · 1) = -1
3330, 32syl6eq 2671 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1)
3425, 33eqtrd 2655 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1)
352, 7, 14psgnodpmr 19855 . . . 4 ((𝐷 ∈ Fin ∧ (𝐹(+g𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g𝑆)𝑓)) = -1) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
361, 13, 34, 35syl3anc 1323 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷)))
37 eqid 2621 . . 3 (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓))
3836, 37fmptd 6340 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷)))
393ad2antrr 761 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp)
40 eqid 2621 . . . . . . . 8 (invg𝑆) = (invg𝑆)
417, 40grpinvcl 17388 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → ((invg𝑆)‘𝐹) ∈ 𝑃)
423, 5, 41syl2an 494 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
4342adantr 481 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) ∈ 𝑃)
44 eldifi 3710 . . . . . 6 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔𝑃)
4544adantl 482 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔𝑃)
467, 11grpcl 17351 . . . . 5 ((𝑆 ∈ Grp ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4739, 43, 45, 46syl3anc 1323 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃)
4816ad2antrr 761 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})))
497, 11, 23ghmlin 17586 . . . . . 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 1323 . . . . 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 17743 . . . . . . . . 9 (𝐹𝑃 → ((invg𝑆)‘𝐹) = 𝐹)
525, 51syl 17 . . . . . . . 8 (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) = 𝐹)
5352ad2antlr 762 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg𝑆)‘𝐹) = 𝐹)
5453fveq2d 6152 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘𝐹))
552, 7, 14psgnodpm 19853 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5655adantlr 750 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1)
5754, 56oveq12d 6622 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘𝐹) · -1))
58 simpll 789 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin)
595ad2antlr 762 . . . . . . . . 9 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹𝑃)
602, 14, 7psgninv 19847 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝐹𝑃) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6158, 59, 60syl2anc 692 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = ((pmSgn‘𝐷)‘𝐹))
6226adantr 481 . . . . . . . 8 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6361, 62eqtrd 2655 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1)
6463oveq1d 6619 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = (-1 · -1))
65 neg1mulneg1e1 11189 . . . . . 6 (-1 · -1) = 1
6664, 65syl6eq 2671 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘𝐹) · -1) = 1)
6750, 57, 663eqtrd 2659 . . . 4 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)
682, 7, 14psgnevpmb 19852 . . . . 5 (𝐷 ∈ Fin → ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 1)))
6968ad2antrr 761 . . . 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 956 . . 3 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) ∈ (pmEven‘𝐷))
71 eqid 2621 . . 3 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔))
7270, 71fmptd 6340 . 2 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg𝑆)‘𝐹)(+g𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷))
73 eqidd 2622 . . . . 5 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g𝑆)𝑓)))
74 eqidd 2622 . . . . 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 6612 . . . . 5 (𝑔 = (𝐹(+g𝑆)𝑓) → (((invg𝑆)‘𝐹)(+g𝑆)𝑔) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
7636, 73, 74, 75fmptco 6351 . . . 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 2621 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
787, 11, 77, 40grplinv 17389 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
794, 6, 78syl2anc 692 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)𝐹) = (0g𝑆))
8079oveq1d 6619 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = ((0g𝑆)(+g𝑆)𝑓))
8142adantr 481 . . . . . . 7 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg𝑆)‘𝐹) ∈ 𝑃)
827, 11grpass 17352 . . . . . . 7 ((𝑆 ∈ Grp ∧ (((invg𝑆)‘𝐹) ∈ 𝑃𝐹𝑃𝑓𝑃)) → ((((invg𝑆)‘𝐹)(+g𝑆)𝐹)(+g𝑆)𝑓) = (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)))
834, 81, 6, 10, 82syl13anc 1325 . . . . . 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 17373 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑓𝑃) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
854, 10, 84syl2anc 692 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g𝑆)(+g𝑆)𝑓) = 𝑓)
8680, 83, 853eqtr3d 2663 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓)) = 𝑓)
8786mpteq2dva 4704 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg𝑆)‘𝐹)(+g𝑆)(𝐹(+g𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓))
8876, 87eqtrd 2655 . . 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 5415 . . 3 (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓) = ( I ↾ (pmEven‘𝐷))
9088, 89syl6eq 2671 . 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 6612 . . . . 5 (𝑓 = (((invg𝑆)‘𝐹)(+g𝑆)𝑔) → (𝐹(+g𝑆)𝑓) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9270, 74, 73, 91fmptco 6351 . . . 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 17390 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝐹𝑃) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
943, 5, 93syl2an 494 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)((invg𝑆)‘𝐹)) = (0g𝑆))
9594oveq1d 6619 . . . . . . 7 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
9695adantr 481 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = ((0g𝑆)(+g𝑆)𝑔))
977, 11grpass 17352 . . . . . . 7 ((𝑆 ∈ Grp ∧ (𝐹𝑃 ∧ ((invg𝑆)‘𝐹) ∈ 𝑃𝑔𝑃)) → ((𝐹(+g𝑆)((invg𝑆)‘𝐹))(+g𝑆)𝑔) = (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)))
9839, 59, 43, 45, 97syl13anc 1325 . . . . . 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 17373 . . . . . . 7 ((𝑆 ∈ Grp ∧ 𝑔𝑃) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
10039, 45, 99syl2anc 692 . . . . . 6 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g𝑆)(+g𝑆)𝑔) = 𝑔)
10196, 98, 1003eqtr3d 2663 . . . . 5 (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔)) = 𝑔)
102101mpteq2dva 4704 . . . 4 ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g𝑆)(((invg𝑆)‘𝐹)(+g𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔))
10392, 102eqtrd 2655 . . 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 5415 . . 3 (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷)))
105103, 104syl6eq 2671 . 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 6503 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 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  {cpr 4150  cmpt 4673   I cid 4984  ccnv 5073  cres 5076  ccom 5078  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Fincfn 7899  1c1 9881   · cmul 9885  -cneg 10211  Basecbs 15781  s cress 15782  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344   GrpHom cghm 17578  SymGrpcsymg 17718  pmSgncpsgn 17830  pmEvencevpm 17831  mulGrpcmgp 18410  fldccnfld 19665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-reverse 13244  df-s2 13530  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-0g 16023  df-gsum 16024  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-subg 17512  df-ghm 17579  df-gim 17622  df-oppg 17697  df-symg 17719  df-pmtr 17783  df-psgn 17832  df-evpm 17833  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-cring 18471  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-invr 18593  df-dvr 18604  df-drng 18670  df-cnfld 19666
This theorem is referenced by:  mdetralt  20333
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