Step | Hyp | Ref
| Expression |
1 | | simpll 790 |
. . . 4
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐷 ∈ Fin) |
2 | | evpmodpmf1o.s |
. . . . . . 7
⊢ 𝑆 = (SymGrp‘𝐷) |
3 | 2 | symggrp 17820 |
. . . . . 6
⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp) |
4 | 3 | ad2antrr 762 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑆 ∈ Grp) |
5 | | eldifi 3732 |
. . . . . 6
⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹 ∈ 𝑃) |
6 | 5 | ad2antlr 763 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝐹 ∈ 𝑃) |
7 | | evpmodpmf1o.p |
. . . . . . . 8
⊢ 𝑃 = (Base‘𝑆) |
8 | 2, 7 | evpmss 19932 |
. . . . . . 7
⊢
(pmEven‘𝐷)
⊆ 𝑃 |
9 | 8 | sseli 3599 |
. . . . . 6
⊢ (𝑓 ∈ (pmEven‘𝐷) → 𝑓 ∈ 𝑃) |
10 | 9 | adantl 482 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → 𝑓 ∈ 𝑃) |
11 | | eqid 2622 |
. . . . . 6
⊢
(+g‘𝑆) = (+g‘𝑆) |
12 | 7, 11 | grpcl 17430 |
. . . . 5
⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃 ∧ 𝑓 ∈ 𝑃) → (𝐹(+g‘𝑆)𝑓) ∈ 𝑃) |
13 | 4, 6, 10, 12 | syl3anc 1326 |
. . . 4
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g‘𝑆)𝑓) ∈ 𝑃) |
14 | | eqid 2622 |
. . . . . . . 8
⊢
(pmSgn‘𝐷) =
(pmSgn‘𝐷) |
15 | | eqid 2622 |
. . . . . . . 8
⊢
((mulGrp‘ℂfld) ↾s {1, -1}) =
((mulGrp‘ℂfld) ↾s {1,
-1}) |
16 | 2, 14, 15 | psgnghm2 19927 |
. . . . . . 7
⊢ (𝐷 ∈ Fin →
(pmSgn‘𝐷) ∈
(𝑆 GrpHom
((mulGrp‘ℂfld) ↾s {1,
-1}))) |
17 | 16 | ad2antrr 762 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom
((mulGrp‘ℂfld) ↾s {1,
-1}))) |
18 | | prex 4909 |
. . . . . . . 8
⊢ {1, -1}
∈ V |
19 | | eqid 2622 |
. . . . . . . . . 10
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
20 | | cnfldmul 19752 |
. . . . . . . . . 10
⊢ ·
= (.r‘ℂfld) |
21 | 19, 20 | mgpplusg 18493 |
. . . . . . . . 9
⊢ ·
= (+g‘(mulGrp‘ℂfld)) |
22 | 15, 21 | ressplusg 15993 |
. . . . . . . 8
⊢ ({1, -1}
∈ V → · =
(+g‘((mulGrp‘ℂfld) ↾s
{1, -1}))) |
23 | 18, 22 | ax-mp 5 |
. . . . . . 7
⊢ ·
= (+g‘((mulGrp‘ℂfld)
↾s {1, -1})) |
24 | 7, 11, 23 | ghmlin 17665 |
. . . . . 6
⊢
(((pmSgn‘𝐷)
∈ (𝑆 GrpHom
((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ 𝑃 ∧ 𝑓 ∈ 𝑃) → ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓))) |
25 | 17, 6, 10, 24 | syl3anc 1326 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓))) |
26 | 2, 7, 14 | psgnodpm 19934 |
. . . . . . . 8
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1) |
27 | 26 | adantr 481 |
. . . . . . 7
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝐹) = -1) |
28 | 2, 7, 14 | psgnevpm 19935 |
. . . . . . . 8
⊢ ((𝐷 ∈ Fin ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1) |
29 | 28 | adantlr 751 |
. . . . . . 7
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑓) = 1) |
30 | 27, 29 | oveq12d 6668 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = (-1 · 1)) |
31 | | ax-1cn 9994 |
. . . . . . 7
⊢ 1 ∈
ℂ |
32 | 31 | mulm1i 10475 |
. . . . . 6
⊢ (-1
· 1) = -1 |
33 | 30, 32 | syl6eq 2672 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((pmSgn‘𝐷)‘𝐹) · ((pmSgn‘𝐷)‘𝑓)) = -1) |
34 | 25, 33 | eqtrd 2656 |
. . . 4
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = -1) |
35 | 2, 7, 14 | psgnodpmr 19936 |
. . . 4
⊢ ((𝐷 ∈ Fin ∧ (𝐹(+g‘𝑆)𝑓) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(𝐹(+g‘𝑆)𝑓)) = -1) → (𝐹(+g‘𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷))) |
36 | 1, 13, 34, 35 | syl3anc 1326 |
. . 3
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (𝐹(+g‘𝑆)𝑓) ∈ (𝑃 ∖ (pmEven‘𝐷))) |
37 | | eqid 2622 |
. . 3
⊢ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) |
38 | 36, 37 | fmptd 6385 |
. 2
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)):(pmEven‘𝐷)⟶(𝑃 ∖ (pmEven‘𝐷))) |
39 | 3 | ad2antrr 762 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑆 ∈ Grp) |
40 | | eqid 2622 |
. . . . . . . 8
⊢
(invg‘𝑆) = (invg‘𝑆) |
41 | 7, 40 | grpinvcl 17467 |
. . . . . . 7
⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃) → ((invg‘𝑆)‘𝐹) ∈ 𝑃) |
42 | 3, 5, 41 | syl2an 494 |
. . . . . 6
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg‘𝑆)‘𝐹) ∈ 𝑃) |
43 | 42 | adantr 481 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg‘𝑆)‘𝐹) ∈ 𝑃) |
44 | | eldifi 3732 |
. . . . . 6
⊢ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝑔 ∈ 𝑃) |
45 | 44 | adantl 482 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑔 ∈ 𝑃) |
46 | 7, 11 | grpcl 17430 |
. . . . 5
⊢ ((𝑆 ∈ Grp ∧
((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝑔 ∈ 𝑃) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃) |
47 | 39, 43, 45, 46 | syl3anc 1326 |
. . . 4
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃) |
48 | 16 | ad2antrr 762 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (pmSgn‘𝐷) ∈ (𝑆 GrpHom
((mulGrp‘ℂfld) ↾s {1,
-1}))) |
49 | 7, 11, 23 | ghmlin 17665 |
. . . . . 6
⊢
(((pmSgn‘𝐷)
∈ (𝑆 GrpHom
((mulGrp‘ℂfld) ↾s {1, -1})) ∧
((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝑔 ∈ 𝑃) → ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔))) |
50 | 48, 43, 45, 49 | syl3anc 1326 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔))) |
51 | 2, 7, 40 | symginv 17822 |
. . . . . . . . 9
⊢ (𝐹 ∈ 𝑃 → ((invg‘𝑆)‘𝐹) = ◡𝐹) |
52 | 5, 51 | syl 17 |
. . . . . . . 8
⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → ((invg‘𝑆)‘𝐹) = ◡𝐹) |
53 | 52 | ad2antlr 763 |
. . . . . . 7
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((invg‘𝑆)‘𝐹) = ◡𝐹) |
54 | 53 | fveq2d 6195 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) = ((pmSgn‘𝐷)‘◡𝐹)) |
55 | 2, 7, 14 | psgnodpm 19934 |
. . . . . . 7
⊢ ((𝐷 ∈ Fin ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1) |
56 | 55 | adantlr 751 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝑔) = -1) |
57 | 54, 56 | oveq12d 6668 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘((invg‘𝑆)‘𝐹)) · ((pmSgn‘𝐷)‘𝑔)) = (((pmSgn‘𝐷)‘◡𝐹) · -1)) |
58 | | simpll 790 |
. . . . . . . . 9
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐷 ∈ Fin) |
59 | 5 | ad2antlr 763 |
. . . . . . . . 9
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹 ∈ 𝑃) |
60 | 2, 14, 7 | psgninv 19928 |
. . . . . . . . 9
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((pmSgn‘𝐷)‘◡𝐹) = ((pmSgn‘𝐷)‘𝐹)) |
61 | 58, 59, 60 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘◡𝐹) = ((pmSgn‘𝐷)‘𝐹)) |
62 | 26 | adantr 481 |
. . . . . . . 8
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘𝐹) = -1) |
63 | 61, 62 | eqtrd 2656 |
. . . . . . 7
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘◡𝐹) = -1) |
64 | 63 | oveq1d 6665 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘◡𝐹) · -1) = (-1 ·
-1)) |
65 | | neg1mulneg1e1 11245 |
. . . . . 6
⊢ (-1
· -1) = 1 |
66 | 64, 65 | syl6eq 2672 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((pmSgn‘𝐷)‘◡𝐹) · -1) = 1) |
67 | 50, 57, 66 | 3eqtrd 2660 |
. . . 4
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 1) |
68 | 2, 7, 14 | psgnevpmb 19933 |
. . . . 5
⊢ (𝐷 ∈ Fin →
((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 1))) |
69 | 68 | ad2antrr 762 |
. . . 4
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ (pmEven‘𝐷) ↔ ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ 𝑃 ∧ ((pmSgn‘𝐷)‘(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 1))) |
70 | 47, 67, 69 | mpbir2and 957 |
. . 3
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) ∈ (pmEven‘𝐷)) |
71 | | eqid 2622 |
. . 3
⊢ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) |
72 | 70, 71 | fmptd 6385 |
. 2
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)):(𝑃 ∖ (pmEven‘𝐷))⟶(pmEven‘𝐷)) |
73 | | eqidd 2623 |
. . . . 5
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) = (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) |
74 | | eqidd 2623 |
. . . . 5
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) |
75 | | oveq2 6658 |
. . . . 5
⊢ (𝑔 = (𝐹(+g‘𝑆)𝑓) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) = (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓))) |
76 | 36, 73, 74, 75 | fmptco 6396 |
. . . 4
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓)))) |
77 | | eqid 2622 |
. . . . . . . . 9
⊢
(0g‘𝑆) = (0g‘𝑆) |
78 | 7, 11, 77, 40 | grplinv 17468 |
. . . . . . . 8
⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹) = (0g‘𝑆)) |
79 | 4, 6, 78 | syl2anc 693 |
. . . . . . 7
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹) = (0g‘𝑆)) |
80 | 79 | oveq1d 6665 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹)(+g‘𝑆)𝑓) = ((0g‘𝑆)(+g‘𝑆)𝑓)) |
81 | 42 | adantr 481 |
. . . . . . 7
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((invg‘𝑆)‘𝐹) ∈ 𝑃) |
82 | 7, 11 | grpass 17431 |
. . . . . . 7
⊢ ((𝑆 ∈ Grp ∧
(((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝐹 ∈ 𝑃 ∧ 𝑓 ∈ 𝑃)) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹)(+g‘𝑆)𝑓) = (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓))) |
83 | 4, 81, 6, 10, 82 | syl13anc 1328 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((((invg‘𝑆)‘𝐹)(+g‘𝑆)𝐹)(+g‘𝑆)𝑓) = (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓))) |
84 | 7, 11, 77 | grplid 17452 |
. . . . . . 7
⊢ ((𝑆 ∈ Grp ∧ 𝑓 ∈ 𝑃) → ((0g‘𝑆)(+g‘𝑆)𝑓) = 𝑓) |
85 | 4, 10, 84 | syl2anc 693 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → ((0g‘𝑆)(+g‘𝑆)𝑓) = 𝑓) |
86 | 80, 83, 85 | 3eqtr3d 2664 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑓 ∈ (pmEven‘𝐷)) → (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓)) = 𝑓) |
87 | 86 | mpteq2dva 4744 |
. . . 4
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)(𝐹(+g‘𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓)) |
88 | 76, 87 | eqtrd 2656 |
. . 3
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) = (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓)) |
89 | | mptresid 5456 |
. . 3
⊢ (𝑓 ∈ (pmEven‘𝐷) ↦ 𝑓) = ( I ↾ (pmEven‘𝐷)) |
90 | 88, 89 | syl6eq 2672 |
. 2
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) ∘ (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓))) = ( I ↾ (pmEven‘𝐷))) |
91 | | oveq2 6658 |
. . . . 5
⊢ (𝑓 =
(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔) → (𝐹(+g‘𝑆)𝑓) = (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) |
92 | 70, 74, 73, 91 | fmptco 6396 |
. . . 4
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)))) |
93 | 7, 11, 77, 40 | grprinv 17469 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Grp ∧ 𝐹 ∈ 𝑃) → (𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹)) = (0g‘𝑆)) |
94 | 3, 5, 93 | syl2an 494 |
. . . . . . . 8
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹)) = (0g‘𝑆)) |
95 | 94 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = ((0g‘𝑆)(+g‘𝑆)𝑔)) |
96 | 95 | adantr 481 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = ((0g‘𝑆)(+g‘𝑆)𝑔)) |
97 | 7, 11 | grpass 17431 |
. . . . . . 7
⊢ ((𝑆 ∈ Grp ∧ (𝐹 ∈ 𝑃 ∧ ((invg‘𝑆)‘𝐹) ∈ 𝑃 ∧ 𝑔 ∈ 𝑃)) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) |
98 | 39, 59, 43, 45, 97 | syl13anc 1328 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝐹(+g‘𝑆)((invg‘𝑆)‘𝐹))(+g‘𝑆)𝑔) = (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) |
99 | 7, 11, 77 | grplid 17452 |
. . . . . . 7
⊢ ((𝑆 ∈ Grp ∧ 𝑔 ∈ 𝑃) → ((0g‘𝑆)(+g‘𝑆)𝑔) = 𝑔) |
100 | 39, 45, 99 | syl2anc 693 |
. . . . . 6
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((0g‘𝑆)(+g‘𝑆)𝑔) = 𝑔) |
101 | 96, 98, 100 | 3eqtr3d 2664 |
. . . . 5
⊢ (((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) ∧ 𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔)) = 𝑔) |
102 | 101 | mpteq2dva 4744 |
. . . 4
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (𝐹(+g‘𝑆)(((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔)) |
103 | 92, 102 | eqtrd 2656 |
. . 3
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔)) |
104 | | mptresid 5456 |
. . 3
⊢ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ 𝑔) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷))) |
105 | 103, 104 | syl6eq 2672 |
. 2
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)) ∘ (𝑔 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↦ (((invg‘𝑆)‘𝐹)(+g‘𝑆)𝑔))) = ( I ↾ (𝑃 ∖ (pmEven‘𝐷)))) |
106 | 38, 72, 90, 105 | fcof1od 6549 |
1
⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)):(pmEven‘𝐷)–1-1-onto→(𝑃 ∖ (pmEven‘𝐷))) |