Metamath Proof Explorer


Theorem imasabl

Description: The image structure of an abelian group is an abelian group ( imasgrp analog). (Contributed by AV, 22-Feb-2025)

Ref Expression
Hypotheses imasabl.u φ U = F 𝑠 R
imasabl.v φ V = Base R
imasabl.p φ + ˙ = + R
imasabl.f φ F : V onto B
imasabl.e φ a V b V p V q V F a = F p F b = F q F a + ˙ b = F p + ˙ q
imasabl.r φ R Abel
imasabl.z 0 ˙ = 0 R
Assertion imasabl φ U Abel F 0 ˙ = 0 U

Proof

Step Hyp Ref Expression
1 imasabl.u φ U = F 𝑠 R
2 imasabl.v φ V = Base R
3 imasabl.p φ + ˙ = + R
4 imasabl.f φ F : V onto B
5 imasabl.e φ a V b V p V q V F a = F p F b = F q F a + ˙ b = F p + ˙ q
6 imasabl.r φ R Abel
7 imasabl.z 0 ˙ = 0 R
8 6 ablgrpd φ R Grp
9 1 2 3 4 5 8 7 imasgrp φ U Grp F 0 ˙ = 0 U
10 1 2 4 6 imasbas φ B = Base U
11 10 eqcomd φ Base U = B
12 11 eleq2d φ x Base U x B
13 11 eleq2d φ y Base U y B
14 12 13 anbi12d φ x Base U y Base U x B y B
15 14 adantr φ U Grp F 0 ˙ = 0 U x Base U y Base U x B y B
16 foelcdmi F : V onto B x B a V F a = x
17 16 ex F : V onto B x B a V F a = x
18 foelcdmi F : V onto B y B b V F b = y
19 18 ex F : V onto B y B b V F b = y
20 17 19 anim12d F : V onto B x B y B a V F a = x b V F b = y
21 4 20 syl φ x B y B a V F a = x b V F b = y
22 21 adantr φ U Grp F 0 ˙ = 0 U x B y B a V F a = x b V F b = y
23 6 ad3antrrr φ U Grp F 0 ˙ = 0 U a V b V R Abel
24 2 eleq2d φ a V a Base R
25 24 biimpd φ a V a Base R
26 25 adantr φ U Grp F 0 ˙ = 0 U a V a Base R
27 26 imp φ U Grp F 0 ˙ = 0 U a V a Base R
28 27 adantr φ U Grp F 0 ˙ = 0 U a V b V a Base R
29 2 eleq2d φ b V b Base R
30 29 biimpd φ b V b Base R
31 30 adantr φ U Grp F 0 ˙ = 0 U b V b Base R
32 31 adantr φ U Grp F 0 ˙ = 0 U a V b V b Base R
33 32 imp φ U Grp F 0 ˙ = 0 U a V b V b Base R
34 eqid Base R = Base R
35 eqid + R = + R
36 34 35 ablcom R Abel a Base R b Base R a + R b = b + R a
37 23 28 33 36 syl3anc φ U Grp F 0 ˙ = 0 U a V b V a + R b = b + R a
38 37 fveq2d φ U Grp F 0 ˙ = 0 U a V b V F a + R b = F b + R a
39 simplll φ U Grp F 0 ˙ = 0 U a V b V φ
40 simpr φ U Grp F 0 ˙ = 0 U a V a V
41 40 adantr φ U Grp F 0 ˙ = 0 U a V b V a V
42 simpr φ U Grp F 0 ˙ = 0 U a V b V b V
43 3 eqcomd φ + R = + ˙
44 43 oveqd φ a + R b = a + ˙ b
45 44 fveq2d φ F a + R b = F a + ˙ b
46 43 oveqd φ p + R q = p + ˙ q
47 46 fveq2d φ F p + R q = F p + ˙ q
48 45 47 eqeq12d φ F a + R b = F p + R q F a + ˙ b = F p + ˙ q
49 48 3ad2ant1 φ a V b V p V q V F a + R b = F p + R q F a + ˙ b = F p + ˙ q
50 5 49 sylibrd φ a V b V p V q V F a = F p F b = F q F a + R b = F p + R q
51 eqid + U = + U
52 4 50 1 2 6 35 51 imasaddval φ a V b V F a + U F b = F a + R b
53 39 41 42 52 syl3anc φ U Grp F 0 ˙ = 0 U a V b V F a + U F b = F a + R b
54 4 50 1 2 6 35 51 imasaddval φ b V a V F b + U F a = F b + R a
55 39 42 41 54 syl3anc φ U Grp F 0 ˙ = 0 U a V b V F b + U F a = F b + R a
56 38 53 55 3eqtr4d φ U Grp F 0 ˙ = 0 U a V b V F a + U F b = F b + U F a
57 56 adantr φ U Grp F 0 ˙ = 0 U a V b V F b = y F a = x F a + U F b = F b + U F a
58 oveq12 F a = x F b = y F a + U F b = x + U y
59 58 ancoms F b = y F a = x F a + U F b = x + U y
60 oveq12 F b = y F a = x F b + U F a = y + U x
61 59 60 eqeq12d F b = y F a = x F a + U F b = F b + U F a x + U y = y + U x
62 61 adantl φ U Grp F 0 ˙ = 0 U a V b V F b = y F a = x F a + U F b = F b + U F a x + U y = y + U x
63 57 62 mpbid φ U Grp F 0 ˙ = 0 U a V b V F b = y F a = x x + U y = y + U x
64 63 exp32 φ U Grp F 0 ˙ = 0 U a V b V F b = y F a = x x + U y = y + U x
65 64 rexlimdva φ U Grp F 0 ˙ = 0 U a V b V F b = y F a = x x + U y = y + U x
66 65 com23 φ U Grp F 0 ˙ = 0 U a V F a = x b V F b = y x + U y = y + U x
67 66 rexlimdva φ U Grp F 0 ˙ = 0 U a V F a = x b V F b = y x + U y = y + U x
68 67 impd φ U Grp F 0 ˙ = 0 U a V F a = x b V F b = y x + U y = y + U x
69 22 68 syld φ U Grp F 0 ˙ = 0 U x B y B x + U y = y + U x
70 15 69 sylbid φ U Grp F 0 ˙ = 0 U x Base U y Base U x + U y = y + U x
71 70 imp φ U Grp F 0 ˙ = 0 U x Base U y Base U x + U y = y + U x
72 71 ralrimivva φ U Grp F 0 ˙ = 0 U x Base U y Base U x + U y = y + U x
73 simpr φ U Grp F 0 ˙ = 0 U U Grp F 0 ˙ = 0 U
74 72 73 jca φ U Grp F 0 ˙ = 0 U x Base U y Base U x + U y = y + U x U Grp F 0 ˙ = 0 U
75 9 74 mpdan φ x Base U y Base U x + U y = y + U x U Grp F 0 ˙ = 0 U
76 eqid Base U = Base U
77 76 51 isabl2 U Abel U Grp x Base U y Base U x + U y = y + U x
78 77 anbi1i U Abel F 0 ˙ = 0 U U Grp x Base U y Base U x + U y = y + U x F 0 ˙ = 0 U
79 an21 U Grp x Base U y Base U x + U y = y + U x F 0 ˙ = 0 U x Base U y Base U x + U y = y + U x U Grp F 0 ˙ = 0 U
80 78 79 bitri U Abel F 0 ˙ = 0 U x Base U y Base U x + U y = y + U x U Grp F 0 ˙ = 0 U
81 75 80 sylibr φ U Abel F 0 ˙ = 0 U