Step | Hyp | Ref
| Expression |
1 | | imasgrp.u |
. 2
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
2 | | imasgrp.v |
. 2
⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
3 | | imasgrp.p |
. 2
⊢ (𝜑 → + =
(+g‘𝑅)) |
4 | | imasgrp.f |
. 2
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
5 | | imasgrp.e |
. 2
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) |
6 | | imasgrp.r |
. 2
⊢ (𝜑 → 𝑅 ∈ Grp) |
7 | 6 | 3ad2ant1 1082 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑅 ∈ Grp) |
8 | | simp2 1062 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉) |
9 | 2 | 3ad2ant1 1082 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅)) |
10 | 8, 9 | eleqtrd 2703 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) |
11 | | simp3 1063 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) |
12 | 11, 9 | eleqtrd 2703 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅)) |
13 | | eqid 2622 |
. . . . 5
⊢
(Base‘𝑅) =
(Base‘𝑅) |
14 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝑅) = (+g‘𝑅) |
15 | 13, 14 | grpcl 17430 |
. . . 4
⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
16 | 7, 10, 12, 15 | syl3anc 1326 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
17 | 3 | 3ad2ant1 1082 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → + =
(+g‘𝑅)) |
18 | 17 | oveqd 6667 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦)) |
19 | 16, 18, 9 | 3eltr4d 2716 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) |
20 | 6 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Grp) |
21 | 10 | 3adant3r3 1276 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅)) |
22 | 12 | 3adant3r3 1276 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅)) |
23 | | simpr3 1069 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉) |
24 | 2 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) |
25 | 23, 24 | eleqtrd 2703 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅)) |
26 | 13, 14 | grpass 17431 |
. . . . 5
⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦)(+g‘𝑅)𝑧) = (𝑥(+g‘𝑅)(𝑦(+g‘𝑅)𝑧))) |
27 | 20, 21, 22, 25, 26 | syl13anc 1328 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥(+g‘𝑅)𝑦)(+g‘𝑅)𝑧) = (𝑥(+g‘𝑅)(𝑦(+g‘𝑅)𝑧))) |
28 | 3 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → + =
(+g‘𝑅)) |
29 | 18 | 3adant3r3 1276 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦)) |
30 | | eqidd 2623 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 = 𝑧) |
31 | 28, 29, 30 | oveq123d 6671 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝑅)𝑦)(+g‘𝑅)𝑧)) |
32 | | eqidd 2623 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 = 𝑥) |
33 | 28 | oveqd 6667 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) = (𝑦(+g‘𝑅)𝑧)) |
34 | 28, 32, 33 | oveq123d 6671 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝑅)(𝑦(+g‘𝑅)𝑧))) |
35 | 27, 31, 34 | 3eqtr4d 2666 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
36 | 35 | fveq2d 6195 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧)))) |
37 | | imasgrp.z |
. . . . 5
⊢ 0 =
(0g‘𝑅) |
38 | 13, 37 | grpidcl 17450 |
. . . 4
⊢ (𝑅 ∈ Grp → 0 ∈
(Base‘𝑅)) |
39 | 6, 38 | syl 17 |
. . 3
⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
40 | 39, 2 | eleqtrrd 2704 |
. 2
⊢ (𝜑 → 0 ∈ 𝑉) |
41 | 3 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → + =
(+g‘𝑅)) |
42 | 41 | oveqd 6667 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = ( 0 (+g‘𝑅)𝑥)) |
43 | 6 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ Grp) |
44 | 2 | eleq2d 2687 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅))) |
45 | 44 | biimpa 501 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) |
46 | 13, 14, 37 | grplid 17452 |
. . . . 5
⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
47 | 43, 45, 46 | syl2anc 693 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
48 | 42, 47 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = 𝑥) |
49 | 48 | fveq2d 6195 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) |
50 | | eqid 2622 |
. . . . 5
⊢
(invg‘𝑅) = (invg‘𝑅) |
51 | 13, 50 | grpinvcl 17467 |
. . . 4
⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) →
((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅)) |
52 | 43, 45, 51 | syl2anc 693 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅)) |
53 | 2 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑉 = (Base‘𝑅)) |
54 | 52, 53 | eleqtrrd 2704 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((invg‘𝑅)‘𝑥) ∈ 𝑉) |
55 | 41 | oveqd 6667 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((invg‘𝑅)‘𝑥) + 𝑥) = (((invg‘𝑅)‘𝑥)(+g‘𝑅)𝑥)) |
56 | 13, 14, 37, 50 | grplinv 17468 |
. . . . 5
⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) →
(((invg‘𝑅)‘𝑥)(+g‘𝑅)𝑥) = 0 ) |
57 | 43, 45, 56 | syl2anc 693 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((invg‘𝑅)‘𝑥)(+g‘𝑅)𝑥) = 0 ) |
58 | 55, 57 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((invg‘𝑅)‘𝑥) + 𝑥) = 0 ) |
59 | 58 | fveq2d 6195 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(((invg‘𝑅)‘𝑥) + 𝑥)) = (𝐹‘ 0 )) |
60 | 1, 2, 3, 4, 5, 6, 19, 36, 40, 49, 54, 59 | imasgrp2 17530 |
1
⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈))) |