Step | Hyp | Ref
| Expression |
1 | | cntzsubr.m |
. . . . . 6
⊢ 𝑀 = (mulGrp‘𝑅) |
2 | | cntzsubr.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
3 | 1, 2 | mgpbas 19735 |
. . . . 5
⊢ 𝐵 = (Base‘𝑀) |
4 | | cntzsubr.z |
. . . . 5
⊢ 𝑍 = (Cntz‘𝑀) |
5 | 3, 4 | cntzssv 18943 |
. . . 4
⊢ (𝑍‘𝑆) ⊆ 𝐵 |
6 | 5 | a1i 11 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵) |
7 | | simpll 764 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → 𝑅 ∈ Ring) |
8 | | ssel2 3917 |
. . . . . . . . 9
⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵) |
9 | 8 | adantll 711 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵) |
10 | | eqid 2739 |
. . . . . . . . 9
⊢
(.r‘𝑅) = (.r‘𝑅) |
11 | | eqid 2739 |
. . . . . . . . 9
⊢
(0g‘𝑅) = (0g‘𝑅) |
12 | 2, 10, 11 | ringlz 19835 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐵) → ((0g‘𝑅)(.r‘𝑅)𝑧) = (0g‘𝑅)) |
13 | 7, 9, 12 | syl2anc 584 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → ((0g‘𝑅)(.r‘𝑅)𝑧) = (0g‘𝑅)) |
14 | 2, 10, 11 | ringrz 19836 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐵) → (𝑧(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
15 | 7, 9, 14 | syl2anc 584 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → (𝑧(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
16 | 13, 15 | eqtr4d 2782 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → ((0g‘𝑅)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(0g‘𝑅))) |
17 | 16 | ralrimiva 3104 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ∀𝑧 ∈ 𝑆 ((0g‘𝑅)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(0g‘𝑅))) |
18 | | simpr 485 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵) |
19 | 2, 11 | ring0cl 19817 |
. . . . . . 7
⊢ (𝑅 ∈ Ring →
(0g‘𝑅)
∈ 𝐵) |
20 | 19 | adantr 481 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑅) ∈ 𝐵) |
21 | 1, 10 | mgpplusg 19733 |
. . . . . . 7
⊢
(.r‘𝑅) = (+g‘𝑀) |
22 | 3, 21, 4 | cntzel 18938 |
. . . . . 6
⊢ ((𝑆 ⊆ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) → ((0g‘𝑅) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((0g‘𝑅)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(0g‘𝑅)))) |
23 | 18, 20, 22 | syl2anc 584 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((0g‘𝑅) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((0g‘𝑅)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(0g‘𝑅)))) |
24 | 17, 23 | mpbird 256 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑅) ∈ (𝑍‘𝑆)) |
25 | 24 | ne0d 4270 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ≠ ∅) |
26 | | simpl2 1191 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ (𝑍‘𝑆)) |
27 | | simpr 485 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆) |
28 | 21, 4 | cntzi 18944 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)𝑥)) |
29 | 26, 27, 28 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)𝑥)) |
30 | | simpl3 1192 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ (𝑍‘𝑆)) |
31 | 21, 4 | cntzi 18944 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑦(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)𝑦)) |
32 | 30, 27, 31 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑦(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)𝑦)) |
33 | 29, 32 | oveq12d 7302 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧)) = ((𝑧(.r‘𝑅)𝑥)(+g‘𝑅)(𝑧(.r‘𝑅)𝑦))) |
34 | | simpl1l 1223 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑅 ∈ Ring) |
35 | 5, 26 | sselid 3920 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ 𝐵) |
36 | 5, 30 | sselid 3920 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝐵) |
37 | | simp1r 1197 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵) |
38 | 37 | sselda 3922 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵) |
39 | | eqid 2739 |
. . . . . . . . . . . 12
⊢
(+g‘𝑅) = (+g‘𝑅) |
40 | 2, 39, 10 | ringdir 19815 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))) |
41 | 34, 35, 36, 38, 40 | syl13anc 1371 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))) |
42 | 2, 39, 10 | ringdi 19814 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)) = ((𝑧(.r‘𝑅)𝑥)(+g‘𝑅)(𝑧(.r‘𝑅)𝑦))) |
43 | 34, 38, 35, 36, 42 | syl13anc 1371 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)) = ((𝑧(.r‘𝑅)𝑥)(+g‘𝑅)(𝑧(.r‘𝑅)𝑦))) |
44 | 33, 41, 43 | 3eqtr4d 2789 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦))) |
45 | 44 | ralrimiva 3104 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → ∀𝑧 ∈ 𝑆 ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦))) |
46 | | simp1l 1196 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑅 ∈ Ring) |
47 | | simp2 1136 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑥 ∈ (𝑍‘𝑆)) |
48 | 5, 47 | sselid 3920 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑥 ∈ 𝐵) |
49 | | simp3 1137 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑦 ∈ (𝑍‘𝑆)) |
50 | 5, 49 | sselid 3920 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑦 ∈ 𝐵) |
51 | 2, 39 | ringacl 19826 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑅)𝑦) ∈ 𝐵) |
52 | 46, 48, 50, 51 | syl3anc 1370 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → (𝑥(+g‘𝑅)𝑦) ∈ 𝐵) |
53 | 3, 21, 4 | cntzel 18938 |
. . . . . . . . 9
⊢ ((𝑆 ⊆ 𝐵 ∧ (𝑥(+g‘𝑅)𝑦) ∈ 𝐵) → ((𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)))) |
54 | 37, 52, 53 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → ((𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)))) |
55 | 45, 54 | mpbird 256 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → (𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆)) |
56 | 55 | 3expa 1117 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑦 ∈ (𝑍‘𝑆)) → (𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆)) |
57 | 56 | ralrimiva 3104 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆)) |
58 | 28 | adantll 711 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)𝑥)) |
59 | 58 | fveq2d 6787 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((invg‘𝑅)‘(𝑥(.r‘𝑅)𝑧)) = ((invg‘𝑅)‘(𝑧(.r‘𝑅)𝑥))) |
60 | | eqid 2739 |
. . . . . . . . 9
⊢
(invg‘𝑅) = (invg‘𝑅) |
61 | | simplll 772 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑅 ∈ Ring) |
62 | | simplr 766 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ (𝑍‘𝑆)) |
63 | 5, 62 | sselid 3920 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ 𝐵) |
64 | | simplr 766 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵) |
65 | 64 | sselda 3922 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵) |
66 | 2, 10, 60, 61, 63, 65 | ringmneg1 19844 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (((invg‘𝑅)‘𝑥)(.r‘𝑅)𝑧) = ((invg‘𝑅)‘(𝑥(.r‘𝑅)𝑧))) |
67 | 2, 10, 60, 61, 65, 63 | ringmneg2 19845 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑧(.r‘𝑅)((invg‘𝑅)‘𝑥)) = ((invg‘𝑅)‘(𝑧(.r‘𝑅)𝑥))) |
68 | 59, 66, 67 | 3eqtr4d 2789 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (((invg‘𝑅)‘𝑥)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)((invg‘𝑅)‘𝑥))) |
69 | 68 | ralrimiva 3104 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑧 ∈ 𝑆 (((invg‘𝑅)‘𝑥)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)((invg‘𝑅)‘𝑥))) |
70 | | ringgrp 19797 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
71 | 70 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑅 ∈ Grp) |
72 | | simpr 485 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ (𝑍‘𝑆)) |
73 | 5, 72 | sselid 3920 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ 𝐵) |
74 | 2, 60 | grpinvcl 18636 |
. . . . . . . 8
⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑅)‘𝑥) ∈ 𝐵) |
75 | 71, 73, 74 | syl2anc 584 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑅)‘𝑥) ∈ 𝐵) |
76 | 3, 21, 4 | cntzel 18938 |
. . . . . . 7
⊢ ((𝑆 ⊆ 𝐵 ∧ ((invg‘𝑅)‘𝑥) ∈ 𝐵) → (((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 (((invg‘𝑅)‘𝑥)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)((invg‘𝑅)‘𝑥)))) |
77 | 64, 75, 76 | syl2anc 584 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → (((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 (((invg‘𝑅)‘𝑥)(.r‘𝑅)𝑧) = (𝑧(.r‘𝑅)((invg‘𝑅)‘𝑥)))) |
78 | 69, 77 | mpbird 256 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆)) |
79 | 57, 78 | jca 512 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → (∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆))) |
80 | 79 | ralrimiva 3104 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)(∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆))) |
81 | 70 | adantr 481 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → 𝑅 ∈ Grp) |
82 | 2, 39, 60 | issubg2 18779 |
. . . 4
⊢ (𝑅 ∈ Grp → ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ↔ ((𝑍‘𝑆) ⊆ 𝐵 ∧ (𝑍‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ (𝑍‘𝑆)(∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆))))) |
83 | 81, 82 | syl 17 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ↔ ((𝑍‘𝑆) ⊆ 𝐵 ∧ (𝑍‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ (𝑍‘𝑆)(∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((invg‘𝑅)‘𝑥) ∈ (𝑍‘𝑆))))) |
84 | 6, 25, 80, 83 | mpbir3and 1341 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑅)) |
85 | 1 | ringmgp 19798 |
. . 3
⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
86 | 3, 4 | cntzsubm 18951 |
. . 3
⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀)) |
87 | 85, 86 | sylan 580 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀)) |
88 | 1 | issubrg3 20061 |
. . 3
⊢ (𝑅 ∈ Ring → ((𝑍‘𝑆) ∈ (SubRing‘𝑅) ↔ ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ∧ (𝑍‘𝑆) ∈ (SubMnd‘𝑀)))) |
89 | 88 | adantr 481 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((𝑍‘𝑆) ∈ (SubRing‘𝑅) ↔ ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ∧ (𝑍‘𝑆) ∈ (SubMnd‘𝑀)))) |
90 | 84, 87, 89 | mpbir2and 710 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRing‘𝑅)) |