Step |
Hyp |
Ref |
Expression |
1 |
|
invrvald.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
invrvald.t |
⊢ · = ( .r ‘ 𝑅 ) |
3 |
|
invrvald.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
4 |
|
invrvald.u |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
5 |
|
invrvald.i |
⊢ 𝐼 = ( invr ‘ 𝑅 ) |
6 |
|
invrvald.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
7 |
|
invrvald.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
|
invrvald.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
9 |
|
invrvald.xy |
⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = 1 ) |
10 |
|
invrvald.yx |
⊢ ( 𝜑 → ( 𝑌 · 𝑋 ) = 1 ) |
11 |
|
eqid |
⊢ ( ∥r ‘ 𝑅 ) = ( ∥r ‘ 𝑅 ) |
12 |
1 11 2
|
dvdsrmul |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ( ∥r ‘ 𝑅 ) ( 𝑌 · 𝑋 ) ) |
13 |
7 8 12
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ( ∥r ‘ 𝑅 ) ( 𝑌 · 𝑋 ) ) |
14 |
13 10
|
breqtrd |
⊢ ( 𝜑 → 𝑋 ( ∥r ‘ 𝑅 ) 1 ) |
15 |
|
eqid |
⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 ) |
16 |
15 1
|
opprbas |
⊢ 𝐵 = ( Base ‘ ( oppr ‘ 𝑅 ) ) |
17 |
|
eqid |
⊢ ( ∥r ‘ ( oppr ‘ 𝑅 ) ) = ( ∥r ‘ ( oppr ‘ 𝑅 ) ) |
18 |
|
eqid |
⊢ ( .r ‘ ( oppr ‘ 𝑅 ) ) = ( .r ‘ ( oppr ‘ 𝑅 ) ) |
19 |
16 17 18
|
dvdsrmul |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) ) |
20 |
7 8 19
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) ) |
21 |
1 2 15 18
|
opprmul |
⊢ ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) = ( 𝑋 · 𝑌 ) |
22 |
21 9
|
eqtrid |
⊢ ( 𝜑 → ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) = 1 ) |
23 |
20 22
|
breqtrd |
⊢ ( 𝜑 → 𝑋 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) 1 ) |
24 |
4 3 11 15 17
|
isunit |
⊢ ( 𝑋 ∈ 𝑈 ↔ ( 𝑋 ( ∥r ‘ 𝑅 ) 1 ∧ 𝑋 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) 1 ) ) |
25 |
14 23 24
|
sylanbrc |
⊢ ( 𝜑 → 𝑋 ∈ 𝑈 ) |
26 |
|
eqid |
⊢ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) = ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) |
27 |
4 26 3
|
unitgrpid |
⊢ ( 𝑅 ∈ Ring → 1 = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
28 |
6 27
|
syl |
⊢ ( 𝜑 → 1 = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
29 |
9 28
|
eqtrd |
⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
30 |
4 26
|
unitgrp |
⊢ ( 𝑅 ∈ Ring → ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ∈ Grp ) |
31 |
6 30
|
syl |
⊢ ( 𝜑 → ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ∈ Grp ) |
32 |
1 11 2
|
dvdsrmul |
⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑌 ( ∥r ‘ 𝑅 ) ( 𝑋 · 𝑌 ) ) |
33 |
8 7 32
|
syl2anc |
⊢ ( 𝜑 → 𝑌 ( ∥r ‘ 𝑅 ) ( 𝑋 · 𝑌 ) ) |
34 |
33 9
|
breqtrd |
⊢ ( 𝜑 → 𝑌 ( ∥r ‘ 𝑅 ) 1 ) |
35 |
16 17 18
|
dvdsrmul |
⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑌 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 𝑋 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑌 ) ) |
36 |
8 7 35
|
syl2anc |
⊢ ( 𝜑 → 𝑌 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 𝑋 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑌 ) ) |
37 |
1 2 15 18
|
opprmul |
⊢ ( 𝑋 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑌 ) = ( 𝑌 · 𝑋 ) |
38 |
37 10
|
eqtrid |
⊢ ( 𝜑 → ( 𝑋 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑌 ) = 1 ) |
39 |
36 38
|
breqtrd |
⊢ ( 𝜑 → 𝑌 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) 1 ) |
40 |
4 3 11 15 17
|
isunit |
⊢ ( 𝑌 ∈ 𝑈 ↔ ( 𝑌 ( ∥r ‘ 𝑅 ) 1 ∧ 𝑌 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) 1 ) ) |
41 |
34 39 40
|
sylanbrc |
⊢ ( 𝜑 → 𝑌 ∈ 𝑈 ) |
42 |
4 26
|
unitgrpbas |
⊢ 𝑈 = ( Base ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
43 |
4
|
fvexi |
⊢ 𝑈 ∈ V |
44 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
45 |
44 2
|
mgpplusg |
⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
46 |
26 45
|
ressplusg |
⊢ ( 𝑈 ∈ V → · = ( +g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
47 |
43 46
|
ax-mp |
⊢ · = ( +g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
48 |
|
eqid |
⊢ ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
49 |
4 26 5
|
invrfval |
⊢ 𝐼 = ( invg ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
50 |
42 47 48 49
|
grpinvid1 |
⊢ ( ( ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ∈ Grp ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈 ) → ( ( 𝐼 ‘ 𝑋 ) = 𝑌 ↔ ( 𝑋 · 𝑌 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
51 |
31 25 41 50
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) = 𝑌 ↔ ( 𝑋 · 𝑌 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
52 |
29 51
|
mpbird |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) = 𝑌 ) |
53 |
25 52
|
jca |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝑈 ∧ ( 𝐼 ‘ 𝑋 ) = 𝑌 ) ) |