Step |
Hyp |
Ref |
Expression |
1 |
|
mulgass2.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
mulgass2.m |
⊢ · = ( .g ‘ 𝑅 ) |
3 |
|
mulgass2.t |
⊢ × = ( .r ‘ 𝑅 ) |
4 |
|
oveq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 · 𝑋 ) = ( 0 · 𝑋 ) ) |
5 |
4
|
oveq1d |
⊢ ( 𝑥 = 0 → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( 0 · 𝑋 ) × 𝑌 ) ) |
6 |
|
oveq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( 0 · ( 𝑋 × 𝑌 ) ) ) |
7 |
5 6
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( 0 · 𝑋 ) × 𝑌 ) = ( 0 · ( 𝑋 × 𝑌 ) ) ) ) |
8 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 · 𝑋 ) = ( 𝑦 · 𝑋 ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( 𝑦 · 𝑋 ) × 𝑌 ) ) |
10 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
12 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · 𝑋 ) = ( ( 𝑦 + 1 ) · 𝑋 ) ) |
13 |
12
|
oveq1d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) ) |
14 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) |
15 |
13 14
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) |
16 |
|
oveq1 |
⊢ ( 𝑥 = - 𝑦 → ( 𝑥 · 𝑋 ) = ( - 𝑦 · 𝑋 ) ) |
17 |
16
|
oveq1d |
⊢ ( 𝑥 = - 𝑦 → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( - 𝑦 · 𝑋 ) × 𝑌 ) ) |
18 |
|
oveq1 |
⊢ ( 𝑥 = - 𝑦 → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( - 𝑦 · ( 𝑋 × 𝑌 ) ) ) |
19 |
17 18
|
eqeq12d |
⊢ ( 𝑥 = - 𝑦 → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( - 𝑦 · 𝑋 ) × 𝑌 ) = ( - 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
20 |
|
oveq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 · 𝑋 ) = ( 𝑁 · 𝑋 ) ) |
21 |
20
|
oveq1d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( 𝑁 · 𝑋 ) × 𝑌 ) ) |
22 |
|
oveq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) |
23 |
21 22
|
eqeq12d |
⊢ ( 𝑥 = 𝑁 → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) |
24 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
25 |
1 3 24
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ) → ( ( 0g ‘ 𝑅 ) × 𝑌 ) = ( 0g ‘ 𝑅 ) ) |
26 |
25
|
3adant3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 0g ‘ 𝑅 ) × 𝑌 ) = ( 0g ‘ 𝑅 ) ) |
27 |
|
simp3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
28 |
1 24 2
|
mulg0 |
⊢ ( 𝑋 ∈ 𝐵 → ( 0 · 𝑋 ) = ( 0g ‘ 𝑅 ) ) |
29 |
27 28
|
syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 0 · 𝑋 ) = ( 0g ‘ 𝑅 ) ) |
30 |
29
|
oveq1d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 · 𝑋 ) × 𝑌 ) = ( ( 0g ‘ 𝑅 ) × 𝑌 ) ) |
31 |
1 3
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
32 |
31
|
3com23 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
33 |
1 24 2
|
mulg0 |
⊢ ( ( 𝑋 × 𝑌 ) ∈ 𝐵 → ( 0 · ( 𝑋 × 𝑌 ) ) = ( 0g ‘ 𝑅 ) ) |
34 |
32 33
|
syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 0 · ( 𝑋 × 𝑌 ) ) = ( 0g ‘ 𝑅 ) ) |
35 |
26 30 34
|
3eqtr4d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 · 𝑋 ) × 𝑌 ) = ( 0 · ( 𝑋 × 𝑌 ) ) ) |
36 |
|
oveq1 |
⊢ ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
37 |
|
simpl1 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑅 ∈ Ring ) |
38 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
39 |
37 38
|
syl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑅 ∈ Grp ) |
40 |
|
nn0z |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ ) |
41 |
40
|
adantl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑦 ∈ ℤ ) |
42 |
27
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑋 ∈ 𝐵 ) |
43 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
44 |
1 2 43
|
mulgp1 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑦 + 1 ) · 𝑋 ) = ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) ) |
45 |
39 41 42 44
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 + 1 ) · 𝑋 ) = ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) ) |
46 |
45
|
oveq1d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) × 𝑌 ) ) |
47 |
38
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑅 ∈ Grp ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑅 ∈ Grp ) |
49 |
1 2
|
mulgcl |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑦 · 𝑋 ) ∈ 𝐵 ) |
50 |
48 41 42 49
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( 𝑦 · 𝑋 ) ∈ 𝐵 ) |
51 |
|
simpl2 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑌 ∈ 𝐵 ) |
52 |
1 43 3
|
ringdir |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑦 · 𝑋 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
53 |
37 50 42 51 52
|
syl13anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
54 |
46 53
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
55 |
32
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
56 |
1 2 43
|
mulgp1 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ ( 𝑋 × 𝑌 ) ∈ 𝐵 ) → ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
57 |
39 41 55 56
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
58 |
54 57
|
eqeq12d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ↔ ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) ) |
59 |
36 58
|
syl5ibr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) |
60 |
59
|
ex |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑦 ∈ ℕ0 → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) ) |
61 |
|
fveq2 |
⊢ ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( invg ‘ 𝑅 ) ‘ ( ( 𝑦 · 𝑋 ) × 𝑌 ) ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
62 |
47
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → 𝑅 ∈ Grp ) |
63 |
|
nnz |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℤ ) |
64 |
63
|
adantl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → 𝑦 ∈ ℤ ) |
65 |
27
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → 𝑋 ∈ 𝐵 ) |
66 |
|
eqid |
⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 ) |
67 |
1 2 66
|
mulgneg |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( - 𝑦 · 𝑋 ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · 𝑋 ) ) ) |
68 |
62 64 65 67
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( - 𝑦 · 𝑋 ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · 𝑋 ) ) ) |
69 |
68
|
oveq1d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( ( - 𝑦 · 𝑋 ) × 𝑌 ) = ( ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · 𝑋 ) ) × 𝑌 ) ) |
70 |
|
simpl1 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → 𝑅 ∈ Ring ) |
71 |
62 64 65 49
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( 𝑦 · 𝑋 ) ∈ 𝐵 ) |
72 |
|
simpl2 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → 𝑌 ∈ 𝐵 ) |
73 |
1 3 66 70 71 72
|
ringmneg1 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · 𝑋 ) ) × 𝑌 ) = ( ( invg ‘ 𝑅 ) ‘ ( ( 𝑦 · 𝑋 ) × 𝑌 ) ) ) |
74 |
69 73
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( ( - 𝑦 · 𝑋 ) × 𝑌 ) = ( ( invg ‘ 𝑅 ) ‘ ( ( 𝑦 · 𝑋 ) × 𝑌 ) ) ) |
75 |
32
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
76 |
1 2 66
|
mulgneg |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ ( 𝑋 × 𝑌 ) ∈ 𝐵 ) → ( - 𝑦 · ( 𝑋 × 𝑌 ) ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
77 |
62 64 75 76
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( - 𝑦 · ( 𝑋 × 𝑌 ) ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
78 |
74 77
|
eqeq12d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( ( ( - 𝑦 · 𝑋 ) × 𝑌 ) = ( - 𝑦 · ( 𝑋 × 𝑌 ) ) ↔ ( ( invg ‘ 𝑅 ) ‘ ( ( 𝑦 · 𝑋 ) × 𝑌 ) ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) ) |
79 |
61 78
|
syl5ibr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑦 ∈ ℕ ) → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( - 𝑦 · 𝑋 ) × 𝑌 ) = ( - 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
80 |
79
|
ex |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑦 ∈ ℕ → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( - 𝑦 · 𝑋 ) × 𝑌 ) = ( - 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) ) |
81 |
7 11 15 19 23 35 60 80
|
zindd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ∈ ℤ → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) |
82 |
81
|
3exp |
⊢ ( 𝑅 ∈ Ring → ( 𝑌 ∈ 𝐵 → ( 𝑋 ∈ 𝐵 → ( 𝑁 ∈ ℤ → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) ) ) |
83 |
82
|
com24 |
⊢ ( 𝑅 ∈ Ring → ( 𝑁 ∈ ℤ → ( 𝑋 ∈ 𝐵 → ( 𝑌 ∈ 𝐵 → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) ) ) |
84 |
83
|
3imp2 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) |