Step |
Hyp |
Ref |
Expression |
1 |
|
srgmulgass.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
srgmulgass.m |
⊢ · = ( .g ‘ 𝑅 ) |
3 |
|
srgmulgass.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 |
7
|
imbi2d |
⊢ ( 𝑥 = 0 → ( ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ) ↔ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 0 · 𝑋 ) × 𝑌 ) = ( 0 · ( 𝑋 × 𝑌 ) ) ) ) ) |
9 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 · 𝑋 ) = ( 𝑦 · 𝑋 ) ) |
10 |
9
|
oveq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( 𝑦 · 𝑋 ) × 𝑌 ) ) |
11 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) |
12 |
10 11
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ) ↔ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) ) ) |
14 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · 𝑋 ) = ( ( 𝑦 + 1 ) · 𝑋 ) ) |
15 |
14
|
oveq1d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) ) |
16 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) |
17 |
15 16
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ) ↔ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) ) |
19 |
|
oveq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 · 𝑋 ) = ( 𝑁 · 𝑋 ) ) |
20 |
19
|
oveq1d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( ( 𝑁 · 𝑋 ) × 𝑌 ) ) |
21 |
|
oveq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 · ( 𝑋 × 𝑌 ) ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) |
22 |
20 21
|
eqeq12d |
⊢ ( 𝑥 = 𝑁 → ( ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ↔ ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) |
23 |
22
|
imbi2d |
⊢ ( 𝑥 = 𝑁 → ( ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑥 · 𝑋 ) × 𝑌 ) = ( 𝑥 · ( 𝑋 × 𝑌 ) ) ) ↔ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) ) |
24 |
|
simpr |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → 𝑅 ∈ SRing ) |
25 |
|
simpr |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
26 |
25
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → 𝑌 ∈ 𝐵 ) |
27 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
28 |
1 3 27
|
srglz |
⊢ ( ( 𝑅 ∈ SRing ∧ 𝑌 ∈ 𝐵 ) → ( ( 0g ‘ 𝑅 ) × 𝑌 ) = ( 0g ‘ 𝑅 ) ) |
29 |
24 26 28
|
syl2anc |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 0g ‘ 𝑅 ) × 𝑌 ) = ( 0g ‘ 𝑅 ) ) |
30 |
|
simpl |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → 𝑋 ∈ 𝐵 ) |
32 |
1 27 2
|
mulg0 |
⊢ ( 𝑋 ∈ 𝐵 → ( 0 · 𝑋 ) = ( 0g ‘ 𝑅 ) ) |
33 |
31 32
|
syl |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( 0 · 𝑋 ) = ( 0g ‘ 𝑅 ) ) |
34 |
33
|
oveq1d |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 0 · 𝑋 ) × 𝑌 ) = ( ( 0g ‘ 𝑅 ) × 𝑌 ) ) |
35 |
1 3
|
srgcl |
⊢ ( ( 𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
36 |
24 31 26 35
|
syl3anc |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
37 |
1 27 2
|
mulg0 |
⊢ ( ( 𝑋 × 𝑌 ) ∈ 𝐵 → ( 0 · ( 𝑋 × 𝑌 ) ) = ( 0g ‘ 𝑅 ) ) |
38 |
36 37
|
syl |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( 0 · ( 𝑋 × 𝑌 ) ) = ( 0g ‘ 𝑅 ) ) |
39 |
29 34 38
|
3eqtr4d |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 0 · 𝑋 ) × 𝑌 ) = ( 0 · ( 𝑋 × 𝑌 ) ) ) |
40 |
|
srgmnd |
⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ Mnd ) |
41 |
40
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → 𝑅 ∈ Mnd ) |
42 |
41
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → 𝑅 ∈ Mnd ) |
43 |
|
simpl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → 𝑦 ∈ ℕ0 ) |
44 |
31
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → 𝑋 ∈ 𝐵 ) |
45 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
46 |
1 2 45
|
mulgnn0p1 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑦 + 1 ) · 𝑋 ) = ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) ) |
47 |
42 43 44 46
|
syl3anc |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( ( 𝑦 + 1 ) · 𝑋 ) = ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) ) |
48 |
47
|
oveq1d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) × 𝑌 ) ) |
49 |
24
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → 𝑅 ∈ SRing ) |
50 |
1 2
|
mulgnn0cl |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑦 · 𝑋 ) ∈ 𝐵 ) |
51 |
42 43 44 50
|
syl3anc |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( 𝑦 · 𝑋 ) ∈ 𝐵 ) |
52 |
26
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → 𝑌 ∈ 𝐵 ) |
53 |
1 45 3
|
srgdir |
⊢ ( ( 𝑅 ∈ SRing ∧ ( ( 𝑦 · 𝑋 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
54 |
49 51 44 52 53
|
syl13anc |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( ( ( 𝑦 · 𝑋 ) ( +g ‘ 𝑅 ) 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
55 |
48 54
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
56 |
55
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) ∧ ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
57 |
|
oveq1 |
⊢ ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
58 |
35
|
3expb |
⊢ ( ( 𝑅 ∈ SRing ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
59 |
58
|
ancoms |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
60 |
59
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
61 |
1 2 45
|
mulgnn0p1 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ ( 𝑋 × 𝑌 ) ∈ 𝐵 ) → ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
62 |
42 43 60 61
|
syl3anc |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
63 |
62
|
eqcomd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) → ( ( 𝑦 · ( 𝑋 × 𝑌 ) ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) |
64 |
57 63
|
sylan9eqr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) ∧ ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) ( +g ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) |
65 |
56 64
|
eqtrd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) ) ∧ ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) |
66 |
65
|
exp31 |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) ) |
67 |
66
|
a2d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑦 · 𝑋 ) × 𝑌 ) = ( 𝑦 · ( 𝑋 × 𝑌 ) ) ) → ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( ( 𝑦 + 1 ) · 𝑋 ) × 𝑌 ) = ( ( 𝑦 + 1 ) · ( 𝑋 × 𝑌 ) ) ) ) ) |
68 |
8 13 18 23 39 67
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑅 ∈ SRing ) → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) |
69 |
68
|
expd |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑅 ∈ SRing → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) ) |
70 |
69
|
3impib |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑅 ∈ SRing → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) ) |
71 |
70
|
impcom |
⊢ ( ( 𝑅 ∈ SRing ∧ ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑁 · 𝑋 ) × 𝑌 ) = ( 𝑁 · ( 𝑋 × 𝑌 ) ) ) |