Step |
Hyp |
Ref |
Expression |
1 |
|
sraassab.a |
⊢ 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) |
2 |
|
sraassab.z |
⊢ 𝑍 = ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) |
3 |
|
sraassab.w |
⊢ ( 𝜑 → 𝑊 ∈ Ring ) |
4 |
|
sraassab.s |
⊢ ( 𝜑 → 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
6 |
5
|
subrgss |
⊢ ( 𝑆 ∈ ( SubRing ‘ 𝑊 ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
7 |
4 6
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
9 |
8
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ ( Base ‘ 𝑊 ) ) |
10 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → 𝐴 ∈ AssAlg ) |
11 |
|
eqid |
⊢ ( 𝑊 ↾s 𝑆 ) = ( 𝑊 ↾s 𝑆 ) |
12 |
11
|
subrgbas |
⊢ ( 𝑆 ∈ ( SubRing ‘ 𝑊 ) → 𝑆 = ( Base ‘ ( 𝑊 ↾s 𝑆 ) ) ) |
13 |
4 12
|
syl |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( 𝑊 ↾s 𝑆 ) ) ) |
14 |
1
|
a1i |
⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) ) |
15 |
14 7
|
srasca |
⊢ ( 𝜑 → ( 𝑊 ↾s 𝑆 ) = ( Scalar ‘ 𝐴 ) ) |
16 |
15
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ ( 𝑊 ↾s 𝑆 ) ) = ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
17 |
13 16
|
eqtrd |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
18 |
17
|
eqimssd |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
19 |
18
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
20 |
19
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
21 |
14 7
|
srabase |
⊢ ( 𝜑 → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐴 ) ) |
22 |
21
|
eqimssd |
⊢ ( 𝜑 → ( Base ‘ 𝑊 ) ⊆ ( Base ‘ 𝐴 ) ) |
23 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) → ( Base ‘ 𝑊 ) ⊆ ( Base ‘ 𝐴 ) ) |
24 |
23
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → 𝑥 ∈ ( Base ‘ 𝐴 ) ) |
25 |
|
eqid |
⊢ ( 1r ‘ 𝑊 ) = ( 1r ‘ 𝑊 ) |
26 |
5 25
|
ringidcl |
⊢ ( 𝑊 ∈ Ring → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
27 |
3 26
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
28 |
27 21
|
eleqtrd |
⊢ ( 𝜑 → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝐴 ) ) |
29 |
28
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝐴 ) ) |
30 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
31 |
|
eqid |
⊢ ( Scalar ‘ 𝐴 ) = ( Scalar ‘ 𝐴 ) |
32 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐴 ) ) = ( Base ‘ ( Scalar ‘ 𝐴 ) ) |
33 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐴 ) |
34 |
|
eqid |
⊢ ( .r ‘ 𝐴 ) = ( .r ‘ 𝐴 ) |
35 |
30 31 32 33 34
|
assaassr |
⊢ ( ( 𝐴 ∈ AssAlg ∧ ( 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝐴 ) ) ) → ( 𝑥 ( .r ‘ 𝐴 ) ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) ) |
36 |
10 20 24 29 35
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝐴 ) ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) ) |
37 |
14 7
|
sramulr |
⊢ ( 𝜑 → ( .r ‘ 𝑊 ) = ( .r ‘ 𝐴 ) ) |
38 |
37
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( .r ‘ 𝑊 ) = ( .r ‘ 𝐴 ) ) |
39 |
38
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑥 ( .r ‘ 𝐴 ) ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) ) |
40 |
14 7
|
sravsca |
⊢ ( 𝜑 → ( .r ‘ 𝑊 ) = ( ·𝑠 ‘ 𝐴 ) ) |
41 |
40
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( .r ‘ 𝑊 ) = ( ·𝑠 ‘ 𝐴 ) ) |
42 |
41
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( .r ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) |
43 |
|
eqid |
⊢ ( .r ‘ 𝑊 ) = ( .r ‘ 𝑊 ) |
44 |
3
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → 𝑊 ∈ Ring ) |
45 |
9
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → 𝑦 ∈ ( Base ‘ 𝑊 ) ) |
46 |
5 43 25 44 45
|
ringridmd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( .r ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = 𝑦 ) |
47 |
42 46
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) = 𝑦 ) |
48 |
47
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) |
49 |
39 48
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝐴 ) ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) |
50 |
41
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( .r ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) ) |
51 |
38
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) |
52 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
53 |
5 43 25 44 52
|
ringridmd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = 𝑥 ) |
54 |
51 53
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) = 𝑥 ) |
55 |
54
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( .r ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ) |
56 |
50 55
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( ·𝑠 ‘ 𝐴 ) ( 𝑥 ( .r ‘ 𝐴 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ) |
57 |
36 49 56
|
3eqtr3rd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) |
58 |
57
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) → ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) |
59 |
|
eqid |
⊢ ( mulGrp ‘ 𝑊 ) = ( mulGrp ‘ 𝑊 ) |
60 |
59 5
|
mgpbas |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ ( mulGrp ‘ 𝑊 ) ) |
61 |
59 43
|
mgpplusg |
⊢ ( .r ‘ 𝑊 ) = ( +g ‘ ( mulGrp ‘ 𝑊 ) ) |
62 |
60 61 2
|
elcntr |
⊢ ( 𝑦 ∈ 𝑍 ↔ ( 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) |
63 |
9 58 62
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ 𝑍 ) |
64 |
63
|
ex |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) → ( 𝑦 ∈ 𝑆 → 𝑦 ∈ 𝑍 ) ) |
65 |
64
|
ssrdv |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ AssAlg ) → 𝑆 ⊆ 𝑍 ) |
66 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐴 ) ) |
67 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → ( 𝑊 ↾s 𝑆 ) = ( Scalar ‘ 𝐴 ) ) |
68 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → 𝑆 = ( Base ‘ ( 𝑊 ↾s 𝑆 ) ) ) |
69 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → ( .r ‘ 𝑊 ) = ( ·𝑠 ‘ 𝐴 ) ) |
70 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → ( .r ‘ 𝑊 ) = ( .r ‘ 𝐴 ) ) |
71 |
1
|
sralmod |
⊢ ( 𝑆 ∈ ( SubRing ‘ 𝑊 ) → 𝐴 ∈ LMod ) |
72 |
4 71
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ LMod ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → 𝐴 ∈ LMod ) |
74 |
1 5
|
sraring |
⊢ ( ( 𝑊 ∈ Ring ∧ 𝑆 ⊆ ( Base ‘ 𝑊 ) ) → 𝐴 ∈ Ring ) |
75 |
3 7 74
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ∈ Ring ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → 𝐴 ∈ Ring ) |
77 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → 𝑊 ∈ Ring ) |
78 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
79 |
78
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
80 |
79
|
3ad2antr1 |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
81 |
|
simpr2 |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑊 ) ) |
82 |
|
simpr3 |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑊 ) ) |
83 |
5 43 77 80 81 82
|
ringassd |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ( .r ‘ 𝑊 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑦 ( .r ‘ 𝑊 ) 𝑧 ) ) ) |
84 |
|
ssel2 |
⊢ ( ( 𝑆 ⊆ 𝑍 ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑍 ) |
85 |
84
|
ad2ant2lr |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ) ) → 𝑥 ∈ 𝑍 ) |
86 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑊 ) ) |
87 |
60 61 2
|
cntri |
⊢ ( ( 𝑥 ∈ 𝑍 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ) |
88 |
85 86 87
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ) |
89 |
88
|
3adantr3 |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ) |
90 |
89
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ( .r ‘ 𝑊 ) 𝑧 ) = ( ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑧 ) ) |
91 |
5 43 77 81 80 82
|
ringassd |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑦 ( .r ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑧 ) = ( 𝑦 ( .r ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑧 ) ) ) |
92 |
90 83 91
|
3eqtr3rd |
⊢ ( ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑦 ( .r ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑧 ) ) = ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑦 ( .r ‘ 𝑊 ) 𝑧 ) ) ) |
93 |
66 67 68 69 70 73 76 83 92
|
isassad |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑍 ) → 𝐴 ∈ AssAlg ) |
94 |
65 93
|
impbida |
⊢ ( 𝜑 → ( 𝐴 ∈ AssAlg ↔ 𝑆 ⊆ 𝑍 ) ) |