| Step |
Hyp |
Ref |
Expression |
| 1 |
|
resssra.a |
⊢ 𝐴 = ( Base ‘ 𝑅 ) |
| 2 |
|
resssra.s |
⊢ 𝑆 = ( 𝑅 ↾s 𝐵 ) |
| 3 |
|
resssra.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
| 4 |
|
resssra.c |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐵 ) |
| 5 |
|
resssra.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑉 ) |
| 6 |
|
eqidd |
⊢ ( 𝜑 → ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) = ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) |
| 7 |
4 3
|
sstrd |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 ) |
| 8 |
7 1
|
sseqtrdi |
⊢ ( 𝜑 → 𝐶 ⊆ ( Base ‘ 𝑅 ) ) |
| 9 |
6 8
|
srabase |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) |
| 10 |
1 9
|
eqtrid |
⊢ ( 𝜑 → 𝐴 = ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) |
| 11 |
10
|
oveq2d |
⊢ ( 𝜑 → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐴 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐴 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ) |
| 13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 ) |
| 14 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ⊆ 𝐴 ) |
| 15 |
13 14
|
eqssd |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 = 𝐵 ) |
| 16 |
15
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐴 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) ) |
| 17 |
|
fvex |
⊢ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ∈ V |
| 18 |
|
eqid |
⊢ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) = ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) |
| 19 |
18
|
ressid |
⊢ ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ∈ V → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) = ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) |
| 20 |
17 19
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) = ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) |
| 21 |
12 16 20
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) = ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) |
| 22 |
1
|
oveq2i |
⊢ ( 𝑅 ↾s 𝐴 ) = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) |
| 23 |
5
|
elexd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
| 24 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 25 |
24
|
ressid |
⊢ ( 𝑅 ∈ V → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 ) |
| 26 |
23 25
|
syl |
⊢ ( 𝜑 → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 ) |
| 27 |
22 26
|
eqtrid |
⊢ ( 𝜑 → ( 𝑅 ↾s 𝐴 ) = 𝑅 ) |
| 28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑅 ↾s 𝐴 ) = 𝑅 ) |
| 29 |
15
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑅 ↾s 𝐴 ) = ( 𝑅 ↾s 𝐵 ) ) |
| 30 |
29 2
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑅 ↾s 𝐴 ) = 𝑆 ) |
| 31 |
28 30
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝑅 = 𝑆 ) |
| 32 |
31
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( subringAlg ‘ 𝑅 ) = ( subringAlg ‘ 𝑆 ) ) |
| 33 |
32
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) = ( ( subringAlg ‘ 𝑆 ) ‘ 𝐶 ) ) |
| 34 |
21 33
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( ( subringAlg ‘ 𝑆 ) ‘ 𝐶 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) ) |
| 35 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ¬ 𝐴 ⊆ 𝐵 ) |
| 36 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → 𝑅 ∈ V ) |
| 37 |
1
|
fvexi |
⊢ 𝐴 ∈ V |
| 38 |
37
|
a1i |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
| 39 |
38 3
|
ssexd |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
| 40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ V ) |
| 41 |
2 1
|
ressval2 |
⊢ ( ( ¬ 𝐴 ⊆ 𝐵 ∧ 𝑅 ∈ V ∧ 𝐵 ∈ V ) → 𝑆 = ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ 𝐴 ) ⟩ ) ) |
| 42 |
35 36 40 41
|
syl3anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → 𝑆 = ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ 𝐴 ) ⟩ ) ) |
| 43 |
|
dfss2 |
⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐵 ) |
| 44 |
3 43
|
sylib |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) = 𝐵 ) |
| 45 |
44
|
opeq2d |
⊢ ( 𝜑 → ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ 𝐴 ) ⟩ = ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) |
| 46 |
45
|
oveq2d |
⊢ ( 𝜑 → ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ 𝐴 ) ⟩ ) = ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 47 |
46
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ 𝐴 ) ⟩ ) = ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 48 |
42 47
|
eqtrd |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → 𝑆 = ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 49 |
2
|
oveq1i |
⊢ ( 𝑆 ↾s 𝐶 ) = ( ( 𝑅 ↾s 𝐵 ) ↾s 𝐶 ) |
| 50 |
|
ressabs |
⊢ ( ( 𝐵 ∈ V ∧ 𝐶 ⊆ 𝐵 ) → ( ( 𝑅 ↾s 𝐵 ) ↾s 𝐶 ) = ( 𝑅 ↾s 𝐶 ) ) |
| 51 |
39 4 50
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑅 ↾s 𝐵 ) ↾s 𝐶 ) = ( 𝑅 ↾s 𝐶 ) ) |
| 52 |
49 51
|
eqtrid |
⊢ ( 𝜑 → ( 𝑆 ↾s 𝐶 ) = ( 𝑅 ↾s 𝐶 ) ) |
| 53 |
52
|
opeq2d |
⊢ ( 𝜑 → ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ = ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) |
| 54 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ = ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) |
| 55 |
48 54
|
oveq12d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) = ( ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) ) |
| 56 |
|
scandxnbasendx |
⊢ ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx ) |
| 57 |
56
|
a1i |
⊢ ( 𝜑 → ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx ) ) |
| 58 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑅 ↾s 𝐶 ) ∈ V ) |
| 59 |
|
fvex |
⊢ ( Scalar ‘ ndx ) ∈ V |
| 60 |
|
fvex |
⊢ ( Base ‘ ndx ) ∈ V |
| 61 |
59 60
|
setscom |
⊢ ( ( ( 𝑅 ∈ V ∧ ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx ) ) ∧ ( ( 𝑅 ↾s 𝐶 ) ∈ V ∧ 𝐵 ∈ V ) ) → ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) ) |
| 62 |
23 57 58 39 61
|
syl22anc |
⊢ ( 𝜑 → ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) ) |
| 63 |
62
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( 𝑅 sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) ) |
| 64 |
55 63
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) = ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 65 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
| 66 |
2 65
|
ressmulr |
⊢ ( 𝐵 ∈ V → ( .r ‘ 𝑅 ) = ( .r ‘ 𝑆 ) ) |
| 67 |
39 66
|
syl |
⊢ ( 𝜑 → ( .r ‘ 𝑅 ) = ( .r ‘ 𝑆 ) ) |
| 68 |
67
|
eqcomd |
⊢ ( 𝜑 → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑅 ) ) |
| 69 |
68
|
opeq2d |
⊢ ( 𝜑 → ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ = ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) |
| 70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ = ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) |
| 71 |
64 70
|
oveq12d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 72 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) ∈ V ) |
| 73 |
|
vscandxnbasendx |
⊢ ( ·𝑠 ‘ ndx ) ≠ ( Base ‘ ndx ) |
| 74 |
73
|
a1i |
⊢ ( 𝜑 → ( ·𝑠 ‘ ndx ) ≠ ( Base ‘ ndx ) ) |
| 75 |
|
fvexd |
⊢ ( 𝜑 → ( .r ‘ 𝑅 ) ∈ V ) |
| 76 |
|
fvex |
⊢ ( ·𝑠 ‘ ndx ) ∈ V |
| 77 |
76 60
|
setscom |
⊢ ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) ∈ V ∧ ( ·𝑠 ‘ ndx ) ≠ ( Base ‘ ndx ) ) ∧ ( ( .r ‘ 𝑅 ) ∈ V ∧ 𝐵 ∈ V ) ) → ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 78 |
72 74 75 39 77
|
syl22anc |
⊢ ( 𝜑 → ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 79 |
78
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 80 |
71 79
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 81 |
68
|
opeq2d |
⊢ ( 𝜑 → ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ = ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) |
| 82 |
81
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ = ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) |
| 83 |
80 82
|
oveq12d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 84 |
|
ovexd |
⊢ ( 𝜑 → ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ∈ V ) |
| 85 |
|
ipndxnbasendx |
⊢ ( ·𝑖 ‘ ndx ) ≠ ( Base ‘ ndx ) |
| 86 |
85
|
a1i |
⊢ ( 𝜑 → ( ·𝑖 ‘ ndx ) ≠ ( Base ‘ ndx ) ) |
| 87 |
|
fvex |
⊢ ( ·𝑖 ‘ ndx ) ∈ V |
| 88 |
87 60
|
setscom |
⊢ ( ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ∈ V ∧ ( ·𝑖 ‘ ndx ) ≠ ( Base ‘ ndx ) ) ∧ ( ( .r ‘ 𝑅 ) ∈ V ∧ 𝐵 ∈ V ) ) → ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 89 |
84 86 75 39 88
|
syl22anc |
⊢ ( 𝜑 → ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 90 |
89
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 91 |
83 90
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 92 |
2
|
ovexi |
⊢ 𝑆 ∈ V |
| 93 |
2 1
|
ressbas2 |
⊢ ( 𝐵 ⊆ 𝐴 → 𝐵 = ( Base ‘ 𝑆 ) ) |
| 94 |
3 93
|
syl |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑆 ) ) |
| 95 |
4 94
|
sseqtrd |
⊢ ( 𝜑 → 𝐶 ⊆ ( Base ‘ 𝑆 ) ) |
| 96 |
95
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → 𝐶 ⊆ ( Base ‘ 𝑆 ) ) |
| 97 |
|
sraval |
⊢ ( ( 𝑆 ∈ V ∧ 𝐶 ⊆ ( Base ‘ 𝑆 ) ) → ( ( subringAlg ‘ 𝑆 ) ‘ 𝐶 ) = ( ( ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) ) |
| 98 |
92 96 97
|
sylancr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( subringAlg ‘ 𝑆 ) ‘ 𝐶 ) = ( ( ( 𝑆 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑆 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ ) ) |
| 99 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → 𝐴 = ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) |
| 100 |
99
|
sseq1d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ⊆ 𝐵 ) ) |
| 101 |
35 100
|
mtbid |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ¬ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ⊆ 𝐵 ) |
| 102 |
|
fvexd |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ∈ V ) |
| 103 |
|
eqid |
⊢ ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) |
| 104 |
103 18
|
ressval2 |
⊢ ( ( ¬ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ⊆ 𝐵 ∧ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ∈ V ∧ 𝐵 ∈ V ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ⟩ ) ) |
| 105 |
101 102 40 104
|
syl3anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ⟩ ) ) |
| 106 |
10
|
ineq2d |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) = ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ) |
| 107 |
106 44
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) = 𝐵 ) |
| 108 |
107
|
opeq2d |
⊢ ( 𝜑 → ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ⟩ = ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) |
| 109 |
108
|
oveq2d |
⊢ ( 𝜑 → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ⟩ ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 110 |
109
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , ( 𝐵 ∩ ( Base ‘ ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ) ) ⟩ ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 111 |
|
sraval |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐶 ⊆ ( Base ‘ 𝑅 ) ) → ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 112 |
5 8 111
|
syl2anc |
⊢ ( 𝜑 → ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) = ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) ) |
| 113 |
112
|
oveq1d |
⊢ ( 𝜑 → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 114 |
113
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 115 |
105 110 114
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) = ( ( ( ( 𝑅 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑅 ↾s 𝐶 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ ) sSet ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ ) ) |
| 116 |
91 98 115
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ( subringAlg ‘ 𝑆 ) ‘ 𝐶 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) ) |
| 117 |
34 116
|
pm2.61dan |
⊢ ( 𝜑 → ( ( subringAlg ‘ 𝑆 ) ‘ 𝐶 ) = ( ( ( subringAlg ‘ 𝑅 ) ‘ 𝐶 ) ↾s 𝐵 ) ) |