Step |
Hyp |
Ref |
Expression |
1 |
|
rhmply1vsca.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
rhmply1vsca.q |
⊢ 𝑄 = ( Poly1 ‘ 𝑆 ) |
3 |
|
rhmply1vsca.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
rhmply1vsca.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
5 |
|
rhmply1vsca.f |
⊢ 𝐹 = ( 𝑝 ∈ 𝐵 ↦ ( 𝐻 ∘ 𝑝 ) ) |
6 |
|
rhmply1vsca.t |
⊢ · = ( ·𝑠 ‘ 𝑃 ) |
7 |
|
rhmply1vsca.u |
⊢ ∙ = ( ·𝑠 ‘ 𝑄 ) |
8 |
|
rhmply1vsca.h |
⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) ) |
9 |
|
rhmply1vsca.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐾 ) |
10 |
|
rhmply1vsca.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
11 |
|
fconst6g |
⊢ ( 𝐶 ∈ 𝐾 → ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) : { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
12 |
9 11
|
syl |
⊢ ( 𝜑 → ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) : { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
13 |
|
psr1baslem |
⊢ ( ℕ0 ↑m 1o ) = { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
14 |
13
|
feq2i |
⊢ ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) : ( ℕ0 ↑m 1o ) ⟶ 𝐾 ↔ ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) : { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
15 |
12 14
|
sylibr |
⊢ ( 𝜑 → ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) : ( ℕ0 ↑m 1o ) ⟶ 𝐾 ) |
16 |
1 3 4
|
ply1basf |
⊢ ( 𝑋 ∈ 𝐵 → 𝑋 : ( ℕ0 ↑m 1o ) ⟶ 𝐾 ) |
17 |
10 16
|
syl |
⊢ ( 𝜑 → 𝑋 : ( ℕ0 ↑m 1o ) ⟶ 𝐾 ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
19 |
4 18
|
rhmf |
⊢ ( 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐻 : 𝐾 ⟶ ( Base ‘ 𝑆 ) ) |
20 |
8 19
|
syl |
⊢ ( 𝜑 → 𝐻 : 𝐾 ⟶ ( Base ‘ 𝑆 ) ) |
21 |
20
|
ffnd |
⊢ ( 𝜑 → 𝐻 Fn 𝐾 ) |
22 |
|
ovexd |
⊢ ( 𝜑 → ( ℕ0 ↑m 1o ) ∈ V ) |
23 |
|
rhmrcl1 |
⊢ ( 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring ) |
24 |
8 23
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
25 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
26 |
4 25
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ∈ 𝐾 ) |
27 |
24 26
|
syl3an1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ∈ 𝐾 ) |
28 |
27
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) ) → ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ∈ 𝐾 ) |
29 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
30 |
4 25 29
|
rhmmul |
⊢ ( ( 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝐻 ‘ ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ) = ( ( 𝐻 ‘ 𝑎 ) ( .r ‘ 𝑆 ) ( 𝐻 ‘ 𝑏 ) ) ) |
31 |
8 30
|
syl3an1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝐻 ‘ ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ) = ( ( 𝐻 ‘ 𝑎 ) ( .r ‘ 𝑆 ) ( 𝐻 ‘ 𝑏 ) ) ) |
32 |
31
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) ) → ( 𝐻 ‘ ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ) = ( ( 𝐻 ‘ 𝑎 ) ( .r ‘ 𝑆 ) ( 𝐻 ‘ 𝑏 ) ) ) |
33 |
15 17 21 22 28 32
|
coof |
⊢ ( 𝜑 → ( 𝐻 ∘ ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ∘f ( .r ‘ 𝑅 ) 𝑋 ) ) = ( ( 𝐻 ∘ ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ) ∘f ( .r ‘ 𝑆 ) ( 𝐻 ∘ 𝑋 ) ) ) |
34 |
|
fcoconst |
⊢ ( ( 𝐻 Fn 𝐾 ∧ 𝐶 ∈ 𝐾 ) → ( 𝐻 ∘ ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ) = ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( 𝐻 ‘ 𝐶 ) } ) ) |
35 |
21 9 34
|
syl2anc |
⊢ ( 𝜑 → ( 𝐻 ∘ ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ) = ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( 𝐻 ‘ 𝐶 ) } ) ) |
36 |
35
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐻 ∘ ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ) ∘f ( .r ‘ 𝑆 ) ( 𝐻 ∘ 𝑋 ) ) = ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( 𝐻 ‘ 𝐶 ) } ) ∘f ( .r ‘ 𝑆 ) ( 𝐻 ∘ 𝑋 ) ) ) |
37 |
33 36
|
eqtrd |
⊢ ( 𝜑 → ( 𝐻 ∘ ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ∘f ( .r ‘ 𝑅 ) 𝑋 ) ) = ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( 𝐻 ‘ 𝐶 ) } ) ∘f ( .r ‘ 𝑆 ) ( 𝐻 ∘ 𝑋 ) ) ) |
38 |
|
eqid |
⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 ) |
39 |
|
eqid |
⊢ ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) = ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) |
40 |
1 3
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
41 |
|
eqid |
⊢ { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
42 |
38 39 4 40 25 41 9 10
|
mplvsca |
⊢ ( 𝜑 → ( 𝐶 ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) 𝑋 ) = ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ∘f ( .r ‘ 𝑅 ) 𝑋 ) ) |
43 |
42
|
coeq2d |
⊢ ( 𝜑 → ( 𝐻 ∘ ( 𝐶 ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) 𝑋 ) ) = ( 𝐻 ∘ ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝐶 } ) ∘f ( .r ‘ 𝑅 ) 𝑋 ) ) ) |
44 |
|
eqid |
⊢ ( 1o mPoly 𝑆 ) = ( 1o mPoly 𝑆 ) |
45 |
|
eqid |
⊢ ( ·𝑠 ‘ ( 1o mPoly 𝑆 ) ) = ( ·𝑠 ‘ ( 1o mPoly 𝑆 ) ) |
46 |
|
eqid |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 ) |
47 |
2 46
|
ply1bas |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ ( 1o mPoly 𝑆 ) ) |
48 |
20 9
|
ffvelcdmd |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝐶 ) ∈ ( Base ‘ 𝑆 ) ) |
49 |
|
rhmghm |
⊢ ( 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐻 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
50 |
|
ghmmhm |
⊢ ( 𝐻 ∈ ( 𝑅 GrpHom 𝑆 ) → 𝐻 ∈ ( 𝑅 MndHom 𝑆 ) ) |
51 |
8 49 50
|
3syl |
⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 MndHom 𝑆 ) ) |
52 |
1 2 3 46 51 10
|
mhmcoply1 |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝑋 ) ∈ ( Base ‘ 𝑄 ) ) |
53 |
44 45 18 47 29 41 48 52
|
mplvsca |
⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐶 ) ( ·𝑠 ‘ ( 1o mPoly 𝑆 ) ) ( 𝐻 ∘ 𝑋 ) ) = ( ( { ℎ ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( 𝐻 ‘ 𝐶 ) } ) ∘f ( .r ‘ 𝑆 ) ( 𝐻 ∘ 𝑋 ) ) ) |
54 |
37 43 53
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝐻 ∘ ( 𝐶 ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) 𝑋 ) ) = ( ( 𝐻 ‘ 𝐶 ) ( ·𝑠 ‘ ( 1o mPoly 𝑆 ) ) ( 𝐻 ∘ 𝑋 ) ) ) |
55 |
1 38 6
|
ply1vsca |
⊢ · = ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) |
56 |
55
|
oveqi |
⊢ ( 𝐶 · 𝑋 ) = ( 𝐶 ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) 𝑋 ) |
57 |
56
|
coeq2i |
⊢ ( 𝐻 ∘ ( 𝐶 · 𝑋 ) ) = ( 𝐻 ∘ ( 𝐶 ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) 𝑋 ) ) |
58 |
2 44 7
|
ply1vsca |
⊢ ∙ = ( ·𝑠 ‘ ( 1o mPoly 𝑆 ) ) |
59 |
58
|
oveqi |
⊢ ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐻 ∘ 𝑋 ) ) = ( ( 𝐻 ‘ 𝐶 ) ( ·𝑠 ‘ ( 1o mPoly 𝑆 ) ) ( 𝐻 ∘ 𝑋 ) ) |
60 |
54 57 59
|
3eqtr4g |
⊢ ( 𝜑 → ( 𝐻 ∘ ( 𝐶 · 𝑋 ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐻 ∘ 𝑋 ) ) ) |
61 |
|
coeq2 |
⊢ ( 𝑝 = ( 𝐶 · 𝑋 ) → ( 𝐻 ∘ 𝑝 ) = ( 𝐻 ∘ ( 𝐶 · 𝑋 ) ) ) |
62 |
1 3 4 6 24 9 10
|
ply1vscl |
⊢ ( 𝜑 → ( 𝐶 · 𝑋 ) ∈ 𝐵 ) |
63 |
8 62
|
coexd |
⊢ ( 𝜑 → ( 𝐻 ∘ ( 𝐶 · 𝑋 ) ) ∈ V ) |
64 |
5 61 62 63
|
fvmptd3 |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 · 𝑋 ) ) = ( 𝐻 ∘ ( 𝐶 · 𝑋 ) ) ) |
65 |
|
coeq2 |
⊢ ( 𝑝 = 𝑋 → ( 𝐻 ∘ 𝑝 ) = ( 𝐻 ∘ 𝑋 ) ) |
66 |
8 10
|
coexd |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝑋 ) ∈ V ) |
67 |
5 65 10 66
|
fvmptd3 |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) = ( 𝐻 ∘ 𝑋 ) ) |
68 |
67
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐹 ‘ 𝑋 ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐻 ∘ 𝑋 ) ) ) |
69 |
60 64 68
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 · 𝑋 ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐹 ‘ 𝑋 ) ) ) |