Step |
Hyp |
Ref |
Expression |
1 |
|
psrring.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrring.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
3 |
|
psrring.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
4 |
|
psrass.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
5 |
|
psrass.t |
⊢ × = ( .r ‘ 𝑆 ) |
6 |
|
psrass.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
7 |
|
psrass.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
|
psrass.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
9 |
|
psrcom.c |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
10 |
|
psrass.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
11 |
|
psrass.n |
⊢ · = ( ·𝑠 ‘ 𝑆 ) |
12 |
|
psrass.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐾 ) |
13 |
1 2 3 4 5 6 7 8 10 11 12
|
psrass23l |
⊢ ( 𝜑 → ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
15 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
16 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝐴 ∈ 𝐾 ) |
17 |
16 10
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
19 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑌 ∈ 𝐵 ) |
20 |
|
ssrab2 |
⊢ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ⊆ 𝐷 |
21 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝐼 ∈ 𝑉 ) |
22 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑘 ∈ 𝐷 ) |
23 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) |
24 |
|
eqid |
⊢ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } |
25 |
4 24
|
psrbagconcl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷 ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑥 ) ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) |
26 |
21 22 23 25
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑥 ) ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) |
27 |
20 26
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑥 ) ∈ 𝐷 ) |
28 |
1 11 14 6 15 4 18 19 27
|
psrvscaval |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) = ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) |
29 |
28
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) = ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) |
30 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑋 ∈ 𝐵 ) |
31 |
1 14 4 6 30
|
psrelbas |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
32 |
20 23
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑥 ∈ 𝐷 ) |
33 |
31 32
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑋 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ) |
34 |
1 14 4 6 19
|
psrelbas |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑌 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
35 |
34 27
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ∈ ( Base ‘ 𝑅 ) ) |
36 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑅 ∈ CRing ) |
37 |
14 15
|
crngcom |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑢 ( .r ‘ 𝑅 ) 𝑣 ) = ( 𝑣 ( .r ‘ 𝑅 ) 𝑢 ) ) |
38 |
37
|
3expb |
⊢ ( ( 𝑅 ∈ CRing ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑢 ( .r ‘ 𝑅 ) 𝑣 ) = ( 𝑣 ( .r ‘ 𝑅 ) 𝑢 ) ) |
39 |
36 38
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑢 ( .r ‘ 𝑅 ) 𝑣 ) = ( 𝑣 ( .r ‘ 𝑅 ) 𝑢 ) ) |
40 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑅 ∈ Ring ) |
41 |
14 15
|
ringass |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ ( Base ‘ 𝑅 ) ∧ 𝑤 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑢 ( .r ‘ 𝑅 ) 𝑣 ) ( .r ‘ 𝑅 ) 𝑤 ) = ( 𝑢 ( .r ‘ 𝑅 ) ( 𝑣 ( .r ‘ 𝑅 ) 𝑤 ) ) ) |
42 |
40 41
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ ( Base ‘ 𝑅 ) ∧ 𝑤 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑢 ( .r ‘ 𝑅 ) 𝑣 ) ( .r ‘ 𝑅 ) 𝑤 ) = ( 𝑢 ( .r ‘ 𝑅 ) ( 𝑣 ( .r ‘ 𝑅 ) 𝑤 ) ) ) |
43 |
33 18 35 39 42
|
caov12d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) = ( 𝐴 ( .r ‘ 𝑅 ) ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) |
44 |
29 43
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) = ( 𝐴 ( .r ‘ 𝑅 ) ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) |
45 |
44
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( 𝐴 ( .r ‘ 𝑅 ) ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) |
46 |
45
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( 𝐴 ( .r ‘ 𝑅 ) ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
47 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
48 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
49 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑅 ∈ Ring ) |
50 |
4
|
psrbaglefi |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷 ) → { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ∈ Fin ) |
51 |
2 50
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ∈ Fin ) |
52 |
14 15
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
53 |
40 33 35 52
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
54 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
55 |
4 54
|
rabex2 |
⊢ 𝐷 ∈ V |
56 |
55
|
mptrabex |
⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∈ V |
57 |
|
funmpt |
⊢ Fun ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) |
58 |
|
fvex |
⊢ ( 0g ‘ 𝑅 ) ∈ V |
59 |
56 57 58
|
3pm3.2i |
⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∈ V ∧ Fun ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∧ ( 0g ‘ 𝑅 ) ∈ V ) |
60 |
59
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∈ V ∧ Fun ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∧ ( 0g ‘ 𝑅 ) ∈ V ) ) |
61 |
|
suppssdm |
⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ⊆ dom ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) |
62 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) |
63 |
62
|
dmmptss |
⊢ dom ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ⊆ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } |
64 |
61 63
|
sstri |
⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } |
65 |
64
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) |
66 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∈ V ∧ Fun ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ∧ ( 0g ‘ 𝑅 ) ∈ V ) ∧ ( { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ∈ Fin ∧ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ) ) → ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
67 |
60 51 65 66
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
68 |
14 47 48 15 49 51 17 53 67
|
gsummulc2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( 𝐴 ( .r ‘ 𝑅 ) ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) = ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
69 |
46 68
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) = ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
70 |
69
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ) |
71 |
1 11 10 6 3 12 8
|
psrvscacl |
⊢ ( 𝜑 → ( 𝐴 · 𝑌 ) ∈ 𝐵 ) |
72 |
1 6 15 5 4 7 71
|
psrmulfval |
⊢ ( 𝜑 → ( 𝑋 × ( 𝐴 · 𝑌 ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ( 𝐴 · 𝑌 ) ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
73 |
1 6 5 3 7 8
|
psrmulcl |
⊢ ( 𝜑 → ( 𝑋 × 𝑌 ) ∈ 𝐵 ) |
74 |
1 11 10 6 15 4 12 73
|
psrvsca |
⊢ ( 𝜑 → ( 𝐴 · ( 𝑋 × 𝑌 ) ) = ( ( 𝐷 × { 𝐴 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) ) |
75 |
55
|
a1i |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
76 |
|
ovex |
⊢ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ∈ V |
77 |
76
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ∈ V ) |
78 |
|
fconstmpt |
⊢ ( 𝐷 × { 𝐴 } ) = ( 𝑘 ∈ 𝐷 ↦ 𝐴 ) |
79 |
78
|
a1i |
⊢ ( 𝜑 → ( 𝐷 × { 𝐴 } ) = ( 𝑘 ∈ 𝐷 ↦ 𝐴 ) ) |
80 |
1 6 15 5 4 7 8
|
psrmulfval |
⊢ ( 𝜑 → ( 𝑋 × 𝑌 ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
81 |
75 16 77 79 80
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐷 × { 𝐴 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑋 × 𝑌 ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ) |
82 |
74 81
|
eqtrd |
⊢ ( 𝜑 → ( 𝐴 · ( 𝑋 × 𝑌 ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝐴 ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ) |
83 |
70 72 82
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝑋 × ( 𝐴 · 𝑌 ) ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) ) |
84 |
13 83
|
jca |
⊢ ( 𝜑 → ( ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) ∧ ( 𝑋 × ( 𝐴 · 𝑌 ) ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) ) ) |