Step | Hyp | Ref
| Expression |
1 | | psrring.s |
. . . . . . . . 9
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
2 | | psrass23l.n |
. . . . . . . . 9
⊢ · = (
·𝑠 ‘𝑆) |
3 | | eqid 2622 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘𝑅) |
4 | | psrass.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑆) |
5 | | eqid 2622 |
. . . . . . . . 9
⊢
(.r‘𝑅) = (.r‘𝑅) |
6 | | psrass.d |
. . . . . . . . 9
⊢ 𝐷 = {𝑓 ∈ (ℕ0
↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈
Fin} |
7 | | psrass23l.a |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ 𝐾) |
8 | 7 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐴 ∈ 𝐾) |
9 | | psrass23l.k |
. . . . . . . . . . 11
⊢ 𝐾 = (Base‘𝑅) |
10 | 8, 9 | syl6eleq 2711 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐴 ∈ (Base‘𝑅)) |
11 | 10 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝐴 ∈ (Base‘𝑅)) |
12 | | psrass.x |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
13 | 12 | ad2antrr 762 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ 𝐵) |
14 | | ssrab2 3687 |
. . . . . . . . . 10
⊢ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ⊆ 𝐷 |
15 | | simpr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) |
16 | 14, 15 | sseldi 3601 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑥 ∈ 𝐷) |
17 | 1, 2, 3, 4, 5, 6, 11, 13, 16 | psrvscaval 19392 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → ((𝐴 · 𝑋)‘𝑥) = (𝐴(.r‘𝑅)(𝑋‘𝑥))) |
18 | 17 | oveq1d 6665 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (((𝐴 · 𝑋)‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = ((𝐴(.r‘𝑅)(𝑋‘𝑥))(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) |
19 | | psrring.r |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ Ring) |
20 | 19 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring) |
21 | 1, 3, 6, 4, 13 | psrelbas 19379 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅)) |
22 | 21, 16 | ffvelrnd 6360 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑋‘𝑥) ∈ (Base‘𝑅)) |
23 | | psrass.y |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
24 | 23 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑌 ∈ 𝐵) |
25 | 1, 3, 6, 4, 24 | psrelbas 19379 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅)) |
26 | | psrring.i |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
27 | 26 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑉) |
28 | | simplr 792 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
29 | | eqid 2622 |
. . . . . . . . . . . 12
⊢ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} |
30 | 6, 29 | psrbagconcl 19373 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑥) ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) |
31 | 27, 28, 15, 30 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑥) ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) |
32 | 14, 31 | sseldi 3601 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑥) ∈ 𝐷) |
33 | 25, 32 | ffvelrnd 6360 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑌‘(𝑘 ∘𝑓 − 𝑥)) ∈ (Base‘𝑅)) |
34 | 3, 5 | ringass 18564 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (Base‘𝑅) ∧ (𝑋‘𝑥) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘𝑓 − 𝑥)) ∈ (Base‘𝑅))) → ((𝐴(.r‘𝑅)(𝑋‘𝑥))(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (𝐴(.r‘𝑅)((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))) |
35 | 20, 11, 22, 33, 34 | syl13anc 1328 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → ((𝐴(.r‘𝑅)(𝑋‘𝑥))(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (𝐴(.r‘𝑅)((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))) |
36 | 18, 35 | eqtrd 2656 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (((𝐴 · 𝑋)‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (𝐴(.r‘𝑅)((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))) |
37 | 36 | mpteq2dva 4744 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (((𝐴 · 𝑋)‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (𝐴(.r‘𝑅)((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))))) |
38 | 37 | oveq2d 6666 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (((𝐴 · 𝑋)‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))) = (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (𝐴(.r‘𝑅)((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))))) |
39 | | eqid 2622 |
. . . . 5
⊢
(0g‘𝑅) = (0g‘𝑅) |
40 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝑅) = (+g‘𝑅) |
41 | 19 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring) |
42 | 6 | psrbaglefi 19372 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∈ Fin) |
43 | 26, 42 | sylan 488 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∈ Fin) |
44 | 3, 5 | ringcl 18561 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑥) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘𝑓 − 𝑥)) ∈ (Base‘𝑅)) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) ∈ (Base‘𝑅)) |
45 | 20, 22, 33, 44 | syl3anc 1326 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) ∈ (Base‘𝑅)) |
46 | | ovex 6678 |
. . . . . . . . . 10
⊢
(ℕ0 ↑𝑚 𝐼) ∈ V |
47 | 6, 46 | rabex2 4815 |
. . . . . . . . 9
⊢ 𝐷 ∈ V |
48 | 47 | mptrabex 6488 |
. . . . . . . 8
⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) ∈ V |
49 | | funmpt 5926 |
. . . . . . . 8
⊢ Fun
(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) |
50 | | fvex 6201 |
. . . . . . . 8
⊢
(0g‘𝑅) ∈ V |
51 | 48, 49, 50 | 3pm3.2i 1239 |
. . . . . . 7
⊢ ((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) |
52 | 51 | a1i 11 |
. . . . . 6
⊢ ((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)) |
53 | | suppssdm 7308 |
. . . . . . . 8
⊢ ((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)))) |
54 | | eqid 2622 |
. . . . . . . . 9
⊢ (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)))) |
55 | 54 | dmmptss 5631 |
. . . . . . . 8
⊢ dom
(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) ⊆ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} |
56 | 53, 55 | sstri 3612 |
. . . . . . 7
⊢ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) supp
(0g‘𝑅))
⊆ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} |
57 | 56 | a1i 11 |
. . . . . 6
⊢ ((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}) |
58 | | suppssfifsupp 8290 |
. . . . . 6
⊢ ((((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)) |
59 | 52, 43, 57, 58 | syl12anc 1324 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) finSupp
(0g‘𝑅)) |
60 | 3, 39, 40, 5, 41, 43, 10, 45, 59 | gsummulc2 18607 |
. . . 4
⊢ ((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))))))) |
61 | 38, 60 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (((𝐴 · 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))))))) |
62 | 61 | mpteq2dva 4744 |
. 2
⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (((𝐴 · 𝑋)‘𝑥)(.r‘𝑅)(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)))))))) |
63 | | psrass.t |
. . 3
⊢ × =
(.r‘𝑆) |
64 | 1, 2, 9, 4, 19, 7,
12 | psrvscacl 19393 |
. . 3
⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝐵) |
65 | 1, 4, 5, 63, 6, 64, 23 | psrmulfval 19385 |
. 2
⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (((𝐴 · 𝑋)‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))))) |
66 | 1, 4, 63, 19, 12, 23 | psrmulcl 19388 |
. . . 4
⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵) |
67 | 1, 2, 9, 4, 5, 6, 7, 66 | psrvsca 19391 |
. . 3
⊢ (𝜑 → (𝐴 · (𝑋 × 𝑌)) = ((𝐷 × {𝐴}) ∘𝑓
(.r‘𝑅)(𝑋 × 𝑌))) |
68 | 47 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ V) |
69 | | ovexd 6680 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))) ∈ V) |
70 | | fconstmpt 5163 |
. . . . 5
⊢ (𝐷 × {𝐴}) = (𝑘 ∈ 𝐷 ↦ 𝐴) |
71 | 70 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝐷 × {𝐴}) = (𝑘 ∈ 𝐷 ↦ 𝐴)) |
72 | 1, 4, 5, 63, 6, 12, 23 | psrmulfval 19385 |
. . . 4
⊢ (𝜑 → (𝑋 × 𝑌) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))))) |
73 | 68, 8, 69, 71, 72 | offval2 6914 |
. . 3
⊢ (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)))))))) |
74 | 67, 73 | eqtrd 2656 |
. 2
⊢ (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)))))))) |
75 | 62, 65, 74 | 3eqtr4d 2666 |
1
⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))) |