Step | Hyp | Ref
| Expression |
1 | | psrring.s |
. . . 4
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
2 | | eqid 2726 |
. . . 4
⊢
(Base‘𝑅) =
(Base‘𝑅) |
3 | | psrass.d |
. . . 4
⊢ 𝐷 = {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin} |
4 | | psrass.b |
. . . 4
⊢ 𝐵 = (Base‘𝑆) |
5 | | psrass.t |
. . . . 5
⊢ × =
(.r‘𝑆) |
6 | | psrring.r |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Ring) |
7 | | psrass.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
8 | | psrass.y |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
9 | 1, 4, 5, 6, 7, 8 | psrmulcl 21951 |
. . . . 5
⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵) |
10 | | psrass.z |
. . . . 5
⊢ (𝜑 → 𝑍 ∈ 𝐵) |
11 | 1, 4, 5, 6, 9, 10 | psrmulcl 21951 |
. . . 4
⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵) |
12 | 1, 2, 3, 4, 11 | psrelbas 21939 |
. . 3
⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅)) |
13 | 12 | ffnd 6721 |
. 2
⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷) |
14 | 1, 4, 5, 6, 8, 10 | psrmulcl 21951 |
. . . . 5
⊢ (𝜑 → (𝑌 × 𝑍) ∈ 𝐵) |
15 | 1, 4, 5, 6, 7, 14 | psrmulcl 21951 |
. . . 4
⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵) |
16 | 1, 2, 3, 4, 15 | psrelbas 21939 |
. . 3
⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅)) |
17 | 16 | ffnd 6721 |
. 2
⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷) |
18 | | eqid 2726 |
. . . . 5
⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} |
19 | | simpr 483 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) |
20 | 6 | ringcmnd 20259 |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ CMnd) |
21 | 20 | adantr 479 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd) |
22 | | eqid 2726 |
. . . . . . 7
⊢
(.r‘𝑅) = (.r‘𝑅) |
23 | 6 | ad3antrrr 728 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑅 ∈ Ring) |
24 | 1, 2, 3, 4, 7 | psrelbas 21939 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
25 | 24 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅)) |
26 | | simpr 483 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) |
27 | | breq1 5148 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑗 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑗 ∘r ≤ 𝑥)) |
28 | 27 | elrab 3680 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥)) |
29 | 26, 28 | sylib 217 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥)) |
30 | 29 | simpld 493 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ 𝐷) |
31 | 25, 30 | ffvelcdmd 7091 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑋‘𝑗) ∈ (Base‘𝑅)) |
32 | 31 | adantr 479 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑋‘𝑗) ∈ (Base‘𝑅)) |
33 | 1, 2, 3, 4, 8 | psrelbas 21939 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) |
34 | 33 | ad3antrrr 728 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅)) |
35 | | simpr 483 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) |
36 | | breq1 5148 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑛 → (ℎ ∘r ≤ (𝑥 ∘f − 𝑗) ↔ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗))) |
37 | 36 | elrab 3680 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↔ (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗))) |
38 | 35, 37 | sylib 217 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗))) |
39 | 38 | simpld 493 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∈ 𝐷) |
40 | 34, 39 | ffvelcdmd 7091 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑌‘𝑛) ∈ (Base‘𝑅)) |
41 | 1, 2, 3, 4, 10 | psrelbas 21939 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍:𝐷⟶(Base‘𝑅)) |
42 | 41 | ad3antrrr 728 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅)) |
43 | | simplr 767 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷) |
44 | 3 | psrbagf 21911 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ 𝐷 → 𝑗:𝐼⟶ℕ0) |
45 | 30, 44 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0) |
46 | 29 | simprd 494 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∘r ≤ 𝑥) |
47 | 3 | psrbagcon 21923 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥)) |
48 | 43, 45, 46, 47 | syl3anc 1368 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥)) |
49 | 48 | simpld 493 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑗) ∈ 𝐷) |
50 | 49 | adantr 479 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑥 ∘f − 𝑗) ∈ 𝐷) |
51 | 3 | psrbagf 21911 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ 𝐷 → 𝑛:𝐼⟶ℕ0) |
52 | 39, 51 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛:𝐼⟶ℕ0) |
53 | 38 | simprd 494 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)) |
54 | 3 | psrbagcon 21923 |
. . . . . . . . . . 11
⊢ (((𝑥 ∘f −
𝑗) ∈ 𝐷 ∧ 𝑛:𝐼⟶ℕ0 ∧ 𝑛 ∘r ≤ (𝑥 ∘f −
𝑗)) → (((𝑥 ∘f −
𝑗) ∘f
− 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘f − 𝑗) ∘f −
𝑛) ∘r ≤
(𝑥 ∘f
− 𝑗))) |
55 | 50, 52, 53, 54 | syl3anc 1368 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (((𝑥 ∘f − 𝑗) ∘f −
𝑛) ∈ 𝐷 ∧ ((𝑥 ∘f − 𝑗) ∘f −
𝑛) ∘r ≤
(𝑥 ∘f
− 𝑗))) |
56 | 55 | simpld 493 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑥 ∘f − 𝑗) ∘f −
𝑛) ∈ 𝐷) |
57 | 42, 56 | ffvelcdmd 7091 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)) ∈
(Base‘𝑅)) |
58 | 2, 22, 23, 40, 57 | ringcld 20238 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))) ∈
(Base‘𝑅)) |
59 | 2, 22, 23, 32, 58 | ringcld 20238 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∈
(Base‘𝑅)) |
60 | 59 | anasss 465 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∈
(Base‘𝑅)) |
61 | | fveq2 6893 |
. . . . . . 7
⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑌‘𝑛) = (𝑌‘(𝑘 ∘f − 𝑗))) |
62 | | oveq2 7424 |
. . . . . . . 8
⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑥 ∘f − 𝑗) ∘f −
𝑛) = ((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))) |
63 | 62 | fveq2d 6897 |
. . . . . . 7
⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)) = (𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))) |
64 | 61, 63 | oveq12d 7434 |
. . . . . 6
⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))) = ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))))) |
65 | 64 | oveq2d 7432 |
. . . . 5
⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))))) |
66 | 3, 18, 19, 2, 21, 60, 65 | psrass1lem 21937 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))))))))) = (𝑅 Σg
(𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))))))))) |
67 | 7 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋 ∈ 𝐵) |
68 | 8 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵) |
69 | | simpr 483 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) |
70 | | breq1 5148 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑘 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑘 ∘r ≤ 𝑥)) |
71 | 70 | elrab 3680 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥)) |
72 | 69, 71 | sylib 217 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥)) |
73 | 72 | simpld 493 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ 𝐷) |
74 | 1, 4, 22, 5, 3, 67, 68, 73 | psrmulval 21949 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))) |
75 | 74 | oveq1d 7431 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) |
76 | | eqid 2726 |
. . . . . . . 8
⊢
(0g‘𝑅) = (0g‘𝑅) |
77 | 6 | ad2antrr 724 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring) |
78 | 3 | psrbaglefi 21925 |
. . . . . . . . 9
⊢ (𝑘 ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin) |
79 | 73, 78 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin) |
80 | 41 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍:𝐷⟶(Base‘𝑅)) |
81 | | simplr 767 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷) |
82 | 3 | psrbagf 21911 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝐷 → 𝑘:𝐼⟶ℕ0) |
83 | 73, 82 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0) |
84 | 72 | simprd 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∘r ≤ 𝑥) |
85 | 3 | psrbagcon 21923 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝐷 ∧ 𝑘:𝐼⟶ℕ0 ∧ 𝑘 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥)) |
86 | 81, 83, 84, 85 | syl3anc 1368 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥)) |
87 | 86 | simpld 493 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑘) ∈ 𝐷) |
88 | 80, 87 | ffvelcdmd 7091 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅)) |
89 | 6 | ad3antrrr 728 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑅 ∈ Ring) |
90 | 24 | ad3antrrr 728 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅)) |
91 | | simpr 483 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) |
92 | | breq1 5148 |
. . . . . . . . . . . . 13
⊢ (ℎ = 𝑗 → (ℎ ∘r ≤ 𝑘 ↔ 𝑗 ∘r ≤ 𝑘)) |
93 | 92 | elrab 3680 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘)) |
94 | 91, 93 | sylib 217 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘)) |
95 | 94 | simpld 493 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷) |
96 | 90, 95 | ffvelcdmd 7091 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑋‘𝑗) ∈ (Base‘𝑅)) |
97 | 33 | ad3antrrr 728 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅)) |
98 | 73 | adantr 479 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
99 | 95, 44 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗:𝐼⟶ℕ0) |
100 | 94 | simprd 494 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∘r ≤ 𝑘) |
101 | 3 | psrbagcon 21923 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑘) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘)) |
102 | 98, 99, 100, 101 | syl3anc 1368 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘)) |
103 | 102 | simpld 493 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷) |
104 | 97, 103 | ffvelcdmd 7091 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅)) |
105 | 2, 22, 89, 96, 104 | ringcld 20238 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))) ∈ (Base‘𝑅)) |
106 | | eqid 2726 |
. . . . . . . . 9
⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) |
107 | | fvex 6906 |
. . . . . . . . . 10
⊢
(0g‘𝑅) ∈ V |
108 | 107 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (0g‘𝑅) ∈ V) |
109 | 106, 79, 105, 108 | fsuppmptdm 9412 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) finSupp
(0g‘𝑅)) |
110 | 2, 76, 22, 77, 79, 88, 105, 109 | gsummulc1 20291 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) |
111 | 88 | adantr 479 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅)) |
112 | 2, 22 | ringass 20232 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ ((𝑋‘𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) |
113 | 89, 96, 104, 111, 112 | syl13anc 1369 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) |
114 | 3 | psrbagf 21911 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ 𝐷 → 𝑥:𝐼⟶ℕ0) |
115 | 114 | ad3antlr 729 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0) |
116 | 115 | ffvelcdmda 7090 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈
ℕ0) |
117 | 83 | adantr 479 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0) |
118 | 117 | ffvelcdmda 7090 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈
ℕ0) |
119 | 99 | ffvelcdmda 7090 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈
ℕ0) |
120 | | nn0cn 12528 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ) |
121 | | nn0cn 12528 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ) |
122 | | nn0cn 12528 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ) |
123 | | nnncan2 11538 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑘‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧))) |
124 | 120, 121,
122, 123 | syl3an 1157 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧))) |
125 | 116, 118,
119, 124 | syl3anc 1368 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧))) |
126 | 125 | mpteq2dva 5245 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧)))) |
127 | | psrring.i |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
128 | 127 | ad3antrrr 728 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑉) |
129 | | ovexd 7451 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V) |
130 | | ovexd 7451 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V) |
131 | 115 | feqmptd 6963 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧))) |
132 | 99 | feqmptd 6963 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧))) |
133 | 128, 116,
119, 131, 132 | offval2 7702 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑥 ∘f − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑗‘𝑧)))) |
134 | 117 | feqmptd 6963 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧))) |
135 | 128, 118,
119, 134, 132 | offval2 7702 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑗‘𝑧)))) |
136 | 128, 129,
130, 133, 135 | offval2 7702 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)) = (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))))) |
137 | 128, 116,
118, 131, 134 | offval2 7702 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑥 ∘f − 𝑘) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧)))) |
138 | 126, 136,
137 | 3eqtr4d 2776 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)) = (𝑥 ∘f −
𝑘)) |
139 | 138 | fveq2d 6897 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))) = (𝑍‘(𝑥 ∘f − 𝑘))) |
140 | 139 | oveq2d 7432 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))) = ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) |
141 | 140 | oveq2d 7432 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) |
142 | 113, 141 | eqtr4d 2769 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))))) |
143 | 142 | mpteq2dva 5245 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))))))) |
144 | 143 | oveq2d 7432 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))))))) |
145 | 75, 110, 144 | 3eqtr2d 2772 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))))))) |
146 | 145 | mpteq2dva 5245 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) = (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗))))))))) |
147 | 146 | oveq2d 7432 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
(𝑘 ∘f
− 𝑗)))))))))) |
148 | 8 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵) |
149 | 10 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍 ∈ 𝐵) |
150 | 1, 4, 22, 5, 3, 148, 149, 49 | psrmulval 21949 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))))) |
151 | 150 | oveq2d 7432 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))))))) |
152 | 6 | ad2antrr 724 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring) |
153 | 3 | psrbaglefi 21925 |
. . . . . . . . 9
⊢ ((𝑥 ∘f −
𝑗) ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin) |
154 | 49, 153 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin) |
155 | | ovex 7449 |
. . . . . . . . . . . . 13
⊢
(ℕ0 ↑m 𝐼) ∈ V |
156 | 3, 155 | rab2ex 5334 |
. . . . . . . . . . . 12
⊢ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ V |
157 | 156 | mptex 7232 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∈
V |
158 | | funmpt 6589 |
. . . . . . . . . . 11
⊢ Fun
(𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) |
159 | 157, 158,
107 | 3pm3.2i 1336 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∈ V ∧ Fun
(𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∧
(0g‘𝑅)
∈ V) |
160 | 159 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∈ V ∧ Fun
(𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∧
(0g‘𝑅)
∈ V)) |
161 | | suppssdm 8183 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) supp
(0g‘𝑅))
⊆ dom (𝑛 ∈
{ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) |
162 | | eqid 2726 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) = (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) |
163 | 162 | dmmptss 6244 |
. . . . . . . . . . 11
⊢ dom
(𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} |
164 | 161, 163 | sstri 3988 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) supp
(0g‘𝑅))
⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} |
165 | 164 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) supp
(0g‘𝑅))
⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) |
166 | | suppssfifsupp 9416 |
. . . . . . . . 9
⊢ ((((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∈ V ∧ Fun
(𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) ∧
(0g‘𝑅)
∈ V) ∧ ({ℎ ∈
𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f −
𝑗)} ∈ Fin ∧
((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) supp
(0g‘𝑅))
⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) finSupp
(0g‘𝑅)) |
167 | 160, 154,
165, 166 | syl12anc 835 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))) finSupp
(0g‘𝑅)) |
168 | 2, 76, 22, 152, 154, 31, 58, 167 | gsummulc2 20292 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))))))) |
169 | 151, 168 | eqtr4d 2769 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))))))) |
170 | 169 | mpteq2dva 5245 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛)))))))) |
171 | 170 | oveq2d 7432 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f −
𝑛))))))))) |
172 | 66, 147, 171 | 3eqtr4d 2776 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)))))) |
173 | 9 | adantr 479 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋 × 𝑌) ∈ 𝐵) |
174 | 10 | adantr 479 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ∈ 𝐵) |
175 | 1, 4, 22, 5, 3, 173, 174, 19 | psrmulval 21949 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))) |
176 | 7 | adantr 479 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑋 ∈ 𝐵) |
177 | 14 | adantr 479 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 × 𝑍) ∈ 𝐵) |
178 | 1, 4, 22, 5, 3, 176, 177, 19 | psrmulval 21949 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)))))) |
179 | 172, 175,
178 | 3eqtr4d 2776 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥)) |
180 | 13, 17, 179 | eqfnfvd 7039 |
1
⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍))) |