Step | Hyp | Ref
| Expression |
1 | | simpllr 776 |
. . . . . . . 8
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
2 | | simpr 489 |
. . . . . . . 8
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
3 | 1, 2 | uneq12d 4065 |
. . . . . . 7
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) = ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})) |
4 | | unrab 4204 |
. . . . . . . . . 10
⊢ ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} |
5 | | zarclsx.1 |
. . . . . . . . . . 11
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
6 | | eqid 2759 |
. . . . . . . . . . . . 13
⊢
(IDLsrg‘𝑅) =
(IDLsrg‘𝑅) |
7 | | eqid 2759 |
. . . . . . . . . . . . 13
⊢
(LIdeal‘𝑅) =
(LIdeal‘𝑅) |
8 | | eqid 2759 |
. . . . . . . . . . . . 13
⊢
(.r‘(IDLsrg‘𝑅)) =
(.r‘(IDLsrg‘𝑅)) |
9 | | simpll 767 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing) |
10 | 9 | crngringd 19363 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring) |
11 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑙 ∈ (LIdeal‘𝑅)) |
12 | | simpr 489 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (LIdeal‘𝑅)) |
13 | 6, 7, 8, 10, 11, 12 | idlsrgmulrcl 31161 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ∈ (LIdeal‘𝑅)) |
14 | | sseq1 3913 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → (𝑖 ⊆ 𝑗 ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)) |
15 | 14 | rabbidv 3390 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗}) |
16 | 15 | eqeq2d 2770 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})) |
17 | 16 | adantl 486 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})) |
18 | | eqid 2759 |
. . . . . . . . . . . . . . . . 17
⊢
(.r‘𝑅) = (.r‘𝑅) |
19 | 9 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑙 ⊆ 𝑗) → 𝑅 ∈ CRing) |
20 | 11 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑙 ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅)) |
21 | 12 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑙 ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅)) |
22 | 6, 7, 8, 18, 19, 20, 21 | idlsrgmulrss1 31162 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑙 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑙) |
23 | | simpr 489 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑙 ⊆ 𝑗) → 𝑙 ⊆ 𝑗) |
24 | 22, 23 | sstrd 3898 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑙 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) |
25 | 10 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑘 ⊆ 𝑗) → 𝑅 ∈ Ring) |
26 | 11 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑘 ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅)) |
27 | 12 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑘 ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅)) |
28 | 6, 7, 8, 18, 25, 26, 27 | idlsrgmulrss2 31163 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑘 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑘) |
29 | | simpr 489 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝑗) |
30 | 28, 29 | sstrd 3898 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
𝑘 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) |
31 | 24, 30 | jaodan 956 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) |
32 | | eqid 2759 |
. . . . . . . . . . . . . . 15
⊢
(LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅)) |
33 | 10 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑅 ∈ Ring) |
34 | | simplr 769 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑗 ∈ (PrmIdeal‘𝑅)) |
35 | 11 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅)) |
36 | 12 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅)) |
37 | | eqid 2759 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Base‘𝑅) =
(Base‘𝑅) |
38 | | eqid 2759 |
. . . . . . . . . . . . . . . . . . 19
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
39 | 37, 7 | lidlss 20036 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅)) |
40 | 35, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ⊆ (Base‘𝑅)) |
41 | 37, 7 | lidlss 20036 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (LIdeal‘𝑅) → 𝑘 ⊆ (Base‘𝑅)) |
42 | 36, 41 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ⊆ (Base‘𝑅)) |
43 | 37, 38, 32, 33, 40, 42 | ringlsmss 31089 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (Base‘𝑅)) |
44 | | eqid 2759 |
. . . . . . . . . . . . . . . . . . 19
⊢
(RSpan‘𝑅) =
(RSpan‘𝑅) |
45 | 44, 37 | rspssid 20049 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Ring ∧ (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (Base‘𝑅)) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘))) |
46 | 33, 43, 45 | syl2anc 588 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘))) |
47 | 6, 7, 8, 38, 32, 33, 35, 36 | idlsrgmulrval 31160 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) = ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘))) |
48 | 46, 47 | sseqtrrd 3929 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (𝑙(.r‘(IDLsrg‘𝑅))𝑘)) |
49 | | simpr 489 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) |
50 | 48, 49 | sstrd 3898 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ 𝑗) |
51 | 32, 33, 34, 35, 36, 50 | idlmulssprm 31123 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑘 ∈
(LIdeal‘𝑅)) ∧
𝑗 ∈
(PrmIdeal‘𝑅)) ∧
(𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)) |
52 | 31, 51 | impbida 801 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → ((𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗) ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)) |
53 | 52 | rabbidva 3388 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗}) |
54 | 13, 17, 53 | rspcedvd 3542 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ∃𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
55 | | fvex 6664 |
. . . . . . . . . . . . 13
⊢
(PrmIdeal‘𝑅)
∈ V |
56 | 55 | rabex 5195 |
. . . . . . . . . . . 12
⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ V |
57 | 56 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ V) |
58 | 5, 54, 57 | elrnmptd 5795 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ ran 𝑉) |
59 | 4, 58 | eqeltrid 2855 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉) |
60 | 59 | adantlr 715 |
. . . . . . . 8
⊢ ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉) |
61 | 60 | adantr 485 |
. . . . . . 7
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉) |
62 | 3, 61 | eqeltrd 2851 |
. . . . . 6
⊢
(((((𝑅 ∈ CRing
∧ 𝑙 ∈
(LIdeal‘𝑅)) ∧
𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) |
63 | 62 | adantl4r 755 |
. . . . 5
⊢
((((((𝑅 ∈ CRing
∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) |
64 | 55 | rabex 5195 |
. . . . . . . . 9
⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V |
65 | 5, 64 | elrnmpti 5794 |
. . . . . . . 8
⊢ (𝑌 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
66 | | sseq1 3913 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗)) |
67 | 66 | rabbidv 3390 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
68 | 67 | eqeq2d 2770 |
. . . . . . . . 9
⊢ (𝑖 = 𝑘 → (𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})) |
69 | 68 | cbvrexvw 3360 |
. . . . . . . 8
⊢
(∃𝑖 ∈
(LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
70 | | biid 264 |
. . . . . . . 8
⊢
(∃𝑘 ∈
(LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
71 | 65, 69, 70 | 3bitri 301 |
. . . . . . 7
⊢ (𝑌 ∈ ran 𝑉 ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
72 | 71 | biimpi 219 |
. . . . . 6
⊢ (𝑌 ∈ ran 𝑉 → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
73 | 72 | ad3antlr 731 |
. . . . 5
⊢ ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) |
74 | 63, 73 | r19.29a 3211 |
. . . 4
⊢ ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) |
75 | 74 | adantl3r 750 |
. . 3
⊢
(((((𝑅 ∈ CRing
∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) |
76 | 5, 64 | elrnmpti 5794 |
. . . . . 6
⊢ (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
77 | | sseq1 3913 |
. . . . . . . . 9
⊢ (𝑖 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑙 ⊆ 𝑗)) |
78 | 77 | rabbidv 3390 |
. . . . . . . 8
⊢ (𝑖 = 𝑙 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
79 | 78 | eqeq2d 2770 |
. . . . . . 7
⊢ (𝑖 = 𝑙 → (𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})) |
80 | 79 | cbvrexvw 3360 |
. . . . . 6
⊢
(∃𝑖 ∈
(LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
81 | | biid 264 |
. . . . . 6
⊢
(∃𝑙 ∈
(LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
82 | 76, 80, 81 | 3bitri 301 |
. . . . 5
⊢ (𝑋 ∈ ran 𝑉 ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
83 | 82 | biimpi 219 |
. . . 4
⊢ (𝑋 ∈ ran 𝑉 → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
84 | 83 | ad2antlr 727 |
. . 3
⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) |
85 | 75, 84 | r19.29a 3211 |
. 2
⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) |
86 | 85 | 3impa 1108 |
1
⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) |