| Step |
Hyp |
Ref |
Expression |
| 1 |
|
zarclsx.1 |
⊢ 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
| 2 |
|
simpllr |
⊢ ( ( ( ( ( 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 } ) |
| 3 |
|
simpr |
⊢ ( ( ( ( ( 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 } ) |
| 4 |
2 3
|
uneq12d |
⊢ ( ( ( ( ( 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 } ) ) |
| 5 |
|
unrab |
⊢ ( { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ∪ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ ( 𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗 ) } |
| 6 |
|
eqid |
⊢ ( IDLsrg ‘ 𝑅 ) = ( IDLsrg ‘ 𝑅 ) |
| 7 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
| 8 |
|
eqid |
⊢ ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) = ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) |
| 9 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝑅 ∈ CRing ) |
| 10 |
9
|
crngringd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
| 11 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 12 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 13 |
6 7 8 10 11 12
|
idlsrgmulrcl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ∈ ( LIdeal ‘ 𝑅 ) ) |
| 14 |
|
sseq1 |
⊢ ( 𝑖 = ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) → ( 𝑖 ⊆ 𝑗 ↔ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) ) |
| 15 |
14
|
rabbidv |
⊢ ( 𝑖 = ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 } ) |
| 16 |
15
|
eqeq2d |
⊢ ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
| 19 |
9
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑙 ⊆ 𝑗 ) → 𝑅 ∈ CRing ) |
| 20 |
11
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑙 ⊆ 𝑗 ) → 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 21 |
12
|
ad2antrr |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑙 ⊆ 𝑗 ) → ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑙 ) |
| 23 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑙 ⊆ 𝑗 ) → 𝑙 ⊆ 𝑗 ) |
| 24 |
22 23
|
sstrd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑙 ⊆ 𝑗 ) → ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) |
| 25 |
10
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑘 ⊆ 𝑗 ) → 𝑅 ∈ Ring ) |
| 26 |
11
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑘 ⊆ 𝑗 ) → 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 27 |
12
|
ad2antrr |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑘 ⊆ 𝑗 ) → ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑘 ) |
| 29 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑘 ⊆ 𝑗 ) → 𝑘 ⊆ 𝑗 ) |
| 30 |
28 29
|
sstrd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝑘 ⊆ 𝑗 ) → ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) |
| 31 |
24 30
|
jaodan |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ ( 𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗 ) ) → ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) |
| 32 |
|
eqid |
⊢ ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) = ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) |
| 33 |
10
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) → 𝑅 ∈ Ring ) |
| 34 |
|
simplr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) → 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 35 |
11
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) → 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 36 |
12
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) → 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 37 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 38 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
| 39 |
37 7
|
lidlss |
⊢ ( 𝑙 ∈ ( LIdeal ‘ 𝑅 ) → 𝑙 ⊆ ( Base ‘ 𝑅 ) ) |
| 40 |
35 39
|
syl |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) → 𝑙 ⊆ ( Base ‘ 𝑅 ) ) |
| 41 |
37 7
|
lidlss |
⊢ ( 𝑘 ∈ ( LIdeal ‘ 𝑅 ) → 𝑘 ⊆ ( Base ‘ 𝑅 ) ) |
| 42 |
36 41
|
syl |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( RSpan ‘ 𝑅 ) = ( RSpan ‘ 𝑅 ) |
| 45 |
44 37
|
rspssid |
⊢ ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → ( ( 𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗 ) ↔ ( 𝑙 ( .r ‘ ( IDLsrg ‘ 𝑅 ) ) 𝑘 ) ⊆ 𝑗 ) ) |
| 53 |
52
|
rabbidva |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( PrmIdeal ‘ 𝑅 ) ∈ V |
| 56 |
55
|
rabex |
⊢ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ ( 𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗 ) } ∈ V |
| 57 |
56
|
a1i |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ ( 𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗 ) } ∈ V ) |
| 58 |
1 54 57
|
elrnmptd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ ( 𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗 ) } ∈ ran 𝑉 ) |
| 59 |
5 58
|
eqeltrid |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → ( { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ∪ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) ∈ ran 𝑉 ) |
| 60 |
59
|
adantlr |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
4 61
|
eqeltrd |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 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 |
⊢ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ∈ V |
| 65 |
1 64
|
elrnmpti |
⊢ ( 𝑌 ∈ ran 𝑉 ↔ ∃ 𝑖 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
| 66 |
|
sseq1 |
⊢ ( 𝑖 = 𝑘 → ( 𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗 ) ) |
| 67 |
66
|
rabbidv |
⊢ ( 𝑖 = 𝑘 → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) |
| 68 |
67
|
eqeq2d |
⊢ ( 𝑖 = 𝑘 → ( 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ↔ 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) ) |
| 69 |
68
|
cbvrexvw |
⊢ ( ∃ 𝑖 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ↔ ∃ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) |
| 70 |
|
biid |
⊢ ( ∃ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ↔ ∃ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) |
| 71 |
65 69 70
|
3bitri |
⊢ ( 𝑌 ∈ ran 𝑉 ↔ ∃ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) |
| 72 |
71
|
biimpi |
⊢ ( 𝑌 ∈ ran 𝑉 → ∃ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) 𝑌 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑘 ⊆ 𝑗 } ) |
| 73 |
72
|
ad3antlr |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉 ) ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) → ( 𝑋 ∪ 𝑌 ) ∈ ran 𝑉 ) |
| 75 |
74
|
adantl3r |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ) ∧ 𝑌 ∈ ran 𝑉 ) ∧ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) → ( 𝑋 ∪ 𝑌 ) ∈ ran 𝑉 ) |
| 76 |
1 64
|
elrnmpti |
⊢ ( 𝑋 ∈ ran 𝑉 ↔ ∃ 𝑖 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
| 77 |
|
sseq1 |
⊢ ( 𝑖 = 𝑙 → ( 𝑖 ⊆ 𝑗 ↔ 𝑙 ⊆ 𝑗 ) ) |
| 78 |
77
|
rabbidv |
⊢ ( 𝑖 = 𝑙 → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) |
| 79 |
78
|
eqeq2d |
⊢ ( 𝑖 = 𝑙 → ( 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ↔ 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) ) |
| 80 |
79
|
cbvrexvw |
⊢ ( ∃ 𝑖 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ↔ ∃ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) |
| 81 |
|
biid |
⊢ ( ∃ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ↔ ∃ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) |
| 82 |
76 80 81
|
3bitri |
⊢ ( 𝑋 ∈ ran 𝑉 ↔ ∃ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) |
| 83 |
82
|
biimpi |
⊢ ( 𝑋 ∈ ran 𝑉 → ∃ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) |
| 84 |
83
|
ad2antlr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ) ∧ 𝑌 ∈ ran 𝑉 ) → ∃ 𝑙 ∈ ( LIdeal ‘ 𝑅 ) 𝑋 = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑙 ⊆ 𝑗 } ) |
| 85 |
75 84
|
r19.29a |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ) ∧ 𝑌 ∈ ran 𝑉 ) → ( 𝑋 ∪ 𝑌 ) ∈ ran 𝑉 ) |
| 86 |
85
|
3impa |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉 ) → ( 𝑋 ∪ 𝑌 ) ∈ ran 𝑉 ) |