| Step |
Hyp |
Ref |
Expression |
| 1 |
|
isufd.i |
⊢ 𝐼 = ( PrmIdeal ‘ 𝑅 ) |
| 2 |
|
isufd.3 |
⊢ 𝑃 = ( RPrime ‘ 𝑅 ) |
| 3 |
|
isufd.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
| 4 |
|
ufdprmidl.2 |
⊢ ( 𝜑 → 𝑅 ∈ UFD ) |
| 5 |
|
ufdprmidl.3 |
⊢ ( 𝜑 → 𝐽 ∈ 𝐼 ) |
| 6 |
|
ufdprmidl.4 |
⊢ ( 𝜑 → 𝐽 ≠ { 0 } ) |
| 7 |
|
ineq1 |
⊢ ( 𝑗 = 𝐽 → ( 𝑗 ∩ 𝑃 ) = ( 𝐽 ∩ 𝑃 ) ) |
| 8 |
7
|
neeq1d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝑗 ∩ 𝑃 ) ≠ ∅ ↔ ( 𝐽 ∩ 𝑃 ) ≠ ∅ ) ) |
| 9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → ( ( 𝑗 ∩ 𝑃 ) ≠ ∅ ↔ ( 𝐽 ∩ 𝑃 ) ≠ ∅ ) ) |
| 10 |
|
incom |
⊢ ( 𝑃 ∩ 𝐽 ) = ( 𝐽 ∩ 𝑃 ) |
| 11 |
10
|
neeq1i |
⊢ ( ( 𝑃 ∩ 𝐽 ) ≠ ∅ ↔ ( 𝐽 ∩ 𝑃 ) ≠ ∅ ) |
| 12 |
|
inn0 |
⊢ ( ( 𝑃 ∩ 𝐽 ) ≠ ∅ ↔ ∃ 𝑝 ∈ 𝑃 𝑝 ∈ 𝐽 ) |
| 13 |
11 12
|
bitr3i |
⊢ ( ( 𝐽 ∩ 𝑃 ) ≠ ∅ ↔ ∃ 𝑝 ∈ 𝑃 𝑝 ∈ 𝐽 ) |
| 14 |
9 13
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → ( ( 𝑗 ∩ 𝑃 ) ≠ ∅ ↔ ∃ 𝑝 ∈ 𝑃 𝑝 ∈ 𝐽 ) ) |
| 15 |
5 6
|
eldifsnd |
⊢ ( 𝜑 → 𝐽 ∈ ( 𝐼 ∖ { { 0 } } ) ) |
| 16 |
1 2 3
|
isufd |
⊢ ( 𝑅 ∈ UFD ↔ ( 𝑅 ∈ IDomn ∧ ∀ 𝑗 ∈ ( 𝐼 ∖ { { 0 } } ) ( 𝑗 ∩ 𝑃 ) ≠ ∅ ) ) |
| 17 |
16
|
simprbi |
⊢ ( 𝑅 ∈ UFD → ∀ 𝑗 ∈ ( 𝐼 ∖ { { 0 } } ) ( 𝑗 ∩ 𝑃 ) ≠ ∅ ) |
| 18 |
4 17
|
syl |
⊢ ( 𝜑 → ∀ 𝑗 ∈ ( 𝐼 ∖ { { 0 } } ) ( 𝑗 ∩ 𝑃 ) ≠ ∅ ) |
| 19 |
14 15 18
|
rspcdv2 |
⊢ ( 𝜑 → ∃ 𝑝 ∈ 𝑃 𝑝 ∈ 𝐽 ) |