| Step | Hyp | Ref
| Expression |
| 1 | | drngidl.b |
. . . 4
⊢ 𝐵 = (Base‘𝑅) |
| 2 | | drngidl.z |
. . . 4
⊢ 0 =
(0g‘𝑅) |
| 3 | | drngidl.u |
. . . 4
⊢ 𝑈 = (LIdeal‘𝑅) |
| 4 | 1, 2, 3 | drngnidl 21253 |
. . 3
⊢ (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵}) |
| 5 | 4 | adantl 481 |
. 2
⊢ ((𝑅 ∈ NzRing ∧ 𝑅 ∈ DivRing) → 𝑈 = {{ 0 }, 𝐵}) |
| 6 | | eqid 2737 |
. . . . 5
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 7 | 6, 2 | nzrnz 20515 |
. . . 4
⊢ (𝑅 ∈ NzRing →
(1r‘𝑅)
≠ 0
) |
| 8 | 7 | adantr 480 |
. . 3
⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → (1r‘𝑅) ≠ 0 ) |
| 9 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 10 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Unit‘𝑅) =
(Unit‘𝑅) |
| 11 | | nzrring 20516 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
| 12 | 11 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → 𝑅 ∈ Ring) |
| 13 | 12 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ Ring) |
| 14 | 13 | ad4antr 732 |
. . . . . . . . . . 11
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → 𝑅 ∈ Ring) |
| 15 | | simp-4r 784 |
. . . . . . . . . . 11
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → 𝑦 ∈ 𝐵) |
| 16 | | simplr 769 |
. . . . . . . . . . 11
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → 𝑧 ∈ 𝐵) |
| 17 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ (𝐵 ∖ { 0 })) |
| 18 | 17 | eldifad 3963 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ 𝐵) |
| 19 | 18 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → 𝑥 ∈ 𝐵) |
| 20 | 19 | ad2antrr 726 |
. . . . . . . . . . 11
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → 𝑥 ∈ 𝐵) |
| 21 | | simpr 484 |
. . . . . . . . . . . 12
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) |
| 22 | 21 | eqcomd 2743 |
. . . . . . . . . . 11
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → (𝑧(.r‘𝑅)𝑦) = (1r‘𝑅)) |
| 23 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) |
| 24 | 23 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)) |
| 25 | 24 | ad2antrr 726 |
. . . . . . . . . . 11
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)) |
| 26 | 1, 2, 6, 9, 10, 14, 15, 16, 20, 22, 25 | ringinveu 33297 |
. . . . . . . . . 10
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → 𝑥 = 𝑧) |
| 27 | 26 | oveq1d 7446 |
. . . . . . . . 9
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → (𝑥(.r‘𝑅)𝑦) = (𝑧(.r‘𝑅)𝑦)) |
| 28 | 27, 22 | eqtrd 2777 |
. . . . . . . 8
⊢
(((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑧 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) → (𝑥(.r‘𝑅)𝑦) = (1r‘𝑅)) |
| 29 | 13 | ad2antrr 726 |
. . . . . . . . 9
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → 𝑅 ∈ Ring) |
| 30 | | simplr 769 |
. . . . . . . . 9
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → 𝑦 ∈ 𝐵) |
| 31 | 1, 6 | ringidcl 20262 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ 𝐵) |
| 32 | 13, 31 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
(1r‘𝑅)
∈ 𝐵) |
| 33 | 32 | ad2antrr 726 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → (1r‘𝑅) ∈ 𝐵) |
| 34 | 30 | snssd 4809 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → {𝑦} ⊆ 𝐵) |
| 35 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(RSpan‘𝑅) =
(RSpan‘𝑅) |
| 36 | 35, 1, 3 | rspcl 21245 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ {𝑦} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑦}) ∈ 𝑈) |
| 37 | 29, 34, 36 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ((RSpan‘𝑅)‘{𝑦}) ∈ 𝑈) |
| 38 | | simp-4r 784 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → 𝑈 = {{ 0 }, 𝐵}) |
| 39 | 37, 38 | eleqtrd 2843 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ((RSpan‘𝑅)‘{𝑦}) ∈ {{ 0 }, 𝐵}) |
| 40 | | elpri 4649 |
. . . . . . . . . . . 12
⊢
(((RSpan‘𝑅)‘{𝑦}) ∈ {{ 0 }, 𝐵} → (((RSpan‘𝑅)‘{𝑦}) = { 0 } ∨ ((RSpan‘𝑅)‘{𝑦}) = 𝐵)) |
| 41 | 39, 40 | syl 17 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → (((RSpan‘𝑅)‘{𝑦}) = { 0 } ∨ ((RSpan‘𝑅)‘{𝑦}) = 𝐵)) |
| 42 | | simplr 769 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) →
(1r‘𝑅) =
(𝑦(.r‘𝑅)𝑥)) |
| 43 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) → 𝑦 = 0 ) |
| 44 | 43 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) → (𝑦(.r‘𝑅)𝑥) = ( 0 (.r‘𝑅)𝑥)) |
| 45 | 1, 9, 2 | ringlz 20290 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ( 0 (.r‘𝑅)𝑥) = 0 ) |
| 46 | 13, 18, 45 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ( 0
(.r‘𝑅)𝑥) = 0 ) |
| 47 | 46 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) → ( 0
(.r‘𝑅)𝑥) = 0 ) |
| 48 | 42, 44, 47 | 3eqtrd 2781 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) →
(1r‘𝑅) =
0
) |
| 49 | 8 | ad4antr 732 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) →
(1r‘𝑅)
≠ 0
) |
| 50 | 49 | neneqd 2945 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑅 ∈
NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) ∧ 𝑦 = 0 ) → ¬
(1r‘𝑅) =
0
) |
| 51 | 48, 50 | pm2.65da 817 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ¬ 𝑦 = 0 ) |
| 52 | 51 | neqned 2947 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → 𝑦 ≠ 0 ) |
| 53 | 1, 2, 35 | pidlnz 33404 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 ) →
((RSpan‘𝑅)‘{𝑦}) ≠ { 0 }) |
| 54 | 29, 30, 52, 53 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ((RSpan‘𝑅)‘{𝑦}) ≠ { 0 }) |
| 55 | 54 | neneqd 2945 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ¬ ((RSpan‘𝑅)‘{𝑦}) = { 0 }) |
| 56 | 41, 55 | orcnd 879 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ((RSpan‘𝑅)‘{𝑦}) = 𝐵) |
| 57 | 33, 56 | eleqtrrd 2844 |
. . . . . . . . 9
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → (1r‘𝑅) ∈ ((RSpan‘𝑅)‘{𝑦})) |
| 58 | 1, 9, 35 | elrspsn 21250 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → ((1r‘𝑅) ∈ ((RSpan‘𝑅)‘{𝑦}) ↔ ∃𝑧 ∈ 𝐵 (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦))) |
| 59 | 58 | biimpa 476 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) ∈ ((RSpan‘𝑅)‘{𝑦})) → ∃𝑧 ∈ 𝐵 (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) |
| 60 | 29, 30, 57, 59 | syl21anc 838 |
. . . . . . . 8
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ∃𝑧 ∈ 𝐵 (1r‘𝑅) = (𝑧(.r‘𝑅)𝑦)) |
| 61 | 28, 60 | r19.29a 3162 |
. . . . . . 7
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → (𝑥(.r‘𝑅)𝑦) = (1r‘𝑅)) |
| 62 | 61, 24 | jca 511 |
. . . . . 6
⊢
(((((𝑅 ∈ NzRing
∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) → ((𝑥(.r‘𝑅)𝑦) = (1r‘𝑅) ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))) |
| 63 | 62 | anasss 466 |
. . . . 5
⊢ ((((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) ∧ (𝑦 ∈ 𝐵 ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥))) → ((𝑥(.r‘𝑅)𝑦) = (1r‘𝑅) ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))) |
| 64 | 18 | snssd 4809 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → {𝑥} ⊆ 𝐵) |
| 65 | 35, 1, 3 | rspcl 21245 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ {𝑥} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑥}) ∈ 𝑈) |
| 66 | 13, 64, 65 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
((RSpan‘𝑅)‘{𝑥}) ∈ 𝑈) |
| 67 | | simplr 769 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑈 = {{ 0 }, 𝐵}) |
| 68 | 66, 67 | eleqtrd 2843 |
. . . . . . . . 9
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
((RSpan‘𝑅)‘{𝑥}) ∈ {{ 0 }, 𝐵}) |
| 69 | | elpri 4649 |
. . . . . . . . 9
⊢
(((RSpan‘𝑅)‘{𝑥}) ∈ {{ 0 }, 𝐵} → (((RSpan‘𝑅)‘{𝑥}) = { 0 } ∨ ((RSpan‘𝑅)‘{𝑥}) = 𝐵)) |
| 70 | 68, 69 | syl 17 |
. . . . . . . 8
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
(((RSpan‘𝑅)‘{𝑥}) = { 0 } ∨ ((RSpan‘𝑅)‘{𝑥}) = 𝐵)) |
| 71 | | eldifsni 4790 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ≠ 0 ) |
| 72 | 71 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ≠ 0 ) |
| 73 | 1, 2, 35 | pidlnz 33404 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) →
((RSpan‘𝑅)‘{𝑥}) ≠ { 0 }) |
| 74 | 13, 18, 72, 73 | syl3anc 1373 |
. . . . . . . . 9
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
((RSpan‘𝑅)‘{𝑥}) ≠ { 0 }) |
| 75 | 74 | neneqd 2945 |
. . . . . . . 8
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ¬
((RSpan‘𝑅)‘{𝑥}) = { 0 }) |
| 76 | 70, 75 | orcnd 879 |
. . . . . . 7
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
((RSpan‘𝑅)‘{𝑥}) = 𝐵) |
| 77 | 32, 76 | eleqtrrd 2844 |
. . . . . 6
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) →
(1r‘𝑅)
∈ ((RSpan‘𝑅)‘{𝑥})) |
| 78 | 1, 9, 35 | elrspsn 21250 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑅) ∈ ((RSpan‘𝑅)‘{𝑥}) ↔ ∃𝑦 ∈ 𝐵 (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥))) |
| 79 | 78 | biimpa 476 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑅) ∈ ((RSpan‘𝑅)‘{𝑥})) → ∃𝑦 ∈ 𝐵 (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) |
| 80 | 13, 18, 77, 79 | syl21anc 838 |
. . . . 5
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ∃𝑦 ∈ 𝐵 (1r‘𝑅) = (𝑦(.r‘𝑅)𝑥)) |
| 81 | 63, 80 | reximddv 3171 |
. . . 4
⊢ (((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ∃𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (1r‘𝑅) ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))) |
| 82 | 81 | ralrimiva 3146 |
. . 3
⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (1r‘𝑅) ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))) |
| 83 | 1, 2, 6, 9, 10, 12 | isdrng4 33298 |
. . 3
⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → (𝑅 ∈ DivRing ↔
((1r‘𝑅)
≠ 0
∧ ∀𝑥 ∈
(𝐵 ∖ { 0
})∃𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (1r‘𝑅) ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))) |
| 84 | 8, 82, 83 | mpbir2and 713 |
. 2
⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → 𝑅 ∈ DivRing) |
| 85 | 5, 84 | impbida 801 |
1
⊢ (𝑅 ∈ NzRing → (𝑅 ∈ DivRing ↔ 𝑈 = {{ 0 }, 𝐵})) |