| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . 6
⊢
(oppr‘𝑄) = (oppr‘𝑄) |
| 2 | | eqid 2737 |
. . . . . 6
⊢
(1r‘𝑄) = (1r‘𝑄) |
| 3 | 1, 2 | oppr1 20350 |
. . . . 5
⊢
(1r‘𝑄) =
(1r‘(oppr‘𝑄)) |
| 4 | | opprqus.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
| 5 | | opprqus.o |
. . . . . 6
⊢ 𝑂 =
(oppr‘𝑅) |
| 6 | | opprqus.q |
. . . . . 6
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) |
| 7 | | opprqus1r.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 8 | | opprqus1r.i |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
| 9 | 4, 5, 6, 7, 8 | opprqus1r 33520 |
. . . . 5
⊢ (𝜑 →
(1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 10 | 3, 9 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 11 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝑄) = (0g‘𝑄) |
| 12 | 1, 11 | oppr0 20349 |
. . . . 5
⊢
(0g‘𝑄) =
(0g‘(oppr‘𝑄)) |
| 13 | 8 | 2idllidld 21264 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
| 14 | | lidlnsg 21258 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅)) |
| 15 | 7, 13, 14 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) |
| 16 | 4, 5, 6, 15 | opprqus0g 33518 |
. . . . 5
⊢ (𝜑 →
(0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 17 | 12, 16 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → (0g‘𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 18 | 10, 17 | neeq12d 3002 |
. . 3
⊢ (𝜑 →
((1r‘𝑄)
≠ (0g‘𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠
(0g‘(𝑂
/s (𝑂
~QG 𝐼))))) |
| 19 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝑄) =
(Base‘𝑄) |
| 20 | 1, 19 | opprbas 20341 |
. . . . . 6
⊢
(Base‘𝑄) =
(Base‘(oppr‘𝑄)) |
| 21 | | eqid 2737 |
. . . . . . . . 9
⊢
(LIdeal‘𝑅) =
(LIdeal‘𝑅) |
| 22 | 4, 21 | lidlss 21222 |
. . . . . . . 8
⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
| 23 | 13, 22 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
| 24 | 4, 5, 6, 7, 23 | opprqusbas 33516 |
. . . . . 6
⊢ (𝜑 →
(Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 25 | 20, 24 | eqtrid 2789 |
. . . . 5
⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 26 | 17 | sneqd 4638 |
. . . . 5
⊢ (𝜑 →
{(0g‘𝑄)} =
{(0g‘(𝑂
/s (𝑂
~QG 𝐼)))}) |
| 27 | 25, 26 | difeq12d 4127 |
. . . 4
⊢ (𝜑 → ((Base‘𝑄) ∖
{(0g‘𝑄)})
= ((Base‘(𝑂
/s (𝑂
~QG 𝐼)))
∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})) |
| 28 | 25 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 29 | 7 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑅 ∈ Ring) |
| 30 | 8 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝐼 ∈ (2Ideal‘𝑅)) |
| 31 | | simplr 769 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) |
| 32 | 31 | eldifad 3963 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ (Base‘𝑄)) |
| 33 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑦 ∈ (Base‘𝑄)) |
| 34 | 4, 5, 6, 29, 30, 19, 32, 33 | opprqusmulr 33519 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑥(.r‘(oppr‘𝑄))𝑦) = (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦)) |
| 35 | 10 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
| 36 | 34, 35 | eqeq12d 2753 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → ((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ↔ (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))) |
| 37 | 4, 5, 6, 29, 30, 19, 33, 32 | opprqusmulr 33519 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑦(.r‘(oppr‘𝑄))𝑥) = (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥)) |
| 38 | 37, 35 | eqeq12d 2753 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → ((𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄) ↔ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))) |
| 39 | 36, 38 | anbi12d 632 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)) ↔ ((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))) |
| 40 | 28, 39 | rexeqbidva 3333 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)) ↔ ∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))) |
| 41 | 27, 40 | raleqbidva 3332 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ ((Base‘𝑄) ∖
{(0g‘𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)) ↔ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))) |
| 42 | 18, 41 | anbi12d 632 |
. 2
⊢ (𝜑 →
(((1r‘𝑄)
≠ (0g‘𝑄) ∧ ∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄))) ↔ ((1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))) |
| 43 | | eqid 2737 |
. . 3
⊢
(.r‘(oppr‘𝑄)) =
(.r‘(oppr‘𝑄)) |
| 44 | | eqid 2737 |
. . . 4
⊢
(Unit‘𝑄) =
(Unit‘𝑄) |
| 45 | 44, 1 | opprunit 20377 |
. . 3
⊢
(Unit‘𝑄) =
(Unit‘(oppr‘𝑄)) |
| 46 | | eqid 2737 |
. . . . . 6
⊢
(2Ideal‘𝑅) =
(2Ideal‘𝑅) |
| 47 | 6, 46 | qusring 21285 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring) |
| 48 | 7, 8, 47 | syl2anc 584 |
. . . 4
⊢ (𝜑 → 𝑄 ∈ Ring) |
| 49 | 1 | opprring 20347 |
. . . 4
⊢ (𝑄 ∈ Ring →
(oppr‘𝑄) ∈ Ring) |
| 50 | 48, 49 | syl 17 |
. . 3
⊢ (𝜑 →
(oppr‘𝑄) ∈ Ring) |
| 51 | 20, 12, 3, 43, 45, 50 | isdrng4 33298 |
. 2
⊢ (𝜑 →
((oppr‘𝑄) ∈ DivRing ↔
((1r‘𝑄)
≠ (0g‘𝑄) ∧ ∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄))))) |
| 52 | | eqid 2737 |
. . 3
⊢
(Base‘(𝑂
/s (𝑂
~QG 𝐼)))
= (Base‘(𝑂
/s (𝑂
~QG 𝐼))) |
| 53 | | eqid 2737 |
. . 3
⊢
(0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) |
| 54 | | eqid 2737 |
. . 3
⊢
(1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) |
| 55 | | eqid 2737 |
. . 3
⊢
(.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) |
| 56 | | eqid 2737 |
. . 3
⊢
(Unit‘(𝑂
/s (𝑂
~QG 𝐼)))
= (Unit‘(𝑂
/s (𝑂
~QG 𝐼))) |
| 57 | 5 | opprring 20347 |
. . . . 5
⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
| 58 | 7, 57 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑂 ∈ Ring) |
| 59 | 5, 7 | oppr2idl 33514 |
. . . . 5
⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂)) |
| 60 | 8, 59 | eleqtrd 2843 |
. . . 4
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑂)) |
| 61 | | eqid 2737 |
. . . . 5
⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼)) |
| 62 | | eqid 2737 |
. . . . 5
⊢
(2Ideal‘𝑂) =
(2Ideal‘𝑂) |
| 63 | 61, 62 | qusring 21285 |
. . . 4
⊢ ((𝑂 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑂)) → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring) |
| 64 | 58, 60, 63 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring) |
| 65 | 52, 53, 54, 55, 56, 64 | isdrng4 33298 |
. 2
⊢ (𝜑 → ((𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing ↔
((1r‘(𝑂
/s (𝑂
~QG 𝐼)))
≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))) |
| 66 | 42, 51, 65 | 3bitr4d 311 |
1
⊢ (𝜑 →
((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing)) |