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| Mirrors > Home > MPE Home > Th. List > issdrg | Structured version Visualization version GIF version | ||
| Description: Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.) |
| Ref | Expression |
|---|---|
| issdrg | ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-sdrg 20788 | . . . 4 ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) | |
| 2 | 1 | mptrcl 7025 | . . 3 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing) |
| 3 | fveq2 6906 | . . . . . . 7 ⊢ (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅)) | |
| 4 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑤 = 𝑅 → (𝑤 ↾s 𝑠) = (𝑅 ↾s 𝑠)) | |
| 5 | 4 | eleq1d 2826 | . . . . . . 7 ⊢ (𝑤 = 𝑅 → ((𝑤 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑠) ∈ DivRing)) |
| 6 | 3, 5 | rabeqbidv 3455 | . . . . . 6 ⊢ (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}) |
| 7 | fvex 6919 | . . . . . . 7 ⊢ (SubRing‘𝑅) ∈ V | |
| 8 | 7 | rabex 5339 | . . . . . 6 ⊢ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ∈ V |
| 9 | 6, 1, 8 | fvmpt 7016 | . . . . 5 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}) |
| 10 | 9 | eleq2d 2827 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})) |
| 11 | oveq2 7439 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑆)) | |
| 12 | 11 | eleq1d 2826 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑆) ∈ DivRing)) |
| 13 | 12 | elrab 3692 | . . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) |
| 14 | 10, 13 | bitrdi 287 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))) |
| 15 | 2, 14 | biadanii 822 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))) |
| 16 | 3anass 1095 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))) | |
| 17 | 15, 16 | bitr4i 278 | 1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 SubRingcsubrg 20569 DivRingcdr 20729 SubDRingcsdrg 20787 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-sdrg 20788 |
| This theorem is referenced by: sdrgrcl 20790 sdrgdrng 20791 sdrgsubrg 20792 sdrgid 20793 sdrgss 20794 issdrg2 20796 fldsdrgfld 20799 sdrgint 20805 primefld 20806 primefld0cl 20807 primefld1cl 20808 sdrgdvcl 33301 sdrginvcl 33302 primefldchr 33303 fldgensdrg 33316 fldgenssp 33320 primefldgen1 33323 1fldgenq 33324 fldextrspunlem2 33727 fldextrspundgdvdslem 33730 fldextrspundgdvds 33731 irngnzply1lem 33740 irngnzply1 33741 ply1annig1p 33747 minplycl 33749 ply1annprmidl 33750 algextdeglem1 33758 algextdeglem2 33759 algextdeglem3 33760 algextdeglem4 33761 algextdeglem5 33762 constrextdg2 33790 2sqr3minply 33791 |
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