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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 20634 | . . . 4 ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) | |
2 | 1 | mptrcl 7007 | . . 3 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing) |
3 | fveq2 6891 | . . . . . . 7 ⊢ (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅)) | |
4 | oveq1 7419 | . . . . . . . 8 ⊢ (𝑤 = 𝑅 → (𝑤 ↾s 𝑠) = (𝑅 ↾s 𝑠)) | |
5 | 4 | eleq1d 2817 | . . . . . . 7 ⊢ (𝑤 = 𝑅 → ((𝑤 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑠) ∈ DivRing)) |
6 | 3, 5 | rabeqbidv 3448 | . . . . . 6 ⊢ (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}) |
7 | fvex 6904 | . . . . . . 7 ⊢ (SubRing‘𝑅) ∈ V | |
8 | 7 | rabex 5332 | . . . . . 6 ⊢ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ∈ V |
9 | 6, 1, 8 | fvmpt 6998 | . . . . 5 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}) |
10 | 9 | eleq2d 2818 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})) |
11 | oveq2 7420 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑆)) | |
12 | 11 | eleq1d 2817 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑆) ∈ DivRing)) |
13 | 12 | elrab 3683 | . . . 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 819 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))) |
16 | 3anass 1094 | . 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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {crab 3431 ‘cfv 6543 (class class class)co 7412 ↾s cress 17180 SubRingcsubrg 20465 DivRingcdr 20583 SubDRingcsdrg 20633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-sdrg 20634 |
This theorem is referenced by: sdrgrcl 20636 sdrgdrng 20637 sdrgsubrg 20638 sdrgid 20639 sdrgss 20640 issdrg2 20642 fldsdrgfld 20645 sdrgint 20651 primefld 20652 primefld0cl 20653 primefld1cl 20654 sdrgdvcl 32835 sdrginvcl 32836 primefldchr 32837 fldgensdrg 32842 fldgenssp 32846 primefldgen1 32849 1fldgenq 32850 irngnzply1lem 33211 irngnzply1 33212 ply1annig1p 33222 minplycl 33224 ply1annprmidl 33225 algextdeglem1 33230 algextdeglem2 33231 algextdeglem3 33232 algextdeglem4 33233 algextdeglem5 33234 |
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