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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 20804 | . . . 4 ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) | |
2 | 1 | mptrcl 7024 | . . 3 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing) |
3 | fveq2 6906 | . . . . . . 7 ⊢ (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅)) | |
4 | oveq1 7437 | . . . . . . . 8 ⊢ (𝑤 = 𝑅 → (𝑤 ↾s 𝑠) = (𝑅 ↾s 𝑠)) | |
5 | 4 | eleq1d 2823 | . . . . . . 7 ⊢ (𝑤 = 𝑅 → ((𝑤 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑠) ∈ DivRing)) |
6 | 3, 5 | rabeqbidv 3451 | . . . . . 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 5344 | . . . . . 6 ⊢ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ∈ V |
9 | 6, 1, 8 | fvmpt 7015 | . . . . 5 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}) |
10 | 9 | eleq2d 2824 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})) |
11 | oveq2 7438 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑆)) | |
12 | 11 | eleq1d 2823 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑆) ∈ DivRing)) |
13 | 12 | elrab 3694 | . . . 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 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 {crab 3432 ‘cfv 6562 (class class class)co 7430 ↾s cress 17273 SubRingcsubrg 20585 DivRingcdr 20745 SubDRingcsdrg 20803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-sdrg 20804 |
This theorem is referenced by: sdrgrcl 20806 sdrgdrng 20807 sdrgsubrg 20808 sdrgid 20809 sdrgss 20810 issdrg2 20812 fldsdrgfld 20815 sdrgint 20821 primefld 20822 primefld0cl 20823 primefld1cl 20824 sdrgdvcl 33280 sdrginvcl 33281 primefldchr 33282 fldgensdrg 33295 fldgenssp 33299 primefldgen1 33302 1fldgenq 33303 irngnzply1lem 33704 irngnzply1 33705 ply1annig1p 33711 minplycl 33713 ply1annprmidl 33714 algextdeglem1 33722 algextdeglem2 33723 algextdeglem3 33724 algextdeglem4 33725 algextdeglem5 33726 2sqr3minply 33752 |
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