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Mirrors > Home > MPE Home > Th. List > subrgrcl | Structured version Visualization version GIF version |
Description: Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.) |
Ref | Expression |
---|---|
subrgrcl | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2726 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 1, 2 | issubrg 20555 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴))) |
4 | 3 | simplbi 496 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)) |
5 | 4 | simpld 493 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 ↾s cress 17242 1rcur 20164 Ringcrg 20216 SubRingcsubrg 20551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-subrg 20553 |
This theorem is referenced by: subrgsubg 20561 subrg1 20566 subrgsubm 20569 subrginv 20572 subrgunit 20574 subrgugrp 20575 opprsubrg 20577 subrgint 20579 subsubrg 20582 resrhm2b 20586 sralmod 21173 subrgpsr 21987 subrgmpl 22039 subrgmvr 22040 subrgmvrf 22041 subrgascl 22079 subrgasclcl 22080 asclply1subcl 22365 subrdom 33137 idlinsubrg 33306 ressply10g 33439 ressply1invg 33441 |
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