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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 2622 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2622 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 1, 2 | issubrg 18780 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴))) |
4 | 3 | simplbi 476 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)) |
5 | 4 | simpld 475 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 1rcur 18501 Ringcrg 18547 SubRingcsubrg 18776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-subrg 18778 |
This theorem is referenced by: subrgsubg 18786 subrg1 18790 subrgsubm 18793 subrginv 18796 subrgunit 18798 subrgugrp 18799 opprsubrg 18801 subrgint 18802 subsubrg 18806 sralmod 19187 subrgpsr 19419 subrgmpl 19460 subrgmvr 19461 subrgmvrf 19462 subrgascl 19498 subrgasclcl 19499 |
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