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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 2799 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2799 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 1, 2 | issubrg 19098 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴))) |
4 | 3 | simplbi 492 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)) |
5 | 4 | simpld 489 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ⊆ wss 3769 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 ↾s cress 16185 1rcur 18817 Ringcrg 18863 SubRingcsubrg 19094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fv 6109 df-ov 6881 df-subrg 19096 |
This theorem is referenced by: subrgsubg 19104 subrg1 19108 subrgsubm 19111 subrginv 19114 subrgunit 19116 subrgugrp 19117 opprsubrg 19119 subrgint 19120 subsubrg 19124 sralmod 19510 subrgpsr 19742 subrgmpl 19783 subrgmvr 19784 subrgmvrf 19785 subrgascl 19820 subrgasclcl 19821 |
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