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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 2823 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2823 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 1, 2 | issubrg 19537 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴))) |
4 | 3 | simplbi 500 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)) |
5 | 4 | simpld 497 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 1rcur 19253 Ringcrg 19299 SubRingcsubrg 19533 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-subrg 19535 |
This theorem is referenced by: subrgsubg 19543 subrg1 19547 subrgsubm 19550 subrginv 19553 subrgunit 19555 subrgugrp 19556 opprsubrg 19558 subrgint 19559 subsubrg 19563 sralmod 19961 subrgpsr 20201 subrgmpl 20243 subrgmvr 20244 subrgmvrf 20245 subrgascl 20280 subrgasclcl 20281 |
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