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Mirrors > Home > MPE Home > Th. List > subrgbas | Structured version Visualization version GIF version |
Description: Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrgbas.b | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
Ref | Expression |
---|---|
subrgbas | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgsubg 19543 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
2 | subrgbas.b | . . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
3 | 2 | subgbas 18285 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘𝑆)) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 SubGrpcsubg 18275 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 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-subg 18278 df-ring 19301 df-subrg 19535 |
This theorem is referenced by: subrg1 19547 subrgmcl 19549 subrgdvds 19551 subrguss 19552 subrginv 19553 subrgdv 19554 subrgunit 19555 issubdrg 19562 subsubrg 19563 abvres 19612 issubassa3 20099 sraassa 20101 resspsrbas 20197 resspsradd 20198 resspsrmul 20199 resspsrvsca 20200 subrgpsr 20201 subrgascl 20280 subrgasclcl 20281 qsssubdrg 20606 gzrngunitlem 20612 gzrngunit 20613 dmatcrng 21113 scmatcrng 21132 scmatstrbas 21137 sranlm 23295 isclmi 23683 plypf1 24804 |
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