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| Description: Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| subrgss.1 | ⊢ 𝐵 = (Base‘𝑅) | 
| Ref | Expression | 
|---|---|
| subrgid | ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
| 2 | subrgss.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | 2 | ressid 17291 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s 𝐵) = 𝑅) | 
| 4 | 3, 1 | eqeltrd 2840 | . 2 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s 𝐵) ∈ Ring) | 
| 5 | eqid 2736 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 6 | 2, 5 | ringidcl 20263 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) | 
| 7 | ssid 4005 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
| 8 | 6, 7 | jctil 519 | . 2 ⊢ (𝑅 ∈ Ring → (𝐵 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐵)) | 
| 9 | 2, 5 | issubrg 20572 | . 2 ⊢ (𝐵 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐵))) | 
| 10 | 1, 4, 8, 9 | syl21anbrc 1344 | 1 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 1rcur 20179 Ringcrg 20231 SubRingcsubrg 20570 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mgp 20139 df-ur 20180 df-ring 20233 df-subrg 20571 | 
| This theorem is referenced by: subrgmre 20598 rnrhmsubrg 20606 rgspnval 20613 rgspncl 20614 sdrgid 20794 rlmlmod 21211 ring2idlqus 21320 rlmassa 21892 aspval 21894 evlrhm 22121 evlsscasrng 22122 evlsca 22123 evlsvarsrng 22124 evlvar 22125 mpfsubrg 22128 evl1sca 22339 evl1var 22341 evls1scasrng 22344 evls1varsrng 22345 pf1subrg 22353 pf1ind 22360 evl1gsumadd 22363 evl1varpw 22366 ressply1evl 22375 evl1maprhm 22384 rlmnlm 24710 rlmbn 25396 dvply2 26329 dvnply 26331 taylply 26412 evl1fpws 33591 rgmoddimOLD 33662 fldextid 33711 riccrng1 42536 evlsevl 42586 evlvvval 42588 evlvvvallem 42589 mhphf4 42615 mzpmfp 42763 | 
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