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Mirrors > Home > MPE Home > Th. List > subrgid | Structured version Visualization version GIF version |
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 16935 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s 𝐵) = 𝑅) |
4 | 3, 1 | eqeltrd 2840 | . 2 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s 𝐵) ∈ Ring) |
5 | eqid 2739 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | 2, 5 | ringidcl 19788 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
7 | ssid 3947 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
8 | 6, 7 | jctil 519 | . 2 ⊢ (𝑅 ∈ Ring → (𝐵 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐵)) |
9 | 2, 5 | issubrg 20005 | . 2 ⊢ (𝐵 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐵))) |
10 | 1, 4, 8, 9 | syl21anbrc 1342 | 1 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ⊆ wss 3891 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 ↾s cress 16922 1rcur 19718 Ringcrg 19764 SubRingcsubrg 20001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mgp 19702 df-ur 19719 df-ring 19766 df-subrg 20003 |
This theorem is referenced by: subrgmre 20029 rnrhmsubrg 20037 sdrgid 20045 rlmlmod 20456 rlmassa 21056 aspval 21058 evlrhm 21287 evlsscasrng 21288 evlsca 21289 evlsvarsrng 21290 evlvar 21291 mpfsubrg 21294 evl1sca 21481 evl1var 21483 evls1scasrng 21486 evls1varsrng 21487 pf1subrg 21495 pf1ind 21502 evl1gsumadd 21505 evl1varpw 21508 rlmnlm 23833 rlmbn 24506 dvply2 25427 dvnply 25429 taylply 25509 rgmoddim 31672 fldextid 31713 mhphf4 40268 mzpmfp 40549 rgspnval 40973 rgspncl 40974 |
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