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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 17290 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s 𝐵) = 𝑅) |
4 | 3, 1 | eqeltrd 2839 | . 2 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s 𝐵) ∈ Ring) |
5 | eqid 2735 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | 2, 5 | ringidcl 20280 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
7 | ssid 4018 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
8 | 6, 7 | jctil 519 | . 2 ⊢ (𝑅 ∈ Ring → (𝐵 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐵)) |
9 | 2, 5 | issubrg 20588 | . 2 ⊢ (𝐵 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐵))) |
10 | 1, 4, 8, 9 | syl21anbrc 1343 | 1 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 1rcur 20199 Ringcrg 20251 SubRingcsubrg 20586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mgp 20153 df-ur 20200 df-ring 20253 df-subrg 20587 |
This theorem is referenced by: subrgmre 20614 rnrhmsubrg 20622 rgspnval 20629 rgspncl 20630 sdrgid 20810 rlmlmod 21228 ring2idlqus 21337 rlmassa 21909 aspval 21911 evlrhm 22138 evlsscasrng 22139 evlsca 22140 evlsvarsrng 22141 evlvar 22142 mpfsubrg 22145 evl1sca 22354 evl1var 22356 evls1scasrng 22359 evls1varsrng 22360 pf1subrg 22368 pf1ind 22375 evl1gsumadd 22378 evl1varpw 22381 ressply1evl 22390 evl1maprhm 22399 rlmnlm 24725 rlmbn 25409 dvply2 26343 dvnply 26345 taylply 26426 evl1fpws 33570 rgmoddimOLD 33638 fldextid 33687 riccrng1 42508 evlsevl 42558 evlvvval 42560 evlvvvallem 42561 mhphf4 42587 mzpmfp 42735 |
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