ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subrginv GIF version
Theorem subrginv 13363
Description: A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrginv.1 𝑆 = (𝑅 β†Ύs 𝐴)
subrginv.2 𝐼 = (invrβ€˜π‘…)
subrginv.3 π‘ˆ = (Unitβ€˜π‘†)
subrginv.4 𝐽 = (invrβ€˜π‘†)
Assertion
Ref Expression
subrginv ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜π‘‹) = (π½β€˜π‘‹))
Proof of Theorem subrginv
StepHypRef Expression
1 subrgrcl 13352 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
21adantr 276 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑅 ∈ Ring)
3 subrginv.1 . . . . . . . 8 𝑆 = (𝑅 β†Ύs 𝐴)
43subrgbas 13356 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
5 eqid 2177 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
65subrgss 13348 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
74, 6eqsstrrd 3194 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (Baseβ€˜π‘†) βŠ† (Baseβ€˜π‘…))
87adantr 276 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (Baseβ€˜π‘†) βŠ† (Baseβ€˜π‘…))
93subrgring 13350 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
10 subrginv.3 . . . . . . 7 π‘ˆ = (Unitβ€˜π‘†)
11 subrginv.4 . . . . . . 7 𝐽 = (invrβ€˜π‘†)
12 eqid 2177 . . . . . . 7 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
1310, 11, 12ringinvcl 13299 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (π½β€˜π‘‹) ∈ (Baseβ€˜π‘†))
149, 13sylan 283 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (π½β€˜π‘‹) ∈ (Baseβ€˜π‘†))
158, 14sseldd 3158 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (π½β€˜π‘‹) ∈ (Baseβ€˜π‘…))
16 eqidd 2178 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
1710a1i 9 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ π‘ˆ = (Unitβ€˜π‘†))
189adantr 276 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑆 ∈ Ring)
19 ringsrg 13229 . . . . . . 7 (𝑆 ∈ Ring β†’ 𝑆 ∈ SRing)
2018, 19syl 14 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑆 ∈ SRing)
21 simpr 110 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ π‘ˆ)
2216, 17, 20, 21unitcld 13282 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ (Baseβ€˜π‘†))
238, 22sseldd 3158 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ (Baseβ€˜π‘…))
24 eqid 2177 . . . . . . 7 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
253, 24, 10subrguss 13362 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ π‘ˆ βŠ† (Unitβ€˜π‘…))
2625sselda 3157 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ (Unitβ€˜π‘…))
27 subrginv.2 . . . . . 6 𝐼 = (invrβ€˜π‘…)
2824, 27, 5ringinvcl 13299 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unitβ€˜π‘…)) β†’ (πΌβ€˜π‘‹) ∈ (Baseβ€˜π‘…))
291, 26, 28syl2an2r 595 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜π‘‹) ∈ (Baseβ€˜π‘…))
30 eqid 2177 . . . . 5 (.rβ€˜π‘…) = (.rβ€˜π‘…)
315, 30ringass 13204 . . . 4 ((𝑅 ∈ Ring ∧ ((π½β€˜π‘‹) ∈ (Baseβ€˜π‘…) ∧ 𝑋 ∈ (Baseβ€˜π‘…) ∧ (πΌβ€˜π‘‹) ∈ (Baseβ€˜π‘…))) β†’ (((π½β€˜π‘‹)(.rβ€˜π‘…)𝑋)(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = ((π½β€˜π‘‹)(.rβ€˜π‘…)(𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹))))
322, 15, 23, 29, 31syl13anc 1240 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (((π½β€˜π‘‹)(.rβ€˜π‘…)𝑋)(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = ((π½β€˜π‘‹)(.rβ€˜π‘…)(𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹))))
33 eqid 2177 . . . . . . 7 (.rβ€˜π‘†) = (.rβ€˜π‘†)
34 eqid 2177 . . . . . . 7 (1rβ€˜π‘†) = (1rβ€˜π‘†)
3510, 11, 33, 34unitlinv 13300 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘†)𝑋) = (1rβ€˜π‘†))
369, 35sylan 283 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘†)𝑋) = (1rβ€˜π‘†))
373, 30ressmulrg 12605 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑅 ∈ Ring) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
381, 37mpdan 421 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
3938adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
4039oveqd 5894 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘…)𝑋) = ((π½β€˜π‘‹)(.rβ€˜π‘†)𝑋))
41 eqid 2177 . . . . . . 7 (1rβ€˜π‘…) = (1rβ€˜π‘…)
423, 41subrg1 13357 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4342adantr 276 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4436, 40, 433eqtr4d 2220 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘…)𝑋) = (1rβ€˜π‘…))
4544oveq1d 5892 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (((π½β€˜π‘‹)(.rβ€˜π‘…)𝑋)(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = ((1rβ€˜π‘…)(.rβ€˜π‘…)(πΌβ€˜π‘‹)))
4624, 27, 30, 41unitrinv 13301 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unitβ€˜π‘…)) β†’ (𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (1rβ€˜π‘…))
471, 26, 46syl2an2r 595 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (1rβ€˜π‘…))
4847oveq2d 5893 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘…)(𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹))) = ((π½β€˜π‘‹)(.rβ€˜π‘…)(1rβ€˜π‘…)))
4932, 45, 483eqtr3d 2218 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((1rβ€˜π‘…)(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = ((π½β€˜π‘‹)(.rβ€˜π‘…)(1rβ€˜π‘…)))
505, 30, 41ringlidm 13211 . . 3 ((𝑅 ∈ Ring ∧ (πΌβ€˜π‘‹) ∈ (Baseβ€˜π‘…)) β†’ ((1rβ€˜π‘…)(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (πΌβ€˜π‘‹))
511, 29, 50syl2an2r 595 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((1rβ€˜π‘…)(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (πΌβ€˜π‘‹))
525, 30, 41ringridm 13212 . . 3 ((𝑅 ∈ Ring ∧ (π½β€˜π‘‹) ∈ (Baseβ€˜π‘…)) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘…)(1rβ€˜π‘…)) = (π½β€˜π‘‹))
531, 15, 52syl2an2r 595 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ ((π½β€˜π‘‹)(.rβ€˜π‘…)(1rβ€˜π‘…)) = (π½β€˜π‘‹))
5449, 51, 533eqtr3d 2218 1 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜π‘‹) = (π½β€˜π‘‹))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   βŠ† wss 3131  β€˜cfv 5218  (class class class)co 5877  Basecbs 12464   β†Ύs cress 12465  .rcmulr 12539  1rcur 13147  SRingcsrg 13151  Ringcrg 13184  Unitcui 13261  invrcinvr 13294  SubRingcsubrg 13343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-subg 13035  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-oppr 13245  df-dvdsr 13263  df-unit 13264  df-invr 13295  df-subrg 13345
This theorem is referenced by:  subrgdv  13364  subrgunit  13365  subrgugrp  13366
  Copyright terms: Public domain W3C validator