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Theorem subrginv 13296
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 13285 . . . . 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 13289 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
5 eqid 2177 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
65subrgss 13281 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
74, 6eqsstrrd 3192 . . . . . 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 13283 . . . . . 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 13225 . . . . . 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 3156 . . . 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 13155 . . . . . . 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 13208 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ (Baseβ€˜π‘†))
238, 22sseldd 3156 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ (Baseβ€˜π‘…))
24 eqid 2177 . . . . . . 7 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
253, 24, 10subrguss 13295 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ π‘ˆ βŠ† (Unitβ€˜π‘…))
2625sselda 3155 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ (Unitβ€˜π‘…))
27 subrginv.2 . . . . . 6 𝐼 = (invrβ€˜π‘…)
2824, 27, 5ringinvcl 13225 . . . . 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 13130 . . . 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 13226 . . . . . 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 12595 . . . . . . . 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 5889 . . . . 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 13290 . . . . . 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 5887 . . 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 13227 . . . . 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 5888 . . 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 13137 . . 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 13138 . . 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 3129  β€˜cfv 5215  (class class class)co 5872  Basecbs 12454   β†Ύs cress 12455  .rcmulr 12529  1rcur 13073  SRingcsrg 13077  Ringcrg 13110  Unitcui 13187  invrcinvr 13220  SubRingcsubrg 13276
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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-oppr 13171  df-dvdsr 13189  df-unit 13190  df-invr 13221  df-subrg 13278
This theorem is referenced by:  subrgdv  13297  subrgunit  13298  subrgugrp  13299
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