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Theorem subrguss 13317
Description: A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrguss.1 𝑆 = (𝑅 β†Ύs 𝐴)
subrguss.2 π‘ˆ = (Unitβ€˜π‘…)
subrguss.3 𝑉 = (Unitβ€˜π‘†)
Assertion
Ref Expression
subrguss (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑉 βŠ† π‘ˆ)
Proof of Theorem subrguss
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 subrguss.3 . . . . . . . . 9 𝑉 = (Unitβ€˜π‘†)
21a1i 9 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑉 = (Unitβ€˜π‘†))
3 eqidd 2178 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘†) = (1rβ€˜π‘†))
4 eqidd 2178 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜π‘†) = (βˆ₯rβ€˜π‘†))
5 eqidd 2178 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (opprβ€˜π‘†) = (opprβ€˜π‘†))
6 eqidd 2178 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜(opprβ€˜π‘†)) = (βˆ₯rβ€˜(opprβ€˜π‘†)))
7 subrguss.1 . . . . . . . . . 10 𝑆 = (𝑅 β†Ύs 𝐴)
87subrgring 13305 . . . . . . . . 9 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
9 ringsrg 13177 . . . . . . . . 9 (𝑆 ∈ Ring β†’ 𝑆 ∈ SRing)
108, 9syl 14 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ SRing)
112, 3, 4, 5, 6, 10isunitd 13228 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (π‘₯ ∈ 𝑉 ↔ (π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘†) ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘†))))
1211simprbda 383 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘†))
13 eqid 2177 . . . . . . . 8 (1rβ€˜π‘…) = (1rβ€˜π‘…)
147, 13subrg1 13312 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
1514adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
1612, 15breqtrrd 4031 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…))
17 eqid 2177 . . . . . . . 8 (βˆ₯rβ€˜π‘…) = (βˆ₯rβ€˜π‘…)
18 eqid 2177 . . . . . . . 8 (βˆ₯rβ€˜π‘†) = (βˆ₯rβ€˜π‘†)
197, 17, 18subrgdvds 13316 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜π‘†) βŠ† (βˆ₯rβ€˜π‘…))
2019adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (βˆ₯rβ€˜π‘†) βŠ† (βˆ₯rβ€˜π‘…))
2120ssbrd 4046 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…) β†’ π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…)))
2216, 21mpd 13 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…))
23 subrgrcl 13307 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
2423adantr 276 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ 𝑅 ∈ Ring)
25 eqid 2177 . . . . . . . 8 (opprβ€˜π‘…) = (opprβ€˜π‘…)
26 eqid 2177 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2725, 26opprbasg 13200 . . . . . . 7 (𝑅 ∈ Ring β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
2824, 27syl 14 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
29 eqidd 2178 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (βˆ₯rβ€˜(opprβ€˜π‘…)) = (βˆ₯rβ€˜(opprβ€˜π‘…)))
3025opprring 13202 . . . . . . 7 (𝑅 ∈ Ring β†’ (opprβ€˜π‘…) ∈ Ring)
31 ringsrg 13177 . . . . . . 7 ((opprβ€˜π‘…) ∈ Ring β†’ (opprβ€˜π‘…) ∈ SRing)
3224, 30, 313syl 17 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (opprβ€˜π‘…) ∈ SRing)
33 eqidd 2178 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…)))
347subrgbas 13311 . . . . . . . . 9 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
3534adantr 276 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ 𝐴 = (Baseβ€˜π‘†))
3626subrgss 13303 . . . . . . . . 9 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
3736adantr 276 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
3835, 37eqsstrrd 3192 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (Baseβ€˜π‘†) βŠ† (Baseβ€˜π‘…))
39 eqidd 2178 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
401a1i 9 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ 𝑉 = (Unitβ€˜π‘†))
4110adantr 276 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ 𝑆 ∈ SRing)
42 simpr 110 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ 𝑉)
4339, 40, 41, 42unitcld 13230 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
4438, 43sseldd 3156 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
45 eqid 2177 . . . . . . . . 9 (invrβ€˜π‘†) = (invrβ€˜π‘†)
46 eqid 2177 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
471, 45, 46ringinvcl 13247 . . . . . . . 8 ((𝑆 ∈ Ring ∧ π‘₯ ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
488, 47sylan 283 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
4938, 48sseldd 3156 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘…))
5028, 29, 32, 33, 44, 49dvdsrmuld 13218 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…))(((invrβ€˜π‘†)β€˜π‘₯)(.rβ€˜(opprβ€˜π‘…))π‘₯))
511, 45unitinvcl 13245 . . . . . . . 8 ((𝑆 ∈ Ring ∧ π‘₯ ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ 𝑉)
528, 51sylan 283 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ 𝑉)
53 eqid 2177 . . . . . . . 8 (.rβ€˜π‘…) = (.rβ€˜π‘…)
54 eqid 2177 . . . . . . . 8 (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…))
5526, 53, 25, 54opprmulg 13196 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((invrβ€˜π‘†)β€˜π‘₯) ∈ 𝑉 ∧ π‘₯ ∈ 𝑉) β†’ (((invrβ€˜π‘†)β€˜π‘₯)(.rβ€˜(opprβ€˜π‘…))π‘₯) = (π‘₯(.rβ€˜π‘…)((invrβ€˜π‘†)β€˜π‘₯)))
5624, 52, 42, 55syl3anc 1238 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (((invrβ€˜π‘†)β€˜π‘₯)(.rβ€˜(opprβ€˜π‘…))π‘₯) = (π‘₯(.rβ€˜π‘…)((invrβ€˜π‘†)β€˜π‘₯)))
57 eqid 2177 . . . . . . . . 9 (.rβ€˜π‘†) = (.rβ€˜π‘†)
58 eqid 2177 . . . . . . . . 9 (1rβ€˜π‘†) = (1rβ€˜π‘†)
591, 45, 57, 58unitrinv 13249 . . . . . . . 8 ((𝑆 ∈ Ring ∧ π‘₯ ∈ 𝑉) β†’ (π‘₯(.rβ€˜π‘†)((invrβ€˜π‘†)β€˜π‘₯)) = (1rβ€˜π‘†))
608, 59sylan 283 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (π‘₯(.rβ€˜π‘†)((invrβ€˜π‘†)β€˜π‘₯)) = (1rβ€˜π‘†))
617, 53ressmulrg 12597 . . . . . . . . . 10 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑅 ∈ Ring) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
6223, 61mpdan 421 . . . . . . . . 9 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
6362adantr 276 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
6463oveqd 5891 . . . . . . 7 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (π‘₯(.rβ€˜π‘…)((invrβ€˜π‘†)β€˜π‘₯)) = (π‘₯(.rβ€˜π‘†)((invrβ€˜π‘†)β€˜π‘₯)))
6560, 64, 153eqtr4d 2220 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (π‘₯(.rβ€˜π‘…)((invrβ€˜π‘†)β€˜π‘₯)) = (1rβ€˜π‘…))
6656, 65eqtrd 2210 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (((invrβ€˜π‘†)β€˜π‘₯)(.rβ€˜(opprβ€˜π‘…))π‘₯) = (1rβ€˜π‘…))
6750, 66breqtrd 4029 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…))(1rβ€˜π‘…))
68 subrguss.2 . . . . . . 7 π‘ˆ = (Unitβ€˜π‘…)
6968a1i 9 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ π‘ˆ = (Unitβ€˜π‘…))
70 eqidd 2178 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘…))
71 eqidd 2178 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜π‘…) = (βˆ₯rβ€˜π‘…))
72 eqidd 2178 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (opprβ€˜π‘…) = (opprβ€˜π‘…))
73 eqidd 2178 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜(opprβ€˜π‘…)) = (βˆ₯rβ€˜(opprβ€˜π‘…)))
74 ringsrg 13177 . . . . . . 7 (𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
7523, 74syl 14 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ SRing)
7669, 70, 71, 72, 73, 75isunitd 13228 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…))(1rβ€˜π‘…))))
7776adantr 276 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…))(1rβ€˜π‘…))))
7822, 67, 77mpbir2and 944 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ π‘ˆ)
7978ex 115 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ π‘ˆ))
8079ssrdv 3161 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑉 βŠ† π‘ˆ)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   βŠ† wss 3129   class class class wbr 4003  β€˜cfv 5216  (class class class)co 5874  Basecbs 12456   β†Ύs cress 12457  .rcmulr 12531  1rcur 13095  SRingcsrg 13099  Ringcrg 13132  opprcoppr 13192  βˆ₯rcdsr 13208  Unitcui 13209  invrcinvr 13242  SubRingcsubrg 13298
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926
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 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-subg 12983  df-cmn 13043  df-abl 13044  df-mgp 13084  df-ur 13096  df-srg 13100  df-ring 13134  df-oppr 13193  df-dvdsr 13211  df-unit 13212  df-invr 13243  df-subrg 13300
This theorem is referenced by:  subrginv  13318  subrgdv  13319  subrgunit  13320  subrgugrp  13321
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