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Theorem subrguss 17571
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  =  ( Rs  A )
subrguss.2  |-  U  =  (Unit `  R )
subrguss.3  |-  V  =  (Unit `  S )
Assertion
Ref Expression
subrguss  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)

Proof of Theorem subrguss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  V )
2 subrguss.3 . . . . . . . . 9  |-  V  =  (Unit `  S )
3 eqid 2457 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
4 eqid 2457 . . . . . . . . 9  |-  ( ||r `  S
)  =  ( ||r `  S
)
5 eqid 2457 . . . . . . . . 9  |-  (oppr `  S
)  =  (oppr `  S
)
6 eqid 2457 . . . . . . . . 9  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
72, 3, 4, 5, 6isunit 17433 . . . . . . . 8  |-  ( x  e.  V  <->  ( x
( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
81, 7sylib 196 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  S )  /\  x ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
98simpld 459 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  S ) )
10 subrguss.1 . . . . . . . 8  |-  S  =  ( Rs  A )
11 eqid 2457 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
1210, 11subrg1 17566 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
1312adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( 1r `  R )  =  ( 1r `  S
) )
149, 13breqtrrd 4482 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 S ) ( 1r `  R ) )
15 eqid 2457 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
1610, 15, 4subrgdvds 17570 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( ||r `  S
)  C_  ( ||r `  R
) )
1716adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( ||r `  S )  C_  ( ||r `  R ) )
1817ssbrd 4497 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( ||r `
 S ) ( 1r `  R )  ->  x ( ||r `  R
) ( 1r `  R ) ) )
1914, 18mpd 15 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 R ) ( 1r `  R ) )
2010subrgbas 17565 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
2120adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  =  ( Base `  S
) )
22 eqid 2457 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2322subrgss 17557 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
2423adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A  C_  ( Base `  R
) )
2521, 24eqsstr3d 3534 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( Base `  S )  C_  ( Base `  R )
)
26 eqid 2457 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
2726, 2unitcl 17435 . . . . . . . 8  |-  ( x  e.  V  ->  x  e.  ( Base `  S
) )
2827adantl 466 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  S
) )
2925, 28sseldd 3500 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
3010subrgring 17559 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
31 eqid 2457 . . . . . . . . 9  |-  ( invr `  S )  =  (
invr `  S )
322, 31, 26ringinvcl 17452 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3330, 32sylan 471 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  S )
)
3425, 33sseldd 3500 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  (
Base `  R )
)
35 eqid 2457 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
3635, 22opprbas 17405 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
37 eqid 2457 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
38 eqid 2457 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3936, 37, 38dvdsrmul 17424 . . . . . 6  |-  ( ( x  e.  ( Base `  R )  /\  (
( invr `  S ) `  x )  e.  (
Base `  R )
)  ->  x ( ||r `  (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
4029, 34, 39syl2anc 661 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( invr `  S ) `  x
) ( .r `  (oppr `  R ) ) x ) )
41 eqid 2457 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4222, 41, 35, 38opprmul 17402 . . . . . 6  |-  ( ( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) ( ( invr `  S
) `  x )
)
43 eqid 2457 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
442, 31, 43, 3unitrinv 17454 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4530, 44sylan 471 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  S ) ( (
invr `  S ) `  x ) )  =  ( 1r `  S
) )
4610, 41ressmulr 14769 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
4746adantr 465 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( .r `  R )  =  ( .r `  S
) )
4847oveqd 6313 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( x ( .r
`  S ) ( ( invr `  S
) `  x )
) )
4945, 48, 133eqtr4d 2508 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
x ( .r `  R ) ( (
invr `  S ) `  x ) )  =  ( 1r `  R
) )
5042, 49syl5eq 2510 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( ( invr `  S
) `  x )
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )
5140, 50breqtrd 4480 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
52 subrguss.2 . . . . 5  |-  U  =  (Unit `  R )
5352, 11, 15, 35, 37isunit 17433 . . . 4  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
5419, 51, 53sylanbrc 664 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  x  e.  U )
5554ex 434 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  V  ->  x  e.  U ) )
5655ssrdv 3505 1  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   ↾s cress 14645   .rcmulr 14713   1rcur 17280   Ringcrg 17325  opprcoppr 17398   ||rcdsr 17414  Unitcui 17415   invrcinvr 17447  SubRingcsubrg 17552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-subg 16325  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-subrg 17554
This theorem is referenced by:  subrginv  17572  subrgdv  17573  subrgunit  17574  subrgugrp  17575  issubdrg  17581  zringunit  18647  zrngunit  18648
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