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Theorem subrgdv 15886
Description: A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgdv.1  |-  S  =  ( Rs  A )
subrgdv.2  |-  ./  =  (/r
`  R )
subrgdv.3  |-  U  =  (Unit `  S )
subrgdv.4  |-  E  =  (/r `  S )
Assertion
Ref Expression
subrgdv  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X E Y ) )

Proof of Theorem subrgdv
StepHypRef Expression
1 subrgdv.1 . . . . . 6  |-  S  =  ( Rs  A )
2 eqid 2437 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
3 subrgdv.3 . . . . . 6  |-  U  =  (Unit `  S )
4 eqid 2437 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
51, 2, 3, 4subrginv 15885 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  =  ( ( invr `  S
) `  Y )
)
653adant2 977 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( ( invr `  R ) `  Y )  =  ( ( invr `  S
) `  Y )
)
76oveq2d 6098 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) )  =  ( X ( .r `  R ) ( (
invr `  S ) `  Y ) ) )
8 eqid 2437 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
91, 8ressmulr 13583 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1093ad2ant1 979 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( .r `  R )  =  ( .r `  S ) )
1110oveqd 6099 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  S ) `  Y
) )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
127, 11eqtrd 2469 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
13 eqid 2437 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
1413subrgss 15870 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
15143ad2ant1 979 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  A  C_  ( Base `  R ) )
16 simp2 959 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  A )
1715, 16sseldd 3350 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  ( Base `  R )
)
18 eqid 2437 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
191, 18, 3subrguss 15884 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  U  C_  (Unit `  R ) )
20193ad2ant1 979 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  U  C_  (Unit `  R ) )
21 simp3 960 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  Y  e.  U )
2220, 21sseldd 3350 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  Y  e.  (Unit `  R ) )
23 subrgdv.2 . . . 4  |-  ./  =  (/r
`  R )
2413, 8, 18, 2, 23dvrval 15791 . . 3  |-  ( ( X  e.  ( Base `  R )  /\  Y  e.  (Unit `  R )
)  ->  ( X  ./  Y )  =  ( X ( .r `  R ) ( (
invr `  R ) `  Y ) ) )
2517, 22, 24syl2anc 644 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X ( .r `  R ) ( (
invr `  R ) `  Y ) ) )
261subrgbas 15878 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
27263ad2ant1 979 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  A  =  ( Base `  S )
)
2816, 27eleqtrd 2513 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  X  e.  ( Base `  S )
)
29 eqid 2437 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
30 eqid 2437 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
31 subrgdv.4 . . . 4  |-  E  =  (/r `  S )
3229, 30, 3, 4, 31dvrval 15791 . . 3  |-  ( ( X  e.  ( Base `  S )  /\  Y  e.  U )  ->  ( X E Y )  =  ( X ( .r
`  S ) ( ( invr `  S
) `  Y )
) )
3328, 21, 32syl2anc 644 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X E Y )  =  ( X ( .r `  S ) ( (
invr `  S ) `  Y ) ) )
3412, 25, 333eqtr4d 2479 1  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  A  /\  Y  e.  U
)  ->  ( X  ./  Y )  =  ( X E Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3321   ` cfv 5455  (class class class)co 6082   Basecbs 13470   ↾s cress 13471   .rcmulr 13531  Unitcui 15745   invrcinvr 15777  /rcdvr 15788  SubRingcsubrg 15865
This theorem is referenced by:  qsssubdrg  16759  qrngdiv  21319  redvr  24278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-0g 13728  df-mnd 14691  df-grp 14813  df-minusg 14814  df-subg 14942  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-dvr 15789  df-subrg 15867
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