Theorem subrgdv 14217

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 = ( R ↾s 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

Step	Hyp	Ref	Expression
1		subrgdv.1	. . . . . 6 |- S = ( R ↾s A )
2		eqid 2229	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
3		subrgdv.3	. . . . . 6 |- U = (Unit ` S )
4		eqid 2229	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
5	1, 2, 3, 4	subrginv 14216	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ Y e. U ) -> ( ( invr ` R ) ` Y ) = ( ( invr ` S ) ` Y ) )
6	5	3adant2 1040	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( ( invr ` R ) ` Y ) = ( ( invr ` S ) ` Y ) )
7	6	oveq2d 6023	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) = ( X ( .r ` R ) ( ( invr ` S ) ` Y ) ) )
8		subrgrcl 14205	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
9		eqid 2229	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
10	1, 9	ressmulrg 13193	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
11	8, 10	mpdan 421	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
12	11	3ad2ant1 1042	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( .r ` R ) = ( .r ` S ) )
13	12	oveqd 6024	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ( .r ` R ) ( ( invr ` S ) ` Y ) ) = ( X ( .r ` S ) ( ( invr ` S ) ` Y ) ) )
14	7, 13	eqtrd 2262	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) = ( X ( .r ` S ) ( ( invr ` S ) ` Y ) ) )
15		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( Base ` R ) = ( Base ` R ) )
16		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( .r ` R ) = ( .r ` R ) )
17		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> (Unit ` R ) = (Unit ` R ) )
18		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( invr ` R ) = ( invr ` R ) )
19		subrgdv.2	. . . 4 |- ./ = (/r ` R )
20	19	a1i 9	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ./ = (/r ` R ) )
21	8	3ad2ant1 1042	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> R e. Ring )
22		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
23	22	subrgss 14201	. . . . 5 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
24	23	3ad2ant1 1042	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> A C_ ( Base ` R ) )
25		simp2 1022	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> X e. A )
26	24, 25	sseldd 3225	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> X e. ( Base ` R ) )
27		eqid 2229	. . . . . 6 |- (Unit ` R ) = (Unit ` R )
28	1, 27, 3	subrguss 14215	. . . . 5 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
29	28	3ad2ant1 1042	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> U C_ (Unit ` R ) )
30		simp3 1023	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> Y e. U )
31	29, 30	sseldd 3225	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> Y e. (Unit ` R ) )
32	15, 16, 17, 18, 20, 21, 26, 31	dvrvald 14113	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ./ Y ) = ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) )
33		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( Base ` S ) = ( Base ` S ) )
34		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( .r ` S ) = ( .r ` S ) )
35	3	a1i 9	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> U = (Unit ` S ) )
36		eqidd 2230	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( invr ` S ) = ( invr ` S ) )
37		subrgdv.4	. . . 4 |- E = (/r ` S )
38	37	a1i 9	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> E = (/r ` S ) )
39	1	subrgring 14203	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
40	39	3ad2ant1 1042	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> S e. Ring )
41	1	subrgbas 14209	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
42	41	3ad2ant1 1042	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> A = ( Base ` S ) )
43	25, 42	eleqtrd 2308	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> X e. ( Base ` S ) )
44	33, 34, 35, 36, 38, 40, 43, 30	dvrvald 14113	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X E Y ) = ( X ( .r ` S ) ( ( invr ` S ) ` Y ) ) )
45	14, 32, 44	3eqtr4d 2272	1 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ./ Y ) = ( X E Y ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   ` cfv 5318  (class class class)co 6007   Basecbs 13047   ↾_s cress 13048   .rcmulr 13126   Ringcrg 13974  Unitcui 14065   invrcinvr 14099  /_rcdvr 14110  SubRingcsubrg 14196

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-tpos 6397  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-mulr 13139  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-subg 13722  df-cmn 13838  df-abl 13839  df-mgp 13899  df-ur 13938  df-srg 13942  df-ring 13976  df-oppr 14046  df-dvdsr 14067  df-unit 14068  df-invr 14100  df-dvr 14111  df-subrg 14198

This theorem is referenced by: (None)