Theorem subrgdv 14251

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 2231	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
3		subrgdv.3	. . . . . 6 |- U = (Unit ` S )
4		eqid 2231	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
5	1, 2, 3, 4	subrginv 14250	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ Y e. U ) -> ( ( invr ` R ) ` Y ) = ( ( invr ` S ) ` Y ) )
6	5	3adant2 1042	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( ( invr ` R ) ` Y ) = ( ( invr ` S ) ` Y ) )
7	6	oveq2d 6033	. . 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 14239	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
9		eqid 2231	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
10	1, 9	ressmulrg 13227	. . . . . 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 1044	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( .r ` R ) = ( .r ` S ) )
13	12	oveqd 6034	. . 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 2264	. 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 2232	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( Base ` R ) = ( Base ` R ) )
16		eqidd 2232	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( .r ` R ) = ( .r ` R ) )
17		eqidd 2232	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> (Unit ` R ) = (Unit ` R ) )
18		eqidd 2232	. . 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 1044	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> R e. Ring )
22		eqid 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
23	22	subrgss 14235	. . . . 5 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
24	23	3ad2ant1 1044	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> A C_ ( Base ` R ) )
25		simp2 1024	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> X e. A )
26	24, 25	sseldd 3228	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> X e. ( Base ` R ) )
27		eqid 2231	. . . . . 6 |- (Unit ` R ) = (Unit ` R )
28	1, 27, 3	subrguss 14249	. . . . 5 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
29	28	3ad2ant1 1044	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> U C_ (Unit ` R ) )
30		simp3 1025	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> Y e. U )
31	29, 30	sseldd 3228	. . 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 14147	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ./ Y ) = ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) )
33		eqidd 2232	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( Base ` S ) = ( Base ` S ) )
34		eqidd 2232	. . 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 2232	. . 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 14237	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
40	39	3ad2ant1 1044	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> S e. Ring )
41	1	subrgbas 14243	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
42	41	3ad2ant1 1044	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> A = ( Base ` S ) )
43	25, 42	eleqtrd 2310	. . 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 14147	. 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 2274	1 |- ( ( A e. (SubRing ` R ) /\ X e. A /\ Y e. U ) -> ( X ./ Y ) = ( X E Y ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   .rcmulr 13160   Ringcrg 14008  Unitcui 14099   invrcinvr 14133  /_rcdvr 14144  SubRingcsubrg 14230

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102  df-invr 14134  df-dvr 14145  df-subrg 14232

This theorem is referenced by: (None)