Theorem unitpropdg 13644

Description: The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

Hypotheses

Ref	Expression
unitpropdg.1	|- ( ph -> B = ( Base ` K ) )
unitpropdg.2	|- ( ph -> B = ( Base ` L ) )
unitpropdg.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
unitpropdg.k	|- ( ph -> K e. Ring )
unitpropdg.l	|- ( ph -> L e. Ring )

Assertion

Ref	Expression
unitpropdg	|- ( ph -> (Unit ` K ) = (Unit ` L ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   ph, x, y

Proof of Theorem unitpropdg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unitpropdg.1	. . . . . . 7 |- ( ph -> B = ( Base ` K ) )
2		unitpropdg.2	. . . . . . 7 |- ( ph -> B = ( Base ` L ) )
3		unitpropdg.3	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
4		unitpropdg.k	. . . . . . 7 |- ( ph -> K e. Ring )
5		unitpropdg.l	. . . . . . 7 |- ( ph -> L e. Ring )
6	1, 2, 3, 4, 5	rngidpropdg 13642	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	6	breq2d 4041	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` K ) <-> z ( ||r ` K ) ( 1r ` L ) ) )
8	6	breq2d 4041	. . . . 5 |- ( ph -> ( z ( ||r ` (oppr ` K ) ) ( 1r ` K ) <-> z ( ||r ` (oppr ` K ) ) ( 1r ` L ) ) )
9	7, 8	anbi12d 473	. . . 4 |- ( ph -> ( ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) <-> ( z ( ||r ` K ) ( 1r ` L ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` L ) ) ) )
10		ringsrg 13543	. . . . . . . 8 |- ( K e. Ring -> K e. SRing )
11	4, 10	syl 14	. . . . . . 7 |- ( ph -> K e. SRing )
12		ringsrg 13543	. . . . . . . 8 |- ( L e. Ring -> L e. SRing )
13	5, 12	syl 14	. . . . . . 7 |- ( ph -> L e. SRing )
14	1, 2, 3, 11, 13	dvdsrpropdg 13643	. . . . . 6 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )
15	14	breqd 4040	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` L ) <-> z ( ||r ` L ) ( 1r ` L ) ) )
16		eqid 2193	. . . . . . . . . 10 |- (oppr ` K ) = (oppr ` K )
17		eqid 2193	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
18	16, 17	opprbasg 13571	. . . . . . . . 9 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (oppr ` K ) ) )
19	4, 18	syl 14	. . . . . . . 8 |- ( ph -> ( Base ` K ) = ( Base ` (oppr ` K ) ) )
20	1, 19	eqtrd 2226	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` K ) ) )
21		eqid 2193	. . . . . . . . . 10 |- (oppr ` L ) = (oppr ` L )
22		eqid 2193	. . . . . . . . . 10 |- ( Base ` L ) = ( Base ` L )
23	21, 22	opprbasg 13571	. . . . . . . . 9 |- ( L e. Ring -> ( Base ` L ) = ( Base ` (oppr ` L ) ) )
24	5, 23	syl 14	. . . . . . . 8 |- ( ph -> ( Base ` L ) = ( Base ` (oppr ` L ) ) )
25	2, 24	eqtrd 2226	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` L ) ) )
26	3	ancom2s 566	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
27	4	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> K e. Ring )
28		simprl 529	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> y e. B )
29		simprr 531	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> x e. B )
30		eqid 2193	. . . . . . . . . 10 |- ( .r ` K ) = ( .r ` K )
31		eqid 2193	. . . . . . . . . 10 |- ( .r ` (oppr ` K ) ) = ( .r ` (oppr ` K ) )
32	17, 30, 16, 31	opprmulg 13567	. . . . . . . . 9 |- ( ( K e. Ring /\ y e. B /\ x e. B ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( x ( .r ` K ) y ) )
33	27, 28, 29, 32	syl3anc 1249	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( x ( .r ` K ) y ) )
34	5	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> L e. Ring )
35		eqid 2193	. . . . . . . . . 10 |- ( .r ` L ) = ( .r ` L )
36		eqid 2193	. . . . . . . . . 10 |- ( .r ` (oppr ` L ) ) = ( .r ` (oppr ` L ) )
37	22, 35, 21, 36	opprmulg 13567	. . . . . . . . 9 |- ( ( L e. Ring /\ y e. B /\ x e. B ) -> ( y ( .r ` (oppr ` L ) ) x ) = ( x ( .r ` L ) y ) )
38	34, 28, 29, 37	syl3anc 1249	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` L ) ) x ) = ( x ( .r ` L ) y ) )
39	26, 33, 38	3eqtr4d 2236	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( y ( .r ` (oppr ` L ) ) x ) )
40	16	opprring 13575	. . . . . . . 8 |- ( K e. Ring -> (oppr ` K ) e. Ring )
41		ringsrg 13543	. . . . . . . 8 |- ( (oppr ` K ) e. Ring -> (oppr ` K ) e. SRing )
42	4, 40, 41	3syl 17	. . . . . . 7 |- ( ph -> (oppr ` K ) e. SRing )
43	21	opprring 13575	. . . . . . . 8 |- ( L e. Ring -> (oppr ` L ) e. Ring )
44		ringsrg 13543	. . . . . . . 8 |- ( (oppr ` L ) e. Ring -> (oppr ` L ) e. SRing )
45	5, 43, 44	3syl 17	. . . . . . 7 |- ( ph -> (oppr ` L ) e. SRing )
46	20, 25, 39, 42, 45	dvdsrpropdg 13643	. . . . . 6 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` L ) ) )
47	46	breqd 4040	. . . . 5 |- ( ph -> ( z ( ||r ` (oppr ` K ) ) ( 1r ` L ) <-> z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) )
48	15, 47	anbi12d 473	. . . 4 |- ( ph -> ( ( z ( ||r ` K ) ( 1r ` L ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` L ) ) <-> ( z ( ||r ` L ) ( 1r ` L ) /\ z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) ) )
49	9, 48	bitrd 188	. . 3 |- ( ph -> ( ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) <-> ( z ( ||r ` L ) ( 1r ` L ) /\ z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) ) )
50		eqidd 2194	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
51		eqidd 2194	. . . 4 |- ( ph -> ( 1r ` K ) = ( 1r ` K ) )
52		eqidd 2194	. . . 4 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
53		eqidd 2194	. . . 4 |- ( ph -> (oppr ` K ) = (oppr ` K ) )
54		eqidd 2194	. . . 4 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` K ) ) )
55	50, 51, 52, 53, 54, 11	isunitd 13602	. . 3 |- ( ph -> ( z e. (Unit ` K ) <-> ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) ) )
56		eqidd 2194	. . . 4 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
57		eqidd 2194	. . . 4 |- ( ph -> ( 1r ` L ) = ( 1r ` L ) )
58		eqidd 2194	. . . 4 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
59		eqidd 2194	. . . 4 |- ( ph -> (oppr ` L ) = (oppr ` L ) )
60		eqidd 2194	. . . 4 |- ( ph -> ( ||r ` (oppr ` L ) ) = ( ||r ` (oppr ` L ) ) )
61	56, 57, 58, 59, 60, 13	isunitd 13602	. . 3 |- ( ph -> ( z e. (Unit ` L ) <-> ( z ( ||r ` L ) ( 1r ` L ) /\ z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) ) )
62	49, 55, 61	3bitr4d 220	. 2 |- ( ph -> ( z e. (Unit ` K ) <-> z e. (Unit ` L ) ) )
63	62	eqrdv 2191	1 |- ( ph -> (Unit ` K ) = (Unit ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   Basecbs 12618   .rcmulr 12696   1rcur 13455  SRingcsrg 13459   Ringcrg 13492  opp_rcoppr 13563   ||_rcdsr 13582  Unitcui 13583

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586

This theorem is referenced by:  invrpropdg  13645