Theorem unitpropd 18697

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
rngidpropd.1	|- ( ph -> B = ( Base ` K ) )
rngidpropd.2	|- ( ph -> B = ( Base ` L ) )
rngidpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )

Assertion

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

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

Proof of Theorem unitpropd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngidpropd.1	. . . . . . 7 |- ( ph -> B = ( Base ` K ) )
2		rngidpropd.2	. . . . . . 7 |- ( ph -> B = ( Base ` L ) )
3		rngidpropd.3	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
4	1, 2, 3	rngidpropd 18695	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
5	4	breq2d 4665	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` K ) <-> z ( ||r ` K ) ( 1r ` L ) ) )
6	4	breq2d 4665	. . . . 5 |- ( ph -> ( z ( ||r ` (oppr ` K ) ) ( 1r ` K ) <-> z ( ||r ` (oppr ` K ) ) ( 1r ` L ) ) )
7	5, 6	anbi12d 747	. . . 4 |- ( ph -> ( ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) <-> ( z ( ||r ` K ) ( 1r ` L ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` L ) ) ) )
8	1, 2, 3	dvdsrpropd 18696	. . . . . 6 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )
9	8	breqd 4664	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` L ) <-> z ( ||r ` L ) ( 1r ` L ) ) )
10		eqid 2622	. . . . . . . . 9 |- (oppr ` K ) = (oppr ` K )
11		eqid 2622	. . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
12	10, 11	opprbas 18629	. . . . . . . 8 |- ( Base ` K ) = ( Base ` (oppr ` K ) )
13	1, 12	syl6eq 2672	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` K ) ) )
14		eqid 2622	. . . . . . . . 9 |- (oppr ` L ) = (oppr ` L )
15		eqid 2622	. . . . . . . . 9 |- ( Base ` L ) = ( Base ` L )
16	14, 15	opprbas 18629	. . . . . . . 8 |- ( Base ` L ) = ( Base ` (oppr ` L ) )
17	2, 16	syl6eq 2672	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` L ) ) )
18	3	ancom2s 844	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
19		eqid 2622	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` K )
20		eqid 2622	. . . . . . . . 9 |- ( .r ` (oppr ` K ) ) = ( .r ` (oppr ` K ) )
21	11, 19, 10, 20	opprmul 18626	. . . . . . . 8 |- ( y ( .r ` (oppr ` K ) ) x ) = ( x ( .r ` K ) y )
22		eqid 2622	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` L )
23		eqid 2622	. . . . . . . . 9 |- ( .r ` (oppr ` L ) ) = ( .r ` (oppr ` L ) )
24	15, 22, 14, 23	opprmul 18626	. . . . . . . 8 |- ( y ( .r ` (oppr ` L ) ) x ) = ( x ( .r ` L ) y )
25	18, 21, 24	3eqtr4g 2681	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( y ( .r ` (oppr ` L ) ) x ) )
26	13, 17, 25	dvdsrpropd 18696	. . . . . 6 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` L ) ) )
27	26	breqd 4664	. . . . 5 |- ( ph -> ( z ( ||r ` (oppr ` K ) ) ( 1r ` L ) <-> z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) )
28	9, 27	anbi12d 747	. . . 4 |- ( ph -> ( ( z ( ||r ` K ) ( 1r ` L ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` L ) ) <-> ( z ( ||r ` L ) ( 1r ` L ) /\ z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) ) )
29	7, 28	bitrd 268	. . 3 |- ( ph -> ( ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) <-> ( z ( ||r ` L ) ( 1r ` L ) /\ z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) ) )
30		eqid 2622	. . . 4 |- (Unit ` K ) = (Unit ` K )
31		eqid 2622	. . . 4 |- ( 1r ` K ) = ( 1r ` K )
32		eqid 2622	. . . 4 |- ( ||r ` K ) = ( ||r ` K )
33		eqid 2622	. . . 4 |- ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` K ) )
34	30, 31, 32, 10, 33	isunit 18657	. . 3 |- ( z e. (Unit ` K ) <-> ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) )
35		eqid 2622	. . . 4 |- (Unit ` L ) = (Unit ` L )
36		eqid 2622	. . . 4 |- ( 1r ` L ) = ( 1r ` L )
37		eqid 2622	. . . 4 |- ( ||r ` L ) = ( ||r ` L )
38		eqid 2622	. . . 4 |- ( ||r ` (oppr ` L ) ) = ( ||r ` (oppr ` L ) )
39	35, 36, 37, 14, 38	isunit 18657	. . 3 |- ( z e. (Unit ` L ) <-> ( z ( ||r ` L ) ( 1r ` L ) /\ z ( ||r ` (oppr ` L ) ) ( 1r ` L ) ) )
40	29, 34, 39	3bitr4g 303	. 2 |- ( ph -> ( z e. (Unit ` K ) <-> z e. (Unit ` L ) ) )
41	40	eqrdv 2620	1 |- ( ph -> (Unit ` K ) = (Unit ` L ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   1rcur 18501  opp_rcoppr 18622   ||_rcdsr 18638  Unitcui 18639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgp 18490  df-ur 18502  df-oppr 18623  df-dvdsr 18641  df-unit 18642

This theorem is referenced by:  invrpropd  18698  drngprop  18758  drngpropd  18774