Theorem unitpropdg 14293

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 14291	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	6	breq2d 4121	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` K ) <-> z ( ||r ` K ) ( 1r ` L ) ) )
8	6	breq2d 4121	. . . . 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 14191	. . . . . . . 8 |- ( K e. Ring -> K e. SRing )
11	4, 10	syl 14	. . . . . . 7 |- ( ph -> K e. SRing )
12		ringsrg 14191	. . . . . . . 8 |- ( L e. Ring -> L e. SRing )
13	5, 12	syl 14	. . . . . . 7 |- ( ph -> L e. SRing )
14	1, 2, 3, 11, 13	dvdsrpropdg 14292	. . . . . 6 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )
15	14	breqd 4120	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` L ) <-> z ( ||r ` L ) ( 1r ` L ) ) )
16		eqid 2232	. . . . . . . . . 10 |- (oppr ` K ) = (oppr ` K )
17		eqid 2232	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
18	16, 17	opprbasg 14219	. . . . . . . . 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 2265	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` K ) ) )
21		eqid 2232	. . . . . . . . . 10 |- (oppr ` L ) = (oppr ` L )
22		eqid 2232	. . . . . . . . . 10 |- ( Base ` L ) = ( Base ` L )
23	21, 22	opprbasg 14219	. . . . . . . . 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 2265	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` L ) ) )
26	3	ancom2s 568	. . . . . . . 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 531	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> y e. B )
29		simprr 533	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> x e. B )
30		eqid 2232	. . . . . . . . . 10 |- ( .r ` K ) = ( .r ` K )
31		eqid 2232	. . . . . . . . . 10 |- ( .r ` (oppr ` K ) ) = ( .r ` (oppr ` K ) )
32	17, 30, 16, 31	opprmulg 14215	. . . . . . . . 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 1274	. . . . . . . 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 2232	. . . . . . . . . 10 |- ( .r ` L ) = ( .r ` L )
36		eqid 2232	. . . . . . . . . 10 |- ( .r ` (oppr ` L ) ) = ( .r ` (oppr ` L ) )
37	22, 35, 21, 36	opprmulg 14215	. . . . . . . . 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 1274	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` L ) ) x ) = ( x ( .r ` L ) y ) )
39	26, 33, 38	3eqtr4d 2275	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( y ( .r ` (oppr ` L ) ) x ) )
40	16	opprring 14223	. . . . . . . 8 |- ( K e. Ring -> (oppr ` K ) e. Ring )
41		ringsrg 14191	. . . . . . . 8 |- ( (oppr ` K ) e. Ring -> (oppr ` K ) e. SRing )
42	4, 40, 41	3syl 17	. . . . . . 7 |- ( ph -> (oppr ` K ) e. SRing )
43	21	opprring 14223	. . . . . . . 8 |- ( L e. Ring -> (oppr ` L ) e. Ring )
44		ringsrg 14191	. . . . . . . 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 14292	. . . . . 6 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` L ) ) )
47	46	breqd 4120	. . . . 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 2233	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
51		eqidd 2233	. . . 4 |- ( ph -> ( 1r ` K ) = ( 1r ` K ) )
52		eqidd 2233	. . . 4 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
53		eqidd 2233	. . . 4 |- ( ph -> (oppr ` K ) = (oppr ` K ) )
54		eqidd 2233	. . . 4 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` K ) ) )
55	50, 51, 52, 53, 54, 11	isunitd 14251	. . 3 |- ( ph -> ( z e. (Unit ` K ) <-> ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) ) )
56		eqidd 2233	. . . 4 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
57		eqidd 2233	. . . 4 |- ( ph -> ( 1r ` L ) = ( 1r ` L ) )
58		eqidd 2233	. . . 4 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
59		eqidd 2233	. . . 4 |- ( ph -> (oppr ` L ) = (oppr ` L ) )
60		eqidd 2233	. . . 4 |- ( ph -> ( ||r ` (oppr ` L ) ) = ( ||r ` (oppr ` L ) ) )
61	56, 57, 58, 59, 60, 13	isunitd 14251	. . 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 2230	1 |- ( ph -> (Unit ` K ) = (Unit ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Basecbs 13212   .rcmulr 13291   1rcur 14103  SRingcsrg 14107   Ringcrg 14140  opp_rcoppr 14211   ||_rcdsr 14230  Unitcui 14231

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234

This theorem is referenced by:  invrpropdg  14294