Theorem unitpropdg 13317

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 13315	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	6	breq2d 4016	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` K ) <-> z ( ||r ` K ) ( 1r ` L ) ) )
8	6	breq2d 4016	. . . . 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 13224	. . . . . . . 8 |- ( K e. Ring -> K e. SRing )
11	4, 10	syl 14	. . . . . . 7 |- ( ph -> K e. SRing )
12		ringsrg 13224	. . . . . . . 8 |- ( L e. Ring -> L e. SRing )
13	5, 12	syl 14	. . . . . . 7 |- ( ph -> L e. SRing )
14	1, 2, 3, 11, 13	dvdsrpropdg 13316	. . . . . 6 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )
15	14	breqd 4015	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` L ) <-> z ( ||r ` L ) ( 1r ` L ) ) )
16		eqid 2177	. . . . . . . . . 10 |- (oppr ` K ) = (oppr ` K )
17		eqid 2177	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
18	16, 17	opprbasg 13247	. . . . . . . . 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 2210	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` K ) ) )
21		eqid 2177	. . . . . . . . . 10 |- (oppr ` L ) = (oppr ` L )
22		eqid 2177	. . . . . . . . . 10 |- ( Base ` L ) = ( Base ` L )
23	21, 22	opprbasg 13247	. . . . . . . . 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 2210	. . . . . . 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 2177	. . . . . . . . . 10 |- ( .r ` K ) = ( .r ` K )
31		eqid 2177	. . . . . . . . . 10 |- ( .r ` (oppr ` K ) ) = ( .r ` (oppr ` K ) )
32	17, 30, 16, 31	opprmulg 13243	. . . . . . . . 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 1238	. . . . . . . 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 2177	. . . . . . . . . 10 |- ( .r ` L ) = ( .r ` L )
36		eqid 2177	. . . . . . . . . 10 |- ( .r ` (oppr ` L ) ) = ( .r ` (oppr ` L ) )
37	22, 35, 21, 36	opprmulg 13243	. . . . . . . . 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 1238	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` L ) ) x ) = ( x ( .r ` L ) y ) )
39	26, 33, 38	3eqtr4d 2220	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( y ( .r ` (oppr ` L ) ) x ) )
40	16	opprring 13249	. . . . . . . 8 |- ( K e. Ring -> (oppr ` K ) e. Ring )
41		ringsrg 13224	. . . . . . . 8 |- ( (oppr ` K ) e. Ring -> (oppr ` K ) e. SRing )
42	4, 40, 41	3syl 17	. . . . . . 7 |- ( ph -> (oppr ` K ) e. SRing )
43	21	opprring 13249	. . . . . . . 8 |- ( L e. Ring -> (oppr ` L ) e. Ring )
44		ringsrg 13224	. . . . . . . 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 13316	. . . . . 6 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` L ) ) )
47	46	breqd 4015	. . . . 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 2178	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
51		eqidd 2178	. . . 4 |- ( ph -> ( 1r ` K ) = ( 1r ` K ) )
52		eqidd 2178	. . . 4 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
53		eqidd 2178	. . . 4 |- ( ph -> (oppr ` K ) = (oppr ` K ) )
54		eqidd 2178	. . . 4 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` K ) ) )
55	50, 51, 52, 53, 54, 11	isunitd 13275	. . 3 |- ( ph -> ( z e. (Unit ` K ) <-> ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) ) )
56		eqidd 2178	. . . 4 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
57		eqidd 2178	. . . 4 |- ( ph -> ( 1r ` L ) = ( 1r ` L ) )
58		eqidd 2178	. . . 4 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
59		eqidd 2178	. . . 4 |- ( ph -> (oppr ` L ) = (oppr ` L ) )
60		eqidd 2178	. . . 4 |- ( ph -> ( ||r ` (oppr ` L ) ) = ( ||r ` (oppr ` L ) ) )
61	56, 57, 58, 59, 60, 13	isunitd 13275	. . 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 2175	1 |- ( ph -> (Unit ` K ) = (Unit ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   Basecbs 12462   .rcmulr 12537   1rcur 13142  SRingcsrg 13146   Ringcrg 13179  opp_rcoppr 13239   ||_rcdsr 13255  Unitcui 13256

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-tpos 6246  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-cmn 13090  df-abl 13091  df-mgp 13131  df-ur 13143  df-srg 13147  df-ring 13181  df-oppr 13240  df-dvdsr 13258  df-unit 13259

This theorem is referenced by:  invrpropdg  13318