Theorem unitpropdg 14161

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 14159	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	6	breq2d 4100	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` K ) <-> z ( ||r ` K ) ( 1r ` L ) ) )
8	6	breq2d 4100	. . . . 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 14059	. . . . . . . 8 |- ( K e. Ring -> K e. SRing )
11	4, 10	syl 14	. . . . . . 7 |- ( ph -> K e. SRing )
12		ringsrg 14059	. . . . . . . 8 |- ( L e. Ring -> L e. SRing )
13	5, 12	syl 14	. . . . . . 7 |- ( ph -> L e. SRing )
14	1, 2, 3, 11, 13	dvdsrpropdg 14160	. . . . . 6 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )
15	14	breqd 4099	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` L ) <-> z ( ||r ` L ) ( 1r ` L ) ) )
16		eqid 2231	. . . . . . . . . 10 |- (oppr ` K ) = (oppr ` K )
17		eqid 2231	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
18	16, 17	opprbasg 14087	. . . . . . . . 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 2264	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` K ) ) )
21		eqid 2231	. . . . . . . . . 10 |- (oppr ` L ) = (oppr ` L )
22		eqid 2231	. . . . . . . . . 10 |- ( Base ` L ) = ( Base ` L )
23	21, 22	opprbasg 14087	. . . . . . . . 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 2264	. . . . . . 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 2231	. . . . . . . . . 10 |- ( .r ` K ) = ( .r ` K )
31		eqid 2231	. . . . . . . . . 10 |- ( .r ` (oppr ` K ) ) = ( .r ` (oppr ` K ) )
32	17, 30, 16, 31	opprmulg 14083	. . . . . . . . 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 1273	. . . . . . . 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 2231	. . . . . . . . . 10 |- ( .r ` L ) = ( .r ` L )
36		eqid 2231	. . . . . . . . . 10 |- ( .r ` (oppr ` L ) ) = ( .r ` (oppr ` L ) )
37	22, 35, 21, 36	opprmulg 14083	. . . . . . . . 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 1273	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` L ) ) x ) = ( x ( .r ` L ) y ) )
39	26, 33, 38	3eqtr4d 2274	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( y ( .r ` (oppr ` L ) ) x ) )
40	16	opprring 14091	. . . . . . . 8 |- ( K e. Ring -> (oppr ` K ) e. Ring )
41		ringsrg 14059	. . . . . . . 8 |- ( (oppr ` K ) e. Ring -> (oppr ` K ) e. SRing )
42	4, 40, 41	3syl 17	. . . . . . 7 |- ( ph -> (oppr ` K ) e. SRing )
43	21	opprring 14091	. . . . . . . 8 |- ( L e. Ring -> (oppr ` L ) e. Ring )
44		ringsrg 14059	. . . . . . . 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 14160	. . . . . 6 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` L ) ) )
47	46	breqd 4099	. . . . 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 2232	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
51		eqidd 2232	. . . 4 |- ( ph -> ( 1r ` K ) = ( 1r ` K ) )
52		eqidd 2232	. . . 4 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
53		eqidd 2232	. . . 4 |- ( ph -> (oppr ` K ) = (oppr ` K ) )
54		eqidd 2232	. . . 4 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` K ) ) )
55	50, 51, 52, 53, 54, 11	isunitd 14119	. . 3 |- ( ph -> ( z e. (Unit ` K ) <-> ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) ) )
56		eqidd 2232	. . . 4 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
57		eqidd 2232	. . . 4 |- ( ph -> ( 1r ` L ) = ( 1r ` L ) )
58		eqidd 2232	. . . 4 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
59		eqidd 2232	. . . 4 |- ( ph -> (oppr ` L ) = (oppr ` L ) )
60		eqidd 2232	. . . 4 |- ( ph -> ( ||r ` (oppr ` L ) ) = ( ||r ` (oppr ` L ) ) )
61	56, 57, 58, 59, 60, 13	isunitd 14119	. . 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 2229	1 |- ( ph -> (Unit ` K ) = (Unit ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   1rcur 13971  SRingcsrg 13975   Ringcrg 14008  opp_rcoppr 14079   ||_rcdsr 14098  Unitcui 14099

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-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-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  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

This theorem is referenced by:  invrpropdg  14162