Theorem unitpropdg 13704

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 13702	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	6	breq2d 4045	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` K ) <-> z ( ||r ` K ) ( 1r ` L ) ) )
8	6	breq2d 4045	. . . . 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 13603	. . . . . . . 8 |- ( K e. Ring -> K e. SRing )
11	4, 10	syl 14	. . . . . . 7 |- ( ph -> K e. SRing )
12		ringsrg 13603	. . . . . . . 8 |- ( L e. Ring -> L e. SRing )
13	5, 12	syl 14	. . . . . . 7 |- ( ph -> L e. SRing )
14	1, 2, 3, 11, 13	dvdsrpropdg 13703	. . . . . 6 |- ( ph -> ( ||r ` K ) = ( ||r ` L ) )
15	14	breqd 4044	. . . . 5 |- ( ph -> ( z ( ||r ` K ) ( 1r ` L ) <-> z ( ||r ` L ) ( 1r ` L ) ) )
16		eqid 2196	. . . . . . . . . 10 |- (oppr ` K ) = (oppr ` K )
17		eqid 2196	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
18	16, 17	opprbasg 13631	. . . . . . . . 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 2229	. . . . . . 7 |- ( ph -> B = ( Base ` (oppr ` K ) ) )
21		eqid 2196	. . . . . . . . . 10 |- (oppr ` L ) = (oppr ` L )
22		eqid 2196	. . . . . . . . . 10 |- ( Base ` L ) = ( Base ` L )
23	21, 22	opprbasg 13631	. . . . . . . . 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 2229	. . . . . . 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 2196	. . . . . . . . . 10 |- ( .r ` K ) = ( .r ` K )
31		eqid 2196	. . . . . . . . . 10 |- ( .r ` (oppr ` K ) ) = ( .r ` (oppr ` K ) )
32	17, 30, 16, 31	opprmulg 13627	. . . . . . . . 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 2196	. . . . . . . . . 10 |- ( .r ` L ) = ( .r ` L )
36		eqid 2196	. . . . . . . . . 10 |- ( .r ` (oppr ` L ) ) = ( .r ` (oppr ` L ) )
37	22, 35, 21, 36	opprmulg 13627	. . . . . . . . 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 2239	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ x e. B ) ) -> ( y ( .r ` (oppr ` K ) ) x ) = ( y ( .r ` (oppr ` L ) ) x ) )
40	16	opprring 13635	. . . . . . . 8 |- ( K e. Ring -> (oppr ` K ) e. Ring )
41		ringsrg 13603	. . . . . . . 8 |- ( (oppr ` K ) e. Ring -> (oppr ` K ) e. SRing )
42	4, 40, 41	3syl 17	. . . . . . 7 |- ( ph -> (oppr ` K ) e. SRing )
43	21	opprring 13635	. . . . . . . 8 |- ( L e. Ring -> (oppr ` L ) e. Ring )
44		ringsrg 13603	. . . . . . . 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 13703	. . . . . 6 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` L ) ) )
47	46	breqd 4044	. . . . 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 2197	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
51		eqidd 2197	. . . 4 |- ( ph -> ( 1r ` K ) = ( 1r ` K ) )
52		eqidd 2197	. . . 4 |- ( ph -> ( ||r ` K ) = ( ||r ` K ) )
53		eqidd 2197	. . . 4 |- ( ph -> (oppr ` K ) = (oppr ` K ) )
54		eqidd 2197	. . . 4 |- ( ph -> ( ||r ` (oppr ` K ) ) = ( ||r ` (oppr ` K ) ) )
55	50, 51, 52, 53, 54, 11	isunitd 13662	. . 3 |- ( ph -> ( z e. (Unit ` K ) <-> ( z ( ||r ` K ) ( 1r ` K ) /\ z ( ||r ` (oppr ` K ) ) ( 1r ` K ) ) ) )
56		eqidd 2197	. . . 4 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
57		eqidd 2197	. . . 4 |- ( ph -> ( 1r ` L ) = ( 1r ` L ) )
58		eqidd 2197	. . . 4 |- ( ph -> ( ||r ` L ) = ( ||r ` L ) )
59		eqidd 2197	. . . 4 |- ( ph -> (oppr ` L ) = (oppr ` L ) )
60		eqidd 2197	. . . 4 |- ( ph -> ( ||r ` (oppr ` L ) ) = ( ||r ` (oppr ` L ) ) )
61	56, 57, 58, 59, 60, 13	isunitd 13662	. . 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 2194	1 |- ( ph -> (Unit ` K ) = (Unit ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   Basecbs 12678   .rcmulr 12756   1rcur 13515  SRingcsrg 13519   Ringcrg 13552  opp_rcoppr 13623   ||_rcdsr 13642  Unitcui 13643

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646

This theorem is referenced by:  invrpropdg  13705