Theorem subrgpropd 13885

Description: If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypotheses

Ref	Expression
subrgpropd.1	|- ( ph -> B = ( Base ` K ) )
subrgpropd.2	|- ( ph -> B = ( Base ` L ) )
subrgpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
subrgpropd.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )

Assertion

Ref	Expression
subrgpropd	|- ( ph -> (SubRing ` K ) = (SubRing ` L ) )

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

Proof of Theorem subrgpropd

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgrcl 13858	. . . 4 |- ( s e. (SubRing ` K ) -> K e. Ring )
2	1	a1i 9	. . 3 |- ( ph -> ( s e. (SubRing ` K ) -> K e. Ring ) )
3		subrgrcl 13858	. . . 4 |- ( s e. (SubRing ` L ) -> L e. Ring )
4		subrgpropd.1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
5		subrgpropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
6		subrgpropd.3	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
7		subrgpropd.4	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
8	4, 5, 6, 7	ringpropd 13670	. . . 4 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
9	3, 8	imbitrrid 156	. . 3 |- ( ph -> ( s e. (SubRing ` L ) -> K e. Ring ) )
10	8	adantr 276	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> ( K e. Ring <-> L e. Ring ) )
11	4	ineq2d 3365	. . . . . . . . . 10 |- ( ph -> ( s i^i B ) = ( s i^i ( Base ` K ) ) )
12	11	adantr 276	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( s i^i ( Base ` K ) ) )
13		eqidd 2197	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2197	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( Base ` K ) = ( Base ` K ) )
15		simplr 528	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> K e. Ring )
16		simpr 110	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> s e. _V )
17	13, 14, 15, 16	ressbasd 12770	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
18	17	elvd 2768	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2229	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3365	. . . . . . . . . 10 |- ( ph -> ( s i^i B ) = ( s i^i ( Base ` L ) ) )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( s i^i ( Base ` L ) ) )
22		eqidd 2197	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2197	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( Base ` L ) = ( Base ` L ) )
24	8	biimpa 296	. . . . . . . . . . . 12 |- ( ( ph /\ K e. Ring ) -> L e. Ring )
25	24	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> L e. Ring )
26	22, 23, 25, 16	ressbasd 12770	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
27	26	elvd 2768	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
28	21, 27	eqtrd 2229	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
29		elinel2 3351	. . . . . . . . . 10 |- ( x e. ( s i^i B ) -> x e. B )
30		elinel2 3351	. . . . . . . . . 10 |- ( y e. ( s i^i B ) -> y e. B )
31	29, 30	anim12i 338	. . . . . . . . 9 |- ( ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) -> ( x e. B /\ y e. B ) )
32	6	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
33		eqidd 2197	. . . . . . . . . . . . 13 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` K ) = ( +g ` K ) )
34	13, 33, 16, 15	ressplusgd 12831	. . . . . . . . . . . 12 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
35	34	elvd 2768	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
36	35	oveqdr 5953	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
37		eqidd 2197	. . . . . . . . . . . . 13 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` L ) = ( +g ` L ) )
38	22, 37, 16, 25	ressplusgd 12831	. . . . . . . . . . . 12 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
39	38	elvd 2768	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
40	39	oveqdr 5953	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` L ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
41	32, 36, 40	3eqtr3d 2237	. . . . . . . . 9 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` ( K ↾s s ) ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
42	31, 41	sylan2 286	. . . . . . . 8 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) ) -> ( x ( +g ` ( K ↾s s ) ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
43	7	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
44		vex 2766	. . . . . . . . . . . . 13 |- s e. _V
45		eqid 2196	. . . . . . . . . . . . . 14 |- ( K ↾s s ) = ( K ↾s s )
46		eqid 2196	. . . . . . . . . . . . . 14 |- ( .r ` K ) = ( .r ` K )
47	45, 46	ressmulrg 12847	. . . . . . . . . . . . 13 |- ( ( s e. _V /\ K e. Ring ) -> ( .r ` K ) = ( .r ` ( K ↾s s ) ) )
48	44, 47	mpan 424	. . . . . . . . . . . 12 |- ( K e. Ring -> ( .r ` K ) = ( .r ` ( K ↾s s ) ) )
49	48	adantl 277	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( .r ` K ) = ( .r ` ( K ↾s s ) ) )
50	49	oveqdr 5953	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
51		eqid 2196	. . . . . . . . . . . . 13 |- ( L ↾s s ) = ( L ↾s s )
52		eqid 2196	. . . . . . . . . . . . 13 |- ( .r ` L ) = ( .r ` L )
53	51, 52	ressmulrg 12847	. . . . . . . . . . . 12 |- ( ( s e. _V /\ L e. Ring ) -> ( .r ` L ) = ( .r ` ( L ↾s s ) ) )
54	44, 24, 53	sylancr 414	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( .r ` L ) = ( .r ` ( L ↾s s ) ) )
55	54	oveqdr 5953	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
56	43, 50, 55	3eqtr3d 2237	. . . . . . . . 9 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` ( K ↾s s ) ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
57	31, 56	sylan2 286	. . . . . . . 8 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) ) -> ( x ( .r ` ( K ↾s s ) ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
58	19, 28, 42, 57	ringpropd 13670	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> ( ( K ↾s s ) e. Ring <-> ( L ↾s s ) e. Ring ) )
59	10, 58	anbi12d 473	. . . . . 6 |- ( ( ph /\ K e. Ring ) -> ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) <-> ( L e. Ring /\ ( L ↾s s ) e. Ring ) ) )
60	4, 5	eqtr3d 2231	. . . . . . . . 9 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
61	60	sseq2d 3214	. . . . . . . 8 |- ( ph -> ( s C_ ( Base ` K ) <-> s C_ ( Base ` L ) ) )
62	61	adantr 276	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> ( s C_ ( Base ` K ) <-> s C_ ( Base ` L ) ) )
63	4	adantr 276	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> B = ( Base ` K ) )
64	5	adantr 276	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> B = ( Base ` L ) )
65		simpr 110	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> K e. Ring )
66	63, 64, 43, 65, 24	rngidpropdg 13778	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( 1r ` K ) = ( 1r ` L ) )
67	66	eleq1d 2265	. . . . . . 7 |- ( ( ph /\ K e. Ring ) -> ( ( 1r ` K ) e. s <-> ( 1r ` L ) e. s ) )
68	62, 67	anbi12d 473	. . . . . 6 |- ( ( ph /\ K e. Ring ) -> ( ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) <-> ( s C_ ( Base ` L ) /\ ( 1r ` L ) e. s ) ) )
69	59, 68	anbi12d 473	. . . . 5 |- ( ( ph /\ K e. Ring ) -> ( ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) /\ ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) ) <-> ( ( L e. Ring /\ ( L ↾s s ) e. Ring ) /\ ( s C_ ( Base ` L ) /\ ( 1r ` L ) e. s ) ) ) )
70		eqid 2196	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
71		eqid 2196	. . . . . 6 |- ( 1r ` K ) = ( 1r ` K )
72	70, 71	issubrg 13853	. . . . 5 |- ( s e. (SubRing ` K ) <-> ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) /\ ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) ) )
73		eqid 2196	. . . . . 6 |- ( Base ` L ) = ( Base ` L )
74		eqid 2196	. . . . . 6 |- ( 1r ` L ) = ( 1r ` L )
75	73, 74	issubrg 13853	. . . . 5 |- ( s e. (SubRing ` L ) <-> ( ( L e. Ring /\ ( L ↾s s ) e. Ring ) /\ ( s C_ ( Base ` L ) /\ ( 1r ` L ) e. s ) ) )
76	69, 72, 75	3bitr4g 223	. . . 4 |- ( ( ph /\ K e. Ring ) -> ( s e. (SubRing ` K ) <-> s e. (SubRing ` L ) ) )
77	76	ex 115	. . 3 |- ( ph -> ( K e. Ring -> ( s e. (SubRing ` K ) <-> s e. (SubRing ` L ) ) ) )
78	2, 9, 77	pm5.21ndd 706	. 2 |- ( ph -> ( s e. (SubRing ` K ) <-> s e. (SubRing ` L ) ) )
79	78	eqrdv 2194	1 |- ( ph -> (SubRing ` K ) = (SubRing ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   _Vcvv 2763   i^i cin 3156   C_ wss 3157   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   .rcmulr 12781   1rcur 13591   Ringcrg 13628  SubRingcsubrg 13849

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-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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-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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-mgp 13553  df-ur 13592  df-ring 13630  df-subrg 13851

This theorem is referenced by: (None)