Theorem subrgpropd 13380

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 13358	. . . 4 |- ( s e. (SubRing ` K ) -> K e. Ring )
2	1	a1i 9	. . 3 |- ( ph -> ( s e. (SubRing ` K ) -> K e. Ring ) )
3		subrgrcl 13358	. . . 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 13228	. . . 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 3338	. . . . . . . . . 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 2178	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2178	. . . . . . . . . . 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 12530	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
18	17	elvd 2744	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2210	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3338	. . . . . . . . . 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 2178	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2178	. . . . . . . . . . 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 12530	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
27	26	elvd 2744	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
28	21, 27	eqtrd 2210	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
29		elinel2 3324	. . . . . . . . . 10 |- ( x e. ( s i^i B ) -> x e. B )
30		elinel2 3324	. . . . . . . . . 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 2178	. . . . . . . . . . . . 13 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` K ) = ( +g ` K ) )
34	13, 33, 16, 15	ressplusgd 12590	. . . . . . . . . . . 12 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
35	34	elvd 2744	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
36	35	oveqdr 5906	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
37		eqidd 2178	. . . . . . . . . . . . 13 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` L ) = ( +g ` L ) )
38	22, 37, 16, 25	ressplusgd 12590	. . . . . . . . . . . 12 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
39	38	elvd 2744	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
40	39	oveqdr 5906	. . . . . . . . . 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 2218	. . . . . . . . 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 2742	. . . . . . . . . . . . 13 |- s e. _V
45		eqid 2177	. . . . . . . . . . . . . 14 |- ( K ↾s s ) = ( K ↾s s )
46		eqid 2177	. . . . . . . . . . . . . 14 |- ( .r ` K ) = ( .r ` K )
47	45, 46	ressmulrg 12606	. . . . . . . . . . . . 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 5906	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
51		eqid 2177	. . . . . . . . . . . . 13 |- ( L ↾s s ) = ( L ↾s s )
52		eqid 2177	. . . . . . . . . . . . 13 |- ( .r ` L ) = ( .r ` L )
53	51, 52	ressmulrg 12606	. . . . . . . . . . . 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 5906	. . . . . . . . . 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 2218	. . . . . . . . 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 13228	. . . . . . 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 2212	. . . . . . . . 9 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
61	60	sseq2d 3187	. . . . . . . 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 13326	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( 1r ` K ) = ( 1r ` L ) )
67	66	eleq1d 2246	. . . . . . 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 2177	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
71		eqid 2177	. . . . . 6 |- ( 1r ` K ) = ( 1r ` K )
72	70, 71	issubrg 13353	. . . . 5 |- ( s e. (SubRing ` K ) <-> ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) /\ ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) ) )
73		eqid 2177	. . . . . 6 |- ( Base ` L ) = ( Base ` L )
74		eqid 2177	. . . . . 6 |- ( 1r ` L ) = ( 1r ` L )
75	73, 74	issubrg 13353	. . . . 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 705	. 2 |- ( ph -> ( s e. (SubRing ` K ) <-> s e. (SubRing ` L ) ) )
79	78	eqrdv 2175	1 |- ( ph -> (SubRing ` K ) = (SubRing ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2739   i^i cin 3130   C_ wss 3131   ` cfv 5218  (class class class)co 5878   Basecbs 12465   ↾_s cress 12466   +g cplusg 12539   .rcmulr 12540   1rcur 13153   Ringcrg 13190  SubRingcsubrg 13349

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-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930

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-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473  df-plusg 12552  df-mulr 12553  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12825  df-grp 12887  df-mgp 13142  df-ur 13154  df-ring 13192  df-subrg 13351

This theorem is referenced by: (None)