Theorem subrgpropd 13463

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 13441	. . . 4 |- ( s e. (SubRing ` K ) -> K e. Ring )
2	1	a1i 9	. . 3 |- ( ph -> ( s e. (SubRing ` K ) -> K e. Ring ) )
3		subrgrcl 13441	. . . 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 13285	. . . 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 3348	. . . . . . . . . 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 2188	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2188	. . . . . . . . . . 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 12540	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
18	17	elvd 2754	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2220	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3348	. . . . . . . . . 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 2188	. . . . . . . . . . 11 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2188	. . . . . . . . . . 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 12540	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
27	26	elvd 2754	. . . . . . . . 9 |- ( ( ph /\ K e. Ring ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
28	21, 27	eqtrd 2220	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
29		elinel2 3334	. . . . . . . . . 10 |- ( x e. ( s i^i B ) -> x e. B )
30		elinel2 3334	. . . . . . . . . 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 2188	. . . . . . . . . . . . 13 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` K ) = ( +g ` K ) )
34	13, 33, 16, 15	ressplusgd 12601	. . . . . . . . . . . 12 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
35	34	elvd 2754	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
36	35	oveqdr 5916	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
37		eqidd 2188	. . . . . . . . . . . . 13 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` L ) = ( +g ` L ) )
38	22, 37, 16, 25	ressplusgd 12601	. . . . . . . . . . . 12 |- ( ( ( ph /\ K e. Ring ) /\ s e. _V ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
39	38	elvd 2754	. . . . . . . . . . 11 |- ( ( ph /\ K e. Ring ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
40	39	oveqdr 5916	. . . . . . . . . 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 2228	. . . . . . . . 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 2752	. . . . . . . . . . . . 13 |- s e. _V
45		eqid 2187	. . . . . . . . . . . . . 14 |- ( K ↾s s ) = ( K ↾s s )
46		eqid 2187	. . . . . . . . . . . . . 14 |- ( .r ` K ) = ( .r ` K )
47	45, 46	ressmulrg 12617	. . . . . . . . . . . . 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 5916	. . . . . . . . . 10 |- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
51		eqid 2187	. . . . . . . . . . . . 13 |- ( L ↾s s ) = ( L ↾s s )
52		eqid 2187	. . . . . . . . . . . . 13 |- ( .r ` L ) = ( .r ` L )
53	51, 52	ressmulrg 12617	. . . . . . . . . . . 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 5916	. . . . . . . . . 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 2228	. . . . . . . . 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 13285	. . . . . . 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 2222	. . . . . . . . 9 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
61	60	sseq2d 3197	. . . . . . . 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 13389	. . . . . . . 8 |- ( ( ph /\ K e. Ring ) -> ( 1r ` K ) = ( 1r ` L ) )
67	66	eleq1d 2256	. . . . . . 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 2187	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
71		eqid 2187	. . . . . 6 |- ( 1r ` K ) = ( 1r ` K )
72	70, 71	issubrg 13436	. . . . 5 |- ( s e. (SubRing ` K ) <-> ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) /\ ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) ) )
73		eqid 2187	. . . . . 6 |- ( Base ` L ) = ( Base ` L )
74		eqid 2187	. . . . . 6 |- ( 1r ` L ) = ( 1r ` L )
75	73, 74	issubrg 13436	. . . . 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 2185	1 |- ( ph -> (SubRing ` K ) = (SubRing ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   _Vcvv 2749   i^i cin 3140   C_ wss 3141   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾_s cress 12476   +g cplusg 12550   .rcmulr 12551   1rcur 13206   Ringcrg 13243  SubRingcsubrg 13432

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-mgp 13171  df-ur 13207  df-ring 13245  df-subrg 13434

This theorem is referenced by: (None)