Theorem subrgpropd 18814

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		subrgpropd.1	. . . . . 6 |- ( ph -> B = ( Base ` K ) )
2		subrgpropd.2	. . . . . 6 |- ( ph -> B = ( Base ` L ) )
3		subrgpropd.3	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4		subrgpropd.4	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
5	1, 2, 3, 4	ringpropd 18582	. . . . 5 |- ( ph -> ( K e. Ring <-> L e. Ring ) )
6	1	ineq2d 3814	. . . . . . 7 |- ( ph -> ( s i^i B ) = ( s i^i ( Base ` K ) ) )
7		vex 3203	. . . . . . . 8 |- s e. _V
8		eqid 2622	. . . . . . . . 9 |- ( K ↾s s ) = ( K ↾s s )
9		eqid 2622	. . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
10	8, 9	ressbas 15930	. . . . . . . 8 |- ( s e. _V -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
11	7, 10	ax-mp 5	. . . . . . 7 |- ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) )
12	6, 11	syl6eq 2672	. . . . . 6 |- ( ph -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
13	2	ineq2d 3814	. . . . . . 7 |- ( ph -> ( s i^i B ) = ( s i^i ( Base ` L ) ) )
14		eqid 2622	. . . . . . . . 9 |- ( L ↾s s ) = ( L ↾s s )
15		eqid 2622	. . . . . . . . 9 |- ( Base ` L ) = ( Base ` L )
16	14, 15	ressbas 15930	. . . . . . . 8 |- ( s e. _V -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
17	7, 16	ax-mp 5	. . . . . . 7 |- ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) )
18	13, 17	syl6eq 2672	. . . . . 6 |- ( ph -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
19		inss2 3834	. . . . . . . . 9 |- ( s i^i B ) C_ B
20	19	sseli 3599	. . . . . . . 8 |- ( x e. ( s i^i B ) -> x e. B )
21	19	sseli 3599	. . . . . . . 8 |- ( y e. ( s i^i B ) -> y e. B )
22	20, 21	anim12i 590	. . . . . . 7 |- ( ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) -> ( x e. B /\ y e. B ) )
23		eqid 2622	. . . . . . . . . . 11 |- ( +g ` K ) = ( +g ` K )
24	8, 23	ressplusg 15993	. . . . . . . . . 10 |- ( s e. _V -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
25	7, 24	ax-mp 5	. . . . . . . . 9 |- ( +g ` K ) = ( +g ` ( K ↾s s ) )
26	25	oveqi 6663	. . . . . . . 8 |- ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y )
27		eqid 2622	. . . . . . . . . . 11 |- ( +g ` L ) = ( +g ` L )
28	14, 27	ressplusg 15993	. . . . . . . . . 10 |- ( s e. _V -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
29	7, 28	ax-mp 5	. . . . . . . . 9 |- ( +g ` L ) = ( +g ` ( L ↾s s ) )
30	29	oveqi 6663	. . . . . . . 8 |- ( x ( +g ` L ) y ) = ( x ( +g ` ( L ↾s s ) ) y )
31	3, 26, 30	3eqtr3g 2679	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` ( K ↾s s ) ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
32	22, 31	sylan2 491	. . . . . 6 |- ( ( ph /\ ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) ) -> ( x ( +g ` ( K ↾s s ) ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
33		eqid 2622	. . . . . . . . . . 11 |- ( .r ` K ) = ( .r ` K )
34	8, 33	ressmulr 16006	. . . . . . . . . 10 |- ( s e. _V -> ( .r ` K ) = ( .r ` ( K ↾s s ) ) )
35	7, 34	ax-mp 5	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` ( K ↾s s ) )
36	35	oveqi 6663	. . . . . . . 8 |- ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y )
37		eqid 2622	. . . . . . . . . . 11 |- ( .r ` L ) = ( .r ` L )
38	14, 37	ressmulr 16006	. . . . . . . . . 10 |- ( s e. _V -> ( .r ` L ) = ( .r ` ( L ↾s s ) ) )
39	7, 38	ax-mp 5	. . . . . . . . 9 |- ( .r ` L ) = ( .r ` ( L ↾s s ) )
40	39	oveqi 6663	. . . . . . . 8 |- ( x ( .r ` L ) y ) = ( x ( .r ` ( L ↾s s ) ) y )
41	4, 36, 40	3eqtr3g 2679	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` ( K ↾s s ) ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
42	22, 41	sylan2 491	. . . . . 6 |- ( ( ph /\ ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) ) -> ( x ( .r ` ( K ↾s s ) ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
43	12, 18, 32, 42	ringpropd 18582	. . . . 5 |- ( ph -> ( ( K ↾s s ) e. Ring <-> ( L ↾s s ) e. Ring ) )
44	5, 43	anbi12d 747	. . . 4 |- ( ph -> ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) <-> ( L e. Ring /\ ( L ↾s s ) e. Ring ) ) )
45	1, 2	eqtr3d 2658	. . . . . 6 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
46	45	sseq2d 3633	. . . . 5 |- ( ph -> ( s C_ ( Base ` K ) <-> s C_ ( Base ` L ) ) )
47	1, 2, 4	rngidpropd 18695	. . . . . 6 |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
48	47	eleq1d 2686	. . . . 5 |- ( ph -> ( ( 1r ` K ) e. s <-> ( 1r ` L ) e. s ) )
49	46, 48	anbi12d 747	. . . 4 |- ( ph -> ( ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) <-> ( s C_ ( Base ` L ) /\ ( 1r ` L ) e. s ) ) )
50	44, 49	anbi12d 747	. . 3 |- ( ph -> ( ( ( 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 ) ) ) )
51		eqid 2622	. . . 4 |- ( 1r ` K ) = ( 1r ` K )
52	9, 51	issubrg 18780	. . 3 |- ( s e. (SubRing ` K ) <-> ( ( K e. Ring /\ ( K ↾s s ) e. Ring ) /\ ( s C_ ( Base ` K ) /\ ( 1r ` K ) e. s ) ) )
53		eqid 2622	. . . 4 |- ( 1r ` L ) = ( 1r ` L )
54	15, 53	issubrg 18780	. . 3 |- ( s e. (SubRing ` L ) <-> ( ( L e. Ring /\ ( L ↾s s ) e. Ring ) /\ ( s C_ ( Base ` L ) /\ ( 1r ` L ) e. s ) ) )
55	50, 52, 54	3bitr4g 303	. 2 |- ( ph -> ( s e. (SubRing ` K ) <-> s e. (SubRing ` L ) ) )
56	55	eqrdv 2620	1 |- ( ph -> (SubRing ` K ) = (SubRing ` L ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   1rcur 18501   Ringcrg 18547  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778

This theorem is referenced by:  ply1subrg  19567  subrgply1  19603