Theorem subrngpropd 13848

Description: If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)

Hypotheses

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

Assertion

Ref	Expression
subrngpropd	|- ( ph -> (SubRng ` K ) = (SubRng ` L ) )

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

Proof of Theorem subrngpropd

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . . . 5 |- ( ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) -> K e. Rng )
2	1	a1i 9	. . . 4 |- ( ph -> ( ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) -> K e. Rng ) )
3		simp1 999	. . . . 5 |- ( ( L e. Rng /\ ( L ↾s s ) e. Rng /\ s C_ ( Base ` L ) ) -> L e. Rng )
4		subrngpropd.1	. . . . . 6 |- ( ph -> B = ( Base ` K ) )
5		subrngpropd.2	. . . . . 6 |- ( ph -> B = ( Base ` L ) )
6		subrngpropd.3	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
7		subrngpropd.4	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
8	4, 5, 6, 7	rngpropd 13587	. . . . 5 |- ( ph -> ( K e. Rng <-> L e. Rng ) )
9	3, 8	imbitrrid 156	. . . 4 |- ( ph -> ( ( L e. Rng /\ ( L ↾s s ) e. Rng /\ s C_ ( Base ` L ) ) -> K e. Rng ) )
10	8	adantr 276	. . . . . 6 |- ( ( ph /\ K e. Rng ) -> ( K e. Rng <-> L e. Rng ) )
11	4	ineq2d 3365	. . . . . . . . 9 |- ( ph -> ( s i^i B ) = ( s i^i ( Base ` K ) ) )
12	11	adantr 276	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( s i^i ( Base ` K ) ) )
13		eqidd 2197	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2197	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( Base ` K ) = ( Base ` K ) )
15		simpr 110	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> K e. Rng )
16		vex 2766	. . . . . . . . . 10 |- s e. _V
17	16	a1i 9	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> s e. _V )
18	13, 14, 15, 17	ressbasd 12770	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2229	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3365	. . . . . . . . 9 |- ( ph -> ( s i^i B ) = ( s i^i ( Base ` L ) ) )
21	20	adantr 276	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( s i^i ( Base ` L ) ) )
22		eqidd 2197	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2197	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( Base ` L ) = ( Base ` L ) )
24	8	biimpa 296	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> L e. Rng )
25	22, 23, 24, 17	ressbasd 12770	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
26	21, 25	eqtrd 2229	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
27		elinel2 3351	. . . . . . . . 9 |- ( x e. ( s i^i B ) -> x e. B )
28		elinel2 3351	. . . . . . . . 9 |- ( y e. ( s i^i B ) -> y e. B )
29	27, 28	anim12i 338	. . . . . . . 8 |- ( ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) -> ( x e. B /\ y e. B ) )
30	6	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
31		eqidd 2197	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` K ) )
32	13, 31, 17, 15	ressplusgd 12831	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
33	32	oveqdr 5953	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
34		eqidd 2197	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` L ) )
35	22, 34, 17, 24	ressplusgd 12831	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
36	35	oveqdr 5953	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` L ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
37	30, 33, 36	3eqtr3d 2237	. . . . . . . 8 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` ( K ↾s s ) ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
38	29, 37	sylan2 286	. . . . . . 7 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) ) -> ( x ( +g ` ( K ↾s s ) ) y ) = ( x ( +g ` ( L ↾s s ) ) y ) )
39	7	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
40		eqid 2196	. . . . . . . . . . . 12 |- ( K ↾s s ) = ( K ↾s s )
41		eqid 2196	. . . . . . . . . . . 12 |- ( .r ` K ) = ( .r ` K )
42	40, 41	ressmulrg 12847	. . . . . . . . . . 11 |- ( ( s e. _V /\ K e. Rng ) -> ( .r ` K ) = ( .r ` ( K ↾s s ) ) )
43	17, 15, 42	syl2anc 411	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( .r ` K ) = ( .r ` ( K ↾s s ) ) )
44	43	oveqdr 5953	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
45		eqid 2196	. . . . . . . . . . . 12 |- ( L ↾s s ) = ( L ↾s s )
46		eqid 2196	. . . . . . . . . . . 12 |- ( .r ` L ) = ( .r ` L )
47	45, 46	ressmulrg 12847	. . . . . . . . . . 11 |- ( ( s e. _V /\ L e. Rng ) -> ( .r ` L ) = ( .r ` ( L ↾s s ) ) )
48	17, 24, 47	syl2anc 411	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( .r ` L ) = ( .r ` ( L ↾s s ) ) )
49	48	oveqdr 5953	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
50	39, 44, 49	3eqtr3d 2237	. . . . . . . 8 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` ( K ↾s s ) ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
51	29, 50	sylan2 286	. . . . . . 7 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. ( s i^i B ) /\ y e. ( s i^i B ) ) ) -> ( x ( .r ` ( K ↾s s ) ) y ) = ( x ( .r ` ( L ↾s s ) ) y ) )
52	19, 26, 38, 51	rngpropd 13587	. . . . . 6 |- ( ( ph /\ K e. Rng ) -> ( ( K ↾s s ) e. Rng <-> ( L ↾s s ) e. Rng ) )
53	4, 5	eqtr3d 2231	. . . . . . . 8 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
54	53	sseq2d 3214	. . . . . . 7 |- ( ph -> ( s C_ ( Base ` K ) <-> s C_ ( Base ` L ) ) )
55	54	adantr 276	. . . . . 6 |- ( ( ph /\ K e. Rng ) -> ( s C_ ( Base ` K ) <-> s C_ ( Base ` L ) ) )
56	10, 52, 55	3anbi123d 1323	. . . . 5 |- ( ( ph /\ K e. Rng ) -> ( ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) <-> ( L e. Rng /\ ( L ↾s s ) e. Rng /\ s C_ ( Base ` L ) ) ) )
57	56	ex 115	. . . 4 |- ( ph -> ( K e. Rng -> ( ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) <-> ( L e. Rng /\ ( L ↾s s ) e. Rng /\ s C_ ( Base ` L ) ) ) ) )
58	2, 9, 57	pm5.21ndd 706	. . 3 |- ( ph -> ( ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) <-> ( L e. Rng /\ ( L ↾s s ) e. Rng /\ s C_ ( Base ` L ) ) ) )
59		eqid 2196	. . . 4 |- ( Base ` K ) = ( Base ` K )
60	59	issubrng 13831	. . 3 |- ( s e. (SubRng ` K ) <-> ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) )
61		eqid 2196	. . . 4 |- ( Base ` L ) = ( Base ` L )
62	61	issubrng 13831	. . 3 |- ( s e. (SubRng ` L ) <-> ( L e. Rng /\ ( L ↾s s ) e. Rng /\ s C_ ( Base ` L ) ) )
63	58, 60, 62	3bitr4g 223	. 2 |- ( ph -> ( s e. (SubRng ` K ) <-> s e. (SubRng ` L ) ) )
64	63	eqrdv 2194	1 |- ( ph -> (SubRng ` K ) = (SubRng ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = 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  Rngcrng 13564  SubRngcsubrng 13829

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-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-subrng 13830

This theorem is referenced by: (None)