Theorem subrngpropd 13560

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 13306	. . . . 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 3351	. . . . . . . . 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 2190	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2190	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( Base ` K ) = ( Base ` K ) )
15		simpr 110	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> K e. Rng )
16		vex 2755	. . . . . . . . . 10 |- s e. _V
17	16	a1i 9	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> s e. _V )
18	13, 14, 15, 17	ressbasd 12576	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2222	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3351	. . . . . . . . 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 2190	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2190	. . . . . . . . 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 12576	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
26	21, 25	eqtrd 2222	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
27		elinel2 3337	. . . . . . . . 9 |- ( x e. ( s i^i B ) -> x e. B )
28		elinel2 3337	. . . . . . . . 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 2190	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` K ) )
32	13, 31, 17, 15	ressplusgd 12637	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
33	32	oveqdr 5923	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
34		eqidd 2190	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` L ) )
35	22, 34, 17, 24	ressplusgd 12637	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
36	35	oveqdr 5923	. . . . . . . . 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 2230	. . . . . . . 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 2189	. . . . . . . . . . . 12 |- ( K ↾s s ) = ( K ↾s s )
41		eqid 2189	. . . . . . . . . . . 12 |- ( .r ` K ) = ( .r ` K )
42	40, 41	ressmulrg 12653	. . . . . . . . . . 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 5923	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
45		eqid 2189	. . . . . . . . . . . 12 |- ( L ↾s s ) = ( L ↾s s )
46		eqid 2189	. . . . . . . . . . . 12 |- ( .r ` L ) = ( .r ` L )
47	45, 46	ressmulrg 12653	. . . . . . . . . . 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 5923	. . . . . . . . 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 2230	. . . . . . . 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 13306	. . . . . 6 |- ( ( ph /\ K e. Rng ) -> ( ( K ↾s s ) e. Rng <-> ( L ↾s s ) e. Rng ) )
53	4, 5	eqtr3d 2224	. . . . . . . 8 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
54	53	sseq2d 3200	. . . . . . 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 2189	. . . 4 |- ( Base ` K ) = ( Base ` K )
60	59	issubrng 13543	. . 3 |- ( s e. (SubRng ` K ) <-> ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) )
61		eqid 2189	. . . 4 |- ( Base ` L ) = ( Base ` L )
62	61	issubrng 13543	. . 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 2187	1 |- ( ph -> (SubRng ` K ) = (SubRng ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   _Vcvv 2752   i^i cin 3143   C_ wss 3144   ` cfv 5235  (class class class)co 5895   Basecbs 12511   ↾_s cress 12512   +g cplusg 12586   .rcmulr 12587  Rngcrng 13283  SubRngcsubrng 13541

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-0g 12760  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-cmn 13222  df-abl 13223  df-mgp 13272  df-rng 13284  df-subrng 13542

This theorem is referenced by: (None)