Theorem subrngpropd 14093

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 1000	. . . . 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 1000	. . . . 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 13832	. . . . 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 3382	. . . . . . . . 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 2208	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2208	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( Base ` K ) = ( Base ` K ) )
15		simpr 110	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> K e. Rng )
16		vex 2779	. . . . . . . . . 10 |- s e. _V
17	16	a1i 9	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> s e. _V )
18	13, 14, 15, 17	ressbasd 13014	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2240	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3382	. . . . . . . . 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 2208	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2208	. . . . . . . . 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 13014	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
26	21, 25	eqtrd 2240	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
27		elinel2 3368	. . . . . . . . 9 |- ( x e. ( s i^i B ) -> x e. B )
28		elinel2 3368	. . . . . . . . 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 2208	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` K ) )
32	13, 31, 17, 15	ressplusgd 13076	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
33	32	oveqdr 5995	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
34		eqidd 2208	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` L ) )
35	22, 34, 17, 24	ressplusgd 13076	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
36	35	oveqdr 5995	. . . . . . . . 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 2248	. . . . . . . 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 2207	. . . . . . . . . . . 12 |- ( K ↾s s ) = ( K ↾s s )
41		eqid 2207	. . . . . . . . . . . 12 |- ( .r ` K ) = ( .r ` K )
42	40, 41	ressmulrg 13092	. . . . . . . . . . 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 5995	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
45		eqid 2207	. . . . . . . . . . . 12 |- ( L ↾s s ) = ( L ↾s s )
46		eqid 2207	. . . . . . . . . . . 12 |- ( .r ` L ) = ( .r ` L )
47	45, 46	ressmulrg 13092	. . . . . . . . . . 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 5995	. . . . . . . . 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 2248	. . . . . . . 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 13832	. . . . . 6 |- ( ( ph /\ K e. Rng ) -> ( ( K ↾s s ) e. Rng <-> ( L ↾s s ) e. Rng ) )
53	4, 5	eqtr3d 2242	. . . . . . . 8 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
54	53	sseq2d 3231	. . . . . . 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 1325	. . . . 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 707	. . 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 2207	. . . 4 |- ( Base ` K ) = ( Base ` K )
60	59	issubrng 14076	. . 3 |- ( s e. (SubRng ` K ) <-> ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) )
61		eqid 2207	. . . 4 |- ( Base ` L ) = ( Base ` L )
62	61	issubrng 14076	. . 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 2205	1 |- ( ph -> (SubRng ` K ) = (SubRng ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   C_ wss 3174   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   .rcmulr 13025  Rngcrng 13809  SubRngcsubrng 14074

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-cmn 13737  df-abl 13738  df-mgp 13798  df-rng 13810  df-subrng 14075

This theorem is referenced by: (None)