Theorem subrngpropd 14292

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 1024	. . . . 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 1024	. . . . 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 14030	. . . . 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 3410	. . . . . . . . 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 2232	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( K ↾s s ) = ( K ↾s s ) )
14		eqidd 2232	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( Base ` K ) = ( Base ` K ) )
15		simpr 110	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> K e. Rng )
16		vex 2806	. . . . . . . . . 10 |- s e. _V
17	16	a1i 9	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> s e. _V )
18	13, 14, 15, 17	ressbasd 13211	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` K ) ) = ( Base ` ( K ↾s s ) ) )
19	12, 18	eqtrd 2264	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( K ↾s s ) ) )
20	5	ineq2d 3410	. . . . . . . . 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 2232	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> ( L ↾s s ) = ( L ↾s s ) )
23		eqidd 2232	. . . . . . . . 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 13211	. . . . . . . 8 |- ( ( ph /\ K e. Rng ) -> ( s i^i ( Base ` L ) ) = ( Base ` ( L ↾s s ) ) )
26	21, 25	eqtrd 2264	. . . . . . 7 |- ( ( ph /\ K e. Rng ) -> ( s i^i B ) = ( Base ` ( L ↾s s ) ) )
27		elinel2 3396	. . . . . . . . 9 |- ( x e. ( s i^i B ) -> x e. B )
28		elinel2 3396	. . . . . . . . 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 2232	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` K ) )
32	13, 31, 17, 15	ressplusgd 13273	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
33	32	oveqdr 6056	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` ( K ↾s s ) ) y ) )
34		eqidd 2232	. . . . . . . . . . 11 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` L ) )
35	22, 34, 17, 24	ressplusgd 13273	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
36	35	oveqdr 6056	. . . . . . . . 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 2272	. . . . . . . 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 2231	. . . . . . . . . . . 12 |- ( K ↾s s ) = ( K ↾s s )
41		eqid 2231	. . . . . . . . . . . 12 |- ( .r ` K ) = ( .r ` K )
42	40, 41	ressmulrg 13289	. . . . . . . . . . 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 6056	. . . . . . . . 9 |- ( ( ( ph /\ K e. Rng ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` ( K ↾s s ) ) y ) )
45		eqid 2231	. . . . . . . . . . . 12 |- ( L ↾s s ) = ( L ↾s s )
46		eqid 2231	. . . . . . . . . . . 12 |- ( .r ` L ) = ( .r ` L )
47	45, 46	ressmulrg 13289	. . . . . . . . . . 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 6056	. . . . . . . . 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 2272	. . . . . . . 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 14030	. . . . . 6 |- ( ( ph /\ K e. Rng ) -> ( ( K ↾s s ) e. Rng <-> ( L ↾s s ) e. Rng ) )
53	4, 5	eqtr3d 2266	. . . . . . . 8 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
54	53	sseq2d 3258	. . . . . . 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 1349	. . . . 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 713	. . 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 2231	. . . 4 |- ( Base ` K ) = ( Base ` K )
60	59	issubrng 14275	. . 3 |- ( s e. (SubRng ` K ) <-> ( K e. Rng /\ ( K ↾s s ) e. Rng /\ s C_ ( Base ` K ) ) )
61		eqid 2231	. . . 4 |- ( Base ` L ) = ( Base ` L )
62	61	issubrng 14275	. . 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 2229	1 |- ( ph -> (SubRng ` K ) = (SubRng ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13143   ↾_s cress 13144   +g cplusg 13221   .rcmulr 13222  Rngcrng 14007  SubRngcsubrng 14273

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-3 9246  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-mulr 13235  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-cmn 13934  df-abl 13935  df-mgp 13996  df-rng 14008  df-subrng 14274

This theorem is referenced by: (None)