Theorem subrngpropd 14236

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 1023	. . . . 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 1023	. . . . 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 13974	. . . . 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 3408	. . . . . . . . 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 2805	. . . . . . . . . 10 |- s e. _V
17	16	a1i 9	. . . . . . . . 9 |- ( ( ph /\ K e. Rng ) -> s e. _V )
18	13, 14, 15, 17	ressbasd 13155	. . . . . . . 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 3408	. . . . . . . . 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 13155	. . . . . . . 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 3394	. . . . . . . . 9 |- ( x e. ( s i^i B ) -> x e. B )
28		elinel2 3394	. . . . . . . . 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 13217	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` K ) = ( +g ` ( K ↾s s ) ) )
33	32	oveqdr 6046	. . . . . . . . 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 13217	. . . . . . . . . 10 |- ( ( ph /\ K e. Rng ) -> ( +g ` L ) = ( +g ` ( L ↾s s ) ) )
36	35	oveqdr 6046	. . . . . . . . 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 13233	. . . . . . . . . . 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 6046	. . . . . . . . 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 13233	. . . . . . . . . . 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 6046	. . . . . . . . 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 13974	. . . . . 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 3257	. . . . . . 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 1348	. . . . 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 712	. . 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 14219	. . 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 14219	. . 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 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   C_ wss 3200   ` cfv 5326  (class class class)co 6018   Basecbs 13087   ↾_s cress 13088   +g cplusg 13165   .rcmulr 13166  Rngcrng 13951  SubRngcsubrng 14217

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-cmn 13878  df-abl 13879  df-mgp 13940  df-rng 13952  df-subrng 14218

This theorem is referenced by: (None)