Theorem aprprop 14461

Description: If two structures have the same ring components (properties), df-apr 14450 generates the same relation for both of them. (Contributed by Jim Kingdon, 31-May-2026.)

Hypotheses

Ref	Expression
aprprop.b	|- ( Base ` K ) = ( Base ` L )
aprprop.p	|- ( +g ` K ) = ( +g ` L )
aprprop.m	|- ( .r ` K ) = ( .r ` L )

Assertion

Ref	Expression
aprprop	|- ( K e. Ring -> (#r ` K ) = (#r ` L ) )

Proof of Theorem aprprop

Dummy variables x y r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aprprop.b	. . . . . . 7 |- ( Base ` K ) = ( Base ` L )
2	1	a1i 9	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` L ) )
3	2	eleq2d 2304	. . . . 5 |- ( K e. Ring -> ( x e. ( Base ` K ) <-> x e. ( Base ` L ) ) )
4	2	eleq2d 2304	. . . . 5 |- ( K e. Ring -> ( y e. ( Base ` K ) <-> y e. ( Base ` L ) ) )
5	3, 4	anbi12d 473	. . . 4 |- ( K e. Ring -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) <-> ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) ) )
6		aprprop.p	. . . . . . . 8 |- ( +g ` K ) = ( +g ` L )
7	6	a1i 9	. . . . . . 7 |- ( K e. Ring -> ( +g ` K ) = ( +g ` L ) )
8		id 19	. . . . . . 7 |- ( K e. Ring -> K e. Ring )
9		aprprop.m	. . . . . . . . 9 |- ( .r ` K ) = ( .r ` L )
10	1, 6, 9	ringprop 14205	. . . . . . . 8 |- ( K e. Ring <-> L e. Ring )
11	10	biimpi 120	. . . . . . 7 |- ( K e. Ring -> L e. Ring )
12	2, 7, 8, 11	grpsubpropdg 13838	. . . . . 6 |- ( K e. Ring -> ( -g ` K ) = ( -g ` L ) )
13	12	oveqd 6069	. . . . 5 |- ( K e. Ring -> ( x ( -g ` K ) y ) = ( x ( -g ` L ) y ) )
14		eqidd 2235	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` K ) )
15	9	a1i 9	. . . . . . 7 |- ( K e. Ring -> ( .r ` K ) = ( .r ` L ) )
16	15	oveqdr 6080	. . . . . 6 |- ( ( K e. Ring /\ ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
17	14, 2, 16, 8, 11	unitpropdg 14315	. . . . 5 |- ( K e. Ring -> (Unit ` K ) = (Unit ` L ) )
18	13, 17	eleq12d 2305	. . . 4 |- ( K e. Ring -> ( ( x ( -g ` K ) y ) e. (Unit ` K ) <-> ( x ( -g ` L ) y ) e. (Unit ` L ) ) )
19	5, 18	anbi12d 473	. . 3 |- ( K e. Ring -> ( ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) <-> ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) ) )
20	19	opabbidv 4178	. 2 |- ( K e. Ring -> { <. x , y >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) } = { <. x , y >. | ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) } )
21		df-apr 14450	. . 3 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
22		fveq2 5672	. . . . . . 7 |- ( r = K -> ( Base ` r ) = ( Base ` K ) )
23	22	eleq2d 2304	. . . . . 6 |- ( r = K -> ( x e. ( Base ` r ) <-> x e. ( Base ` K ) ) )
24	22	eleq2d 2304	. . . . . 6 |- ( r = K -> ( y e. ( Base ` r ) <-> y e. ( Base ` K ) ) )
25	23, 24	anbi12d 473	. . . . 5 |- ( r = K -> ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) <-> ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) )
26		fveq2 5672	. . . . . . 7 |- ( r = K -> ( -g ` r ) = ( -g ` K ) )
27	26	oveqd 6069	. . . . . 6 |- ( r = K -> ( x ( -g ` r ) y ) = ( x ( -g ` K ) y ) )
28		fveq2 5672	. . . . . 6 |- ( r = K -> (Unit ` r ) = (Unit ` K ) )
29	27, 28	eleq12d 2305	. . . . 5 |- ( r = K -> ( ( x ( -g ` r ) y ) e. (Unit ` r ) <-> ( x ( -g ` K ) y ) e. (Unit ` K ) ) )
30	25, 29	anbi12d 473	. . . 4 |- ( r = K -> ( ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) <-> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) ) )
31	30	opabbidv 4178	. . 3 |- ( r = K -> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } = { <. x , y >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) } )
32		elex 2827	. . 3 |- ( K e. Ring -> K e. _V )
33		basfn 13292	. . . . . 6 |- Base Fn _V
34		funfvex 5689	. . . . . . 7 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
35	34	funfni 5460	. . . . . 6 |- ( ( Base Fn _V /\ K e. _V ) -> ( Base ` K ) e. _V )
36	33, 32, 35	sylancr 414	. . . . 5 |- ( K e. Ring -> ( Base ` K ) e. _V )
37	36, 36	xpexd 4867	. . . 4 |- ( K e. Ring -> ( ( Base ` K ) X. ( Base ` K ) ) e. _V )
38		opabssxp 4826	. . . . 5 |- { <. x , y >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) } C_ ( ( Base ` K ) X. ( Base ` K ) )
39	38	a1i 9	. . . 4 |- ( K e. Ring -> { <. x , y >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) } C_ ( ( Base ` K ) X. ( Base ` K ) ) )
40	37, 39	ssexd 4252	. . 3 |- ( K e. Ring -> { <. x , y >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) } e. _V )
41	21, 31, 32, 40	fvmptd3 5773	. 2 |- ( K e. Ring -> (#r ` K ) = { <. x , y >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ ( x ( -g ` K ) y ) e. (Unit ` K ) ) } )
42		fveq2 5672	. . . . . . 7 |- ( r = L -> ( Base ` r ) = ( Base ` L ) )
43	42	eleq2d 2304	. . . . . 6 |- ( r = L -> ( x e. ( Base ` r ) <-> x e. ( Base ` L ) ) )
44	42	eleq2d 2304	. . . . . 6 |- ( r = L -> ( y e. ( Base ` r ) <-> y e. ( Base ` L ) ) )
45	43, 44	anbi12d 473	. . . . 5 |- ( r = L -> ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) <-> ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) ) )
46		fveq2 5672	. . . . . . 7 |- ( r = L -> ( -g ` r ) = ( -g ` L ) )
47	46	oveqd 6069	. . . . . 6 |- ( r = L -> ( x ( -g ` r ) y ) = ( x ( -g ` L ) y ) )
48		fveq2 5672	. . . . . 6 |- ( r = L -> (Unit ` r ) = (Unit ` L ) )
49	47, 48	eleq12d 2305	. . . . 5 |- ( r = L -> ( ( x ( -g ` r ) y ) e. (Unit ` r ) <-> ( x ( -g ` L ) y ) e. (Unit ` L ) ) )
50	45, 49	anbi12d 473	. . . 4 |- ( r = L -> ( ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) <-> ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) ) )
51	50	opabbidv 4178	. . 3 |- ( r = L -> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } = { <. x , y >. | ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) } )
52	11	elexd 2829	. . 3 |- ( K e. Ring -> L e. _V )
53	1, 36	eqeltrrid 2322	. . . . 5 |- ( K e. Ring -> ( Base ` L ) e. _V )
54	53, 53	xpexd 4867	. . . 4 |- ( K e. Ring -> ( ( Base ` L ) X. ( Base ` L ) ) e. _V )
55		opabssxp 4826	. . . . 5 |- { <. x , y >. | ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) } C_ ( ( Base ` L ) X. ( Base ` L ) )
56	55	a1i 9	. . . 4 |- ( K e. Ring -> { <. x , y >. | ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) } C_ ( ( Base ` L ) X. ( Base ` L ) ) )
57	54, 56	ssexd 4252	. . 3 |- ( K e. Ring -> { <. x , y >. | ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) } e. _V )
58	21, 51, 52, 57	fvmptd3 5773	. 2 |- ( K e. Ring -> (#r ` L ) = { <. x , y >. | ( ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) /\ ( x ( -g ` L ) y ) e. (Unit ` L ) ) } )
59	20, 41, 58	3eqtr4d 2277	1 |- ( K e. Ring -> (#r ` K ) = (#r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   {copab 4172   X. cxp 4749   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13233   +g cplusg 13311   .rcmulr 13312   -gcsg 13736   Ringcrg 14161  Unitcui 14253  #_rcapr 14449

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-tpos 6478  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-sbg 13739  df-cmn 14024  df-abl 14025  df-mgp 14086  df-ur 14125  df-srg 14129  df-ring 14163  df-oppr 14233  df-dvdsr 14255  df-unit 14256  df-apr 14450

This theorem is referenced by:  drngprop  14477