Theorem aprprop 14584

Description: If two structures have the same ring components (properties), df-apr 14573 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 2308	. . . . 5 |- ( K e. Ring -> ( x e. ( Base ` K ) <-> x e. ( Base ` L ) ) )
4	2	eleq2d 2308	. . . . 5 |- ( K e. Ring -> ( y e. ( Base ` K ) <-> y e. ( Base ` L ) ) )
5	3, 4	anbi12d 477	. . . 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 14328	. . . . . . . 8 |- ( K e. Ring <-> L e. Ring )
11	10	biimpi 120	. . . . . . 7 |- ( K e. Ring -> L e. Ring )
12	2, 7, 8, 11	grpsubpropdg 13892	. . . . . 6 |- ( K e. Ring -> ( -g ` K ) = ( -g ` L ) )
13	12	oveqd 6096	. . . . 5 |- ( K e. Ring -> ( x ( -g ` K ) y ) = ( x ( -g ` L ) y ) )
14		eqidd 2239	. . . . . 6 |- ( K e. Ring -> ( Base ` K ) = ( Base ` K ) )
15	9	a1i 9	. . . . . . 7 |- ( K e. Ring -> ( .r ` K ) = ( .r ` L ) )
16	15	oveqdr 6107	. . . . . 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 14438	. . . . 5 |- ( K e. Ring -> (Unit ` K ) = (Unit ` L ) )
18	13, 17	eleq12d 2309	. . . 4 |- ( K e. Ring -> ( ( x ( -g ` K ) y ) e. (Unit ` K ) <-> ( x ( -g ` L ) y ) e. (Unit ` L ) ) )
19	5, 18	anbi12d 477	. . 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 4195	. 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 14573	. . 3 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
22		fveq2 5693	. . . . . . 7 |- ( r = K -> ( Base ` r ) = ( Base ` K ) )
23	22	eleq2d 2308	. . . . . 6 |- ( r = K -> ( x e. ( Base ` r ) <-> x e. ( Base ` K ) ) )
24	22	eleq2d 2308	. . . . . 6 |- ( r = K -> ( y e. ( Base ` r ) <-> y e. ( Base ` K ) ) )
25	23, 24	anbi12d 477	. . . . 5 |- ( r = K -> ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) <-> ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) )
26		fveq2 5693	. . . . . . 7 |- ( r = K -> ( -g ` r ) = ( -g ` K ) )
27	26	oveqd 6096	. . . . . 6 |- ( r = K -> ( x ( -g ` r ) y ) = ( x ( -g ` K ) y ) )
28		fveq2 5693	. . . . . 6 |- ( r = K -> (Unit ` r ) = (Unit ` K ) )
29	27, 28	eleq12d 2309	. . . . 5 |- ( r = K -> ( ( x ( -g ` r ) y ) e. (Unit ` r ) <-> ( x ( -g ` K ) y ) e. (Unit ` K ) ) )
30	25, 29	anbi12d 477	. . . 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 4195	. . 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 2833	. . 3 |- ( K e. Ring -> K e. _V )
33		basfn 13394	. . . . . 6 |- Base Fn _V
34		funfvex 5710	. . . . . . 7 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
35	34	funfni 5481	. . . . . 6 |- ( ( Base Fn _V /\ K e. _V ) -> ( Base ` K ) e. _V )
36	33, 32, 35	sylancr 418	. . . . 5 |- ( K e. Ring -> ( Base ` K ) e. _V )
37	36, 36	xpexd 4888	. . . 4 |- ( K e. Ring -> ( ( Base ` K ) X. ( Base ` K ) ) e. _V )
38		opabssxp 4847	. . . . 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 4271	. . 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 5796	. 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 5693	. . . . . . 7 |- ( r = L -> ( Base ` r ) = ( Base ` L ) )
43	42	eleq2d 2308	. . . . . 6 |- ( r = L -> ( x e. ( Base ` r ) <-> x e. ( Base ` L ) ) )
44	42	eleq2d 2308	. . . . . 6 |- ( r = L -> ( y e. ( Base ` r ) <-> y e. ( Base ` L ) ) )
45	43, 44	anbi12d 477	. . . . 5 |- ( r = L -> ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) <-> ( x e. ( Base ` L ) /\ y e. ( Base ` L ) ) ) )
46		fveq2 5693	. . . . . . 7 |- ( r = L -> ( -g ` r ) = ( -g ` L ) )
47	46	oveqd 6096	. . . . . 6 |- ( r = L -> ( x ( -g ` r ) y ) = ( x ( -g ` L ) y ) )
48		fveq2 5693	. . . . . 6 |- ( r = L -> (Unit ` r ) = (Unit ` L ) )
49	47, 48	eleq12d 2309	. . . . 5 |- ( r = L -> ( ( x ( -g ` r ) y ) e. (Unit ` r ) <-> ( x ( -g ` L ) y ) e. (Unit ` L ) ) )
50	45, 49	anbi12d 477	. . . 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 4195	. . 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 2835	. . 3 |- ( K e. Ring -> L e. _V )
53	1, 36	eqeltrrid 2326	. . . . 5 |- ( K e. Ring -> ( Base ` L ) e. _V )
54	53, 53	xpexd 4888	. . . 4 |- ( K e. Ring -> ( ( Base ` L ) X. ( Base ` L ) ) e. _V )
55		opabssxp 4847	. . . . 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 4271	. . 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 5796	. 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 2281	1 |- ( K e. Ring -> (#r ` K ) = (#r ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {copab 4189   X. cxp 4770   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   Basecbs 13335   +g cplusg 13414   .rcmulr 13415   -gcsg 13790   Ringcrg 14283  Unitcui 14376  #_rcapr 14572

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-tpos 6510  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-cmn 14072  df-abl 14073  df-mgp 14201  df-ur 14246  df-srg 14251  df-ring 14285  df-oppr 14356  df-dvdsr 14378  df-unit 14379  df-apr 14573

This theorem is referenced by:  drngprop  14600