Theorem aprval 14286

Description: Expand Definition df-apr 14285. (Contributed by Jim Kingdon, 17-Feb-2025.)

Hypotheses

Ref	Expression
aprval.b	|- ( ph -> B = ( Base ` R ) )
aprval.ap	|- ( ph -> # = (#r ` R ) )
aprval.s	|- ( ph -> .- = ( -g ` R ) )
aprval.u	|- ( ph -> U = (Unit ` R ) )
aprval.r	|- ( ph -> R e. Ring )
aprval.x	|- ( ph -> X e. B )
aprval.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
aprval	|- ( ph -> ( X # Y <-> ( X .- Y ) e. U ) )

Proof of Theorem aprval

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

Step	Hyp	Ref	Expression
1		df-br 4087	. . 3 |- ( X # Y <-> <. X , Y >. e. # )
2		aprval.ap	. . . . . 6 |- ( ph -> # = (#r ` R ) )
3		df-apr 14285	. . . . . . 7 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
4		fveq2 5635	. . . . . . . . . . 11 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5	4	eleq2d 2299	. . . . . . . . . 10 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
6	4	eleq2d 2299	. . . . . . . . . 10 |- ( r = R -> ( y e. ( Base ` r ) <-> y e. ( Base ` R ) ) )
7	5, 6	anbi12d 473	. . . . . . . . 9 |- ( r = R -> ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) <-> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) )
8		fveq2 5635	. . . . . . . . . . 11 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
9	8	oveqd 6030	. . . . . . . . . 10 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
10		fveq2 5635	. . . . . . . . . 10 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
11	9, 10	eleq12d 2300	. . . . . . . . 9 |- ( r = R -> ( ( x ( -g ` r ) y ) e. (Unit ` r ) <-> ( x ( -g ` R ) y ) e. (Unit ` R ) ) )
12	7, 11	anbi12d 473	. . . . . . . 8 |- ( r = R -> ( ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) <-> ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) ) )
13	12	opabbidv 4153	. . . . . . 7 |- ( r = R -> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
14		aprval.r	. . . . . . . 8 |- ( ph -> R e. Ring )
15	14	elexd 2814	. . . . . . 7 |- ( ph -> R e. _V )
16		basfn 13131	. . . . . . . . . 10 |- Base Fn _V
17		funfvex 5652	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
18	17	funfni 5429	. . . . . . . . . 10 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
19	16, 15, 18	sylancr 414	. . . . . . . . 9 |- ( ph -> ( Base ` R ) e. _V )
20		xpexg 4838	. . . . . . . . 9 |- ( ( ( Base ` R ) e. _V /\ ( Base ` R ) e. _V ) -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
21	19, 19, 20	syl2anc 411	. . . . . . . 8 |- ( ph -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
22		opabssxp 4798	. . . . . . . . 9 |- { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } C_ ( ( Base ` R ) X. ( Base ` R ) )
23	22	a1i 9	. . . . . . . 8 |- ( ph -> { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } C_ ( ( Base ` R ) X. ( Base ` R ) ) )
24	21, 23	ssexd 4227	. . . . . . 7 |- ( ph -> { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } e. _V )
25	3, 13, 15, 24	fvmptd3 5736	. . . . . 6 |- ( ph -> (#r ` R ) = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
26	2, 25	eqtrd 2262	. . . . 5 |- ( ph -> # = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
27	26	eleq2d 2299	. . . 4 |- ( ph -> ( <. X , Y >. e. # <-> <. X , Y >. e. { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } ) )
28		aprval.x	. . . . . 6 |- ( ph -> X e. B )
29		aprval.b	. . . . . 6 |- ( ph -> B = ( Base ` R ) )
30	28, 29	eleqtrd 2308	. . . . 5 |- ( ph -> X e. ( Base ` R ) )
31		aprval.y	. . . . . 6 |- ( ph -> Y e. B )
32	31, 29	eleqtrd 2308	. . . . 5 |- ( ph -> Y e. ( Base ` R ) )
33		oveq12 6022	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( x ( -g ` R ) y ) = ( X ( -g ` R ) Y ) )
34	33	eleq1d 2298	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ( x ( -g ` R ) y ) e. (Unit ` R ) <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
35	34	opelopab2a 4357	. . . . 5 |- ( ( X e. ( Base ` R ) /\ Y e. ( Base ` R ) ) -> ( <. X , Y >. e. { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
36	30, 32, 35	syl2anc 411	. . . 4 |- ( ph -> ( <. X , Y >. e. { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
37	27, 36	bitrd 188	. . 3 |- ( ph -> ( <. X , Y >. e. # <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
38	1, 37	bitrid 192	. 2 |- ( ph -> ( X # Y <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
39		aprval.s	. . . 4 |- ( ph -> .- = ( -g ` R ) )
40	39	oveqd 6030	. . 3 |- ( ph -> ( X .- Y ) = ( X ( -g ` R ) Y ) )
41		aprval.u	. . 3 |- ( ph -> U = (Unit ` R ) )
42	40, 41	eleq12d 2300	. 2 |- ( ph -> ( ( X .- Y ) e. U <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
43	38, 42	bitr4d 191	1 |- ( ph -> ( X # Y <-> ( X .- Y ) e. U ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   C_ wss 3198   <.cop 3670   class class class wbr 4086   {copab 4147   X. cxp 4721   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   Basecbs 13072   -gcsg 13575   Ringcrg 13999  Unitcui 14090  #_rcapr 14284

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-apr 14285

This theorem is referenced by:  aprirr  14287  aprsym  14288  aprcotr  14289