Theorem aprval 13914

Description: Expand Definition df-apr 13913. (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 4035	. . 3 |- ( X # Y <-> <. X , Y >. e. # )
2		aprval.ap	. . . . . 6 |- ( ph -> # = (#r ` R ) )
3		df-apr 13913	. . . . . . 7 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
4		fveq2 5561	. . . . . . . . . . 11 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5	4	eleq2d 2266	. . . . . . . . . 10 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
6	4	eleq2d 2266	. . . . . . . . . 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 5561	. . . . . . . . . . 11 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
9	8	oveqd 5942	. . . . . . . . . 10 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
10		fveq2 5561	. . . . . . . . . 10 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
11	9, 10	eleq12d 2267	. . . . . . . . 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 4100	. . . . . . 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 2776	. . . . . . 7 |- ( ph -> R e. _V )
16		basfn 12761	. . . . . . . . . 10 |- Base Fn _V
17		funfvex 5578	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
18	17	funfni 5361	. . . . . . . . . 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 4778	. . . . . . . . 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 4738	. . . . . . . . 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 4174	. . . . . . 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 5658	. . . . . 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 2229	. . . . 5 |- ( ph -> # = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
27	26	eleq2d 2266	. . . 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 2275	. . . . 5 |- ( ph -> X e. ( Base ` R ) )
31		aprval.y	. . . . . 6 |- ( ph -> Y e. B )
32	31, 29	eleqtrd 2275	. . . . 5 |- ( ph -> Y e. ( Base ` R ) )
33		oveq12 5934	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( x ( -g ` R ) y ) = ( X ( -g ` R ) Y ) )
34	33	eleq1d 2265	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ( x ( -g ` R ) y ) e. (Unit ` R ) <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
35	34	opelopab2a 4300	. . . . 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 5942	. . 3 |- ( ph -> ( X .- Y ) = ( X ( -g ` R ) Y ) )
41		aprval.u	. . 3 |- ( ph -> U = (Unit ` R ) )
42	40, 41	eleq12d 2267	. 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 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   <.cop 3626   class class class wbr 4034   {copab 4094   X. cxp 4662   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Basecbs 12703   -gcsg 13204   Ringcrg 13628  Unitcui 13719  #_rcapr 13912

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-apr 13913

This theorem is referenced by:  aprirr  13915  aprsym  13916  aprcotr  13917