Theorem aprval 14529

Description: Expand Definition df-apr 14528. (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 4115	. . 3 |- ( X # Y <-> <. X , Y >. e. # )
2		aprval.ap	. . . . . 6 |- ( ph -> # = (#r ` R ) )
3		df-apr 14528	. . . . . . 7 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
4		fveq2 5675	. . . . . . . . . . 11 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5	4	eleq2d 2304	. . . . . . . . . 10 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
6	4	eleq2d 2304	. . . . . . . . . 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 5675	. . . . . . . . . . 11 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
9	8	oveqd 6075	. . . . . . . . . 10 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
10		fveq2 5675	. . . . . . . . . 10 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
11	9, 10	eleq12d 2305	. . . . . . . . 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 4181	. . . . . . 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 2829	. . . . . . 7 |- ( ph -> R e. _V )
16		basfn 13355	. . . . . . . . . 10 |- Base Fn _V
17		funfvex 5692	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
18	17	funfni 5463	. . . . . . . . . 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 4869	. . . . . . . . 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 4829	. . . . . . . . 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 4255	. . . . . . 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 5776	. . . . . 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 2267	. . . . 5 |- ( ph -> # = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
27	26	eleq2d 2304	. . . 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 2313	. . . . 5 |- ( ph -> X e. ( Base ` R ) )
31		aprval.y	. . . . . 6 |- ( ph -> Y e. B )
32	31, 29	eleqtrd 2313	. . . . 5 |- ( ph -> Y e. ( Base ` R ) )
33		oveq12 6067	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( x ( -g ` R ) y ) = ( X ( -g ` R ) Y ) )
34	33	eleq1d 2303	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ( x ( -g ` R ) y ) e. (Unit ` R ) <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
35	34	opelopab2a 4388	. . . . 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 6075	. . 3 |- ( ph -> ( X .- Y ) = ( X ( -g ` R ) Y ) )
41		aprval.u	. . 3 |- ( ph -> U = (Unit ` R ) )
42	40, 41	eleq12d 2305	. 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 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   <.cop 3697   class class class wbr 4114   {copab 4175   X. cxp 4752   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   -gcsg 13757   Ringcrg 14239  Unitcui 14331  #_rcapr 14527

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 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-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-apr 14528

This theorem is referenced by:  aprunit  14530  aprirr  14533  aprsym  14534  aprcotr  14535  aprnzr  14537  aprlring  14538  opprdrng  14558