Theorem aprval 14428

Description: Expand Definition df-apr 14427. (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 4110	. . 3 |- ( X # Y <-> <. X , Y >. e. # )
2		aprval.ap	. . . . . 6 |- ( ph -> # = (#r ` R ) )
3		df-apr 14427	. . . . . . 7 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
4		fveq2 5670	. . . . . . . . . . 11 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5	4	eleq2d 2302	. . . . . . . . . 10 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
6	4	eleq2d 2302	. . . . . . . . . 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 5670	. . . . . . . . . . 11 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
9	8	oveqd 6067	. . . . . . . . . 10 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
10		fveq2 5670	. . . . . . . . . 10 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
11	9, 10	eleq12d 2303	. . . . . . . . 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 4176	. . . . . . 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 2827	. . . . . . 7 |- ( ph -> R e. _V )
16		basfn 13271	. . . . . . . . . 10 |- Base Fn _V
17		funfvex 5687	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
18	17	funfni 5458	. . . . . . . . . 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 4864	. . . . . . . . 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 4824	. . . . . . . . 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 4250	. . . . . . 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 5771	. . . . . 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 2265	. . . . 5 |- ( ph -> # = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
27	26	eleq2d 2302	. . . 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 2311	. . . . 5 |- ( ph -> X e. ( Base ` R ) )
31		aprval.y	. . . . . 6 |- ( ph -> Y e. B )
32	31, 29	eleqtrd 2311	. . . . 5 |- ( ph -> Y e. ( Base ` R ) )
33		oveq12 6059	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( x ( -g ` R ) y ) = ( X ( -g ` R ) Y ) )
34	33	eleq1d 2301	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ( x ( -g ` R ) y ) e. (Unit ` R ) <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
35	34	opelopab2a 4383	. . . . 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 6067	. . 3 |- ( ph -> ( X .- Y ) = ( X ( -g ` R ) Y ) )
41		aprval.u	. . 3 |- ( ph -> U = (Unit ` R ) )
42	40, 41	eleq12d 2303	. 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 2203   _Vcvv 2813   C_ wss 3211   <.cop 3692   class class class wbr 4109   {copab 4170   X. cxp 4747   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Basecbs 13212   -gcsg 13715   Ringcrg 14140  Unitcui 14231  #_rcapr 14426

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-apr 14427

This theorem is referenced by:  aprirr  14429  aprsym  14430  aprcotr  14431  aprnzr  14433  aprlring  14434