Theorem aprap 14019

Description: The relation given by df-apr 14014 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)

Assertion

Ref	Expression
aprap	|- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )

Proof of Theorem aprap

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

Step	Hyp	Ref	Expression
1		df-apr 14014	. . . 4 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
2		fveq2 5575	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2274	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4	2	eleq2d 2274	. . . . . . 7 |- ( r = R -> ( y e. ( Base ` r ) <-> y e. ( Base ` R ) ) )
5	3, 4	anbi12d 473	. . . . . 6 |- ( r = R -> ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) <-> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) )
6		fveq2 5575	. . . . . . . 8 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
7	6	oveqd 5960	. . . . . . 7 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
8		fveq2 5575	. . . . . . 7 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
9	7, 8	eleq12d 2275	. . . . . 6 |- ( r = R -> ( ( x ( -g ` r ) y ) e. (Unit ` r ) <-> ( x ( -g ` R ) y ) e. (Unit ` R ) ) )
10	5, 9	anbi12d 473	. . . . 5 |- ( 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 ) ) ) )
11	10	opabbidv 4109	. . . 4 |- ( 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 ) ) } )
12		elex 2782	. . . 4 |- ( R e. LRing -> R e. _V )
13		basfn 12861	. . . . . . . 8 |- Base Fn _V
14	13	a1i 9	. . . . . . 7 |- ( R e. LRing -> Base Fn _V )
15		funfvex 5592	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
16	15	funfni 5375	. . . . . . 7 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
17	14, 12, 16	syl2anc 411	. . . . . 6 |- ( R e. LRing -> ( Base ` R ) e. _V )
18		xpexg 4788	. . . . . 6 |- ( ( ( Base ` R ) e. _V /\ ( Base ` R ) e. _V ) -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
19	17, 17, 18	syl2anc 411	. . . . 5 |- ( R e. LRing -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
20		opabssxp 4748	. . . . . 6 |- { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } C_ ( ( Base ` R ) X. ( Base ` R ) )
21	20	a1i 9	. . . . 5 |- ( R e. LRing -> { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } C_ ( ( Base ` R ) X. ( Base ` R ) ) )
22	19, 21	ssexd 4183	. . . 4 |- ( R e. LRing -> { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } e. _V )
23	1, 11, 12, 22	fvmptd3 5672	. . 3 |- ( R e. LRing -> (#r ` R ) = { <. x , y >. | ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) /\ ( x ( -g ` R ) y ) e. (Unit ` R ) ) } )
24	23, 20	eqsstrdi 3244	. 2 |- ( R e. LRing -> (#r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) )
25		eqidd 2205	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
26		eqidd 2205	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> (#r ` R ) = (#r ` R ) )
27		lringring 13927	. . . . 5 |- ( R e. LRing -> R e. Ring )
28	27	adantr 276	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> R e. Ring )
29		simpr 110	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> x e. ( Base ` R ) )
30		eqid 2204	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
31		eqid 2204	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
32	30, 31	lringnz 13928	. . . . 5 |- ( R e. LRing -> ( 1r ` R ) =/= ( 0g ` R ) )
33	32	adantr 276	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> ( 1r ` R ) =/= ( 0g ` R ) )
34	25, 26, 28, 29, 33	aprirr 14016	. . 3 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> -. x (#r ` R ) x )
35	34	ralrimiva 2578	. 2 |- ( R e. LRing -> A. x e. ( Base ` R ) -. x (#r ` R ) x )
36		eqidd 2205	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( Base ` R ) = ( Base ` R ) )
37		eqidd 2205	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> (#r ` R ) = (#r ` R ) )
38	27	adantr 276	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> R e. Ring )
39		simprl 529	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
40		simprr 531	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
41	36, 37, 38, 39, 40	aprsym 14017	. . . 4 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x (#r ` R ) y -> y (#r ` R ) x ) )
42	41	ralrimivva 2587	. . 3 |- ( R e. LRing -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( x (#r ` R ) y -> y (#r ` R ) x ) )
43		eqidd 2205	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( Base ` R ) = ( Base ` R ) )
44		eqidd 2205	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> (#r ` R ) = (#r ` R ) )
45		simpl 109	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> R e. LRing )
46		simpr1 1005	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
47		simpr2 1006	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
48		simpr3 1007	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> z e. ( Base ` R ) )
49	43, 44, 45, 46, 47, 48	aprcotr 14018	. . . 4 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x (#r ` R ) y -> ( x (#r ` R ) z \/ y (#r ` R ) z ) ) )
50	49	ralrimivvva 2588	. . 3 |- ( R e. LRing -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( x (#r ` R ) y -> ( x (#r ` R ) z \/ y (#r ` R ) z ) ) )
51	42, 50	jca 306	. 2 |- ( R e. LRing -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( x (#r ` R ) y -> y (#r ` R ) x ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( x (#r ` R ) y -> ( x (#r ` R ) z \/ y (#r ` R ) z ) ) ) )
52		df-pap 7359	. 2 |- ( (#r ` R ) Ap ( Base ` R ) <-> ( ( (#r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) /\ A. x e. ( Base ` R ) -. x (#r ` R ) x ) /\ ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( x (#r ` R ) y -> y (#r ` R ) x ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( x (#r ` R ) y -> ( x (#r ` R ) z \/ y (#r ` R ) z ) ) ) ) )
53	24, 35, 51, 52	syl21anbrc 1184	1 |- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1372   e. wcel 2175   =/= wne 2375   A.wral 2483   _Vcvv 2771   C_ wss 3165   class class class wbr 4043   {copab 4103   X. cxp 4672   Fn wfn 5265   ` cfv 5270  (class class class)co 5943   Ap wap 7358   Basecbs 12803   0gc0g 13059   -gcsg 13305   1rcur 13692   Ringcrg 13729  Unitcui 13820  LRingclring 13923  #_rcapr 14013

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-tpos 6330  df-pap 7359  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-iress 12811  df-plusg 12893  df-mulr 12894  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-sbg 13308  df-cmn 13593  df-abl 13594  df-mgp 13654  df-ur 13693  df-srg 13697  df-ring 13731  df-oppr 13801  df-dvdsr 13822  df-unit 13823  df-invr 13854  df-dvr 13865  df-nzr 13913  df-lring 13924  df-apr 14014

This theorem is referenced by: (None)