Theorem aprap 14290

Description: The relation given by df-apr 14285 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 14285	. . . 4 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
2		fveq2 5635	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2299	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4	2	eleq2d 2299	. . . . . . 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 5635	. . . . . . . 8 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
7	6	oveqd 6030	. . . . . . 7 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
8		fveq2 5635	. . . . . . 7 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
9	7, 8	eleq12d 2300	. . . . . 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 4153	. . . 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 2812	. . . 4 |- ( R e. LRing -> R e. _V )
13		basfn 13131	. . . . . . . 8 |- Base Fn _V
14	13	a1i 9	. . . . . . 7 |- ( R e. LRing -> Base Fn _V )
15		funfvex 5652	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
16	15	funfni 5429	. . . . . . 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 4838	. . . . . 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 4798	. . . . . 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 4227	. . . 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 5736	. . 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 3277	. 2 |- ( R e. LRing -> (#r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) )
25		eqidd 2230	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
26		eqidd 2230	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> (#r ` R ) = (#r ` R ) )
27		lringring 14198	. . . . 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 2229	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
31		eqid 2229	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
32	30, 31	lringnz 14199	. . . . 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 14287	. . 3 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> -. x (#r ` R ) x )
35	34	ralrimiva 2603	. 2 |- ( R e. LRing -> A. x e. ( Base ` R ) -. x (#r ` R ) x )
36		eqidd 2230	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( Base ` R ) = ( Base ` R ) )
37		eqidd 2230	. . . . 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 14288	. . . 4 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x (#r ` R ) y -> y (#r ` R ) x ) )
42	41	ralrimivva 2612	. . 3 |- ( R e. LRing -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( x (#r ` R ) y -> y (#r ` R ) x ) )
43		eqidd 2230	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( Base ` R ) = ( Base ` R ) )
44		eqidd 2230	. . . . 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 1027	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
47		simpr2 1028	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
48		simpr3 1029	. . . . 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 14289	. . . 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 2613	. . 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 7457	. 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 1206	1 |- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   _Vcvv 2800   C_ wss 3198   class class class wbr 4086   {copab 4147   X. cxp 4721   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   Ap wap 7456   Basecbs 13072   0gc0g 13329   -gcsg 13575   1rcur 13962   Ringcrg 13999  Unitcui 14090  LRingclring 14194  #_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-in1 617  ax-in2 618  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-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  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-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-tpos 6406  df-pap 7457  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-sbg 13578  df-cmn 13863  df-abl 13864  df-mgp 13924  df-ur 13963  df-srg 13967  df-ring 14001  df-oppr 14071  df-dvdsr 14092  df-unit 14093  df-invr 14125  df-dvr 14136  df-nzr 14184  df-lring 14195  df-apr 14285

This theorem is referenced by: (None)