Theorem aprap 14244

Description: The relation given by df-apr 14239 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 14239	. . . 4 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
2		fveq2 5626	. . . . . . . 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 5626	. . . . . . . 8 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
7	6	oveqd 6017	. . . . . . 7 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
8		fveq2 5626	. . . . . . 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 4149	. . . 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 2811	. . . 4 |- ( R e. LRing -> R e. _V )
13		basfn 13086	. . . . . . . 8 |- Base Fn _V
14	13	a1i 9	. . . . . . 7 |- ( R e. LRing -> Base Fn _V )
15		funfvex 5643	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
16	15	funfni 5422	. . . . . . 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 4832	. . . . . 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 4792	. . . . . 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 4223	. . . 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 5727	. . 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 3276	. 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 14152	. . . . 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 14153	. . . . 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 14241	. . 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 14242	. . . 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 14243	. . . 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 7430	. 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 2799   C_ wss 3197   class class class wbr 4082   {copab 4143   X. cxp 4716   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   Ap wap 7429   Basecbs 13027   0gc0g 13284   -gcsg 13530   1rcur 13917   Ringcrg 13954  Unitcui 14045  LRingclring 14148  #_rcapr 14238

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-tpos 6389  df-pap 7430  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-cmn 13818  df-abl 13819  df-mgp 13879  df-ur 13918  df-srg 13922  df-ring 13956  df-oppr 14026  df-dvdsr 14047  df-unit 14048  df-invr 14079  df-dvr 14090  df-nzr 14138  df-lring 14149  df-apr 14239

This theorem is referenced by: (None)