Theorem aprap 14133

Description: The relation given by df-apr 14128 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 14128	. . . 4 |- #r = ( r e. _V |-> { <. x , y >. | ( ( x e. ( Base ` r ) /\ y e. ( Base ` r ) ) /\ ( x ( -g ` r ) y ) e. (Unit ` r ) ) } )
2		fveq2 5594	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2276	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4	2	eleq2d 2276	. . . . . . 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 5594	. . . . . . . 8 |- ( r = R -> ( -g ` r ) = ( -g ` R ) )
7	6	oveqd 5979	. . . . . . 7 |- ( r = R -> ( x ( -g ` r ) y ) = ( x ( -g ` R ) y ) )
8		fveq2 5594	. . . . . . 7 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
9	7, 8	eleq12d 2277	. . . . . 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 4121	. . . 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 2785	. . . 4 |- ( R e. LRing -> R e. _V )
13		basfn 12975	. . . . . . . 8 |- Base Fn _V
14	13	a1i 9	. . . . . . 7 |- ( R e. LRing -> Base Fn _V )
15		funfvex 5611	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
16	15	funfni 5390	. . . . . . 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 4802	. . . . . 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 4762	. . . . . 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 4195	. . . 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 5691	. . 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 3249	. 2 |- ( R e. LRing -> (#r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) )
25		eqidd 2207	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
26		eqidd 2207	. . . 4 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> (#r ` R ) = (#r ` R ) )
27		lringring 14041	. . . . 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 2206	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
31		eqid 2206	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
32	30, 31	lringnz 14042	. . . . 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 14130	. . 3 |- ( ( R e. LRing /\ x e. ( Base ` R ) ) -> -. x (#r ` R ) x )
35	34	ralrimiva 2580	. 2 |- ( R e. LRing -> A. x e. ( Base ` R ) -. x (#r ` R ) x )
36		eqidd 2207	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( Base ` R ) = ( Base ` R ) )
37		eqidd 2207	. . . . 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 14131	. . . 4 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x (#r ` R ) y -> y (#r ` R ) x ) )
42	41	ralrimivva 2589	. . 3 |- ( R e. LRing -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( x (#r ` R ) y -> y (#r ` R ) x ) )
43		eqidd 2207	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( Base ` R ) = ( Base ` R ) )
44		eqidd 2207	. . . . 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 1006	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
47		simpr2 1007	. . . . 5 |- ( ( R e. LRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
48		simpr3 1008	. . . . 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 14132	. . . 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 2590	. . 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 7390	. 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 1185	1 |- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   A.wral 2485   _Vcvv 2773   C_ wss 3170   class class class wbr 4054   {copab 4115   X. cxp 4686   Fn wfn 5280   ` cfv 5285  (class class class)co 5962   Ap wap 7389   Basecbs 12917   0gc0g 13173   -gcsg 13419   1rcur 13806   Ringcrg 13843  Unitcui 13934  LRingclring 14037  #_rcapr 14127

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-tpos 6349  df-pap 7390  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-iress 12925  df-plusg 13007  df-mulr 13008  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-grp 13420  df-minusg 13421  df-sbg 13422  df-cmn 13707  df-abl 13708  df-mgp 13768  df-ur 13807  df-srg 13811  df-ring 13845  df-oppr 13915  df-dvdsr 13936  df-unit 13937  df-invr 13968  df-dvr 13979  df-nzr 14027  df-lring 14038  df-apr 14128

This theorem is referenced by: (None)