Theorem reldvdsrsrg 13261

Description: The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)

Assertion

Ref	Expression
reldvdsrsrg	|- ( R e. SRing -> Rel ( ||r ` R ) )

Proof of Theorem reldvdsrsrg

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

Step	Hyp	Ref	Expression
1		df-dvdsr 13258	. . . . 5 |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
2		fveq2 5516	. . . . . . . 8 |- ( w = R -> ( Base ` w ) = ( Base ` R ) )
3	2	eleq2d 2247	. . . . . . 7 |- ( w = R -> ( x e. ( Base ` w ) <-> x e. ( Base ` R ) ) )
4		fveq2 5516	. . . . . . . . . 10 |- ( w = R -> ( .r ` w ) = ( .r ` R ) )
5	4	oveqd 5892	. . . . . . . . 9 |- ( w = R -> ( z ( .r ` w ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2186	. . . . . . . 8 |- ( w = R -> ( ( z ( .r ` w ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2686	. . . . . . 7 |- ( w = R -> ( E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y <-> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
8	3, 7	anbi12d 473	. . . . . 6 |- ( w = R -> ( ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) <-> ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) ) )
9	8	opabbidv 4070	. . . . 5 |- ( w = R -> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
10		elex 2749	. . . . 5 |- ( R e. SRing -> R e. _V )
11		basfn 12520	. . . . . . . 8 |- Base Fn _V
12		funfvex 5533	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
13	12	funfni 5317	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
14	11, 10, 13	sylancr 414	. . . . . . 7 |- ( R e. SRing -> ( Base ` R ) e. _V )
15		xpexg 4741	. . . . . . 7 |- ( ( ( Base ` R ) e. _V /\ ( Base ` R ) e. _V ) -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
16	14, 14, 15	syl2anc 411	. . . . . 6 |- ( R e. SRing -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
17		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> ( z ( .r ` R ) x ) = y )
18		simplll 533	. . . . . . . . . . . 12 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> R e. SRing )
19		simplr 528	. . . . . . . . . . . 12 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> z e. ( Base ` R ) )
20		simpllr 534	. . . . . . . . . . . 12 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> x e. ( Base ` R ) )
21		eqid 2177	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
22		eqid 2177	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
23	21, 22	srgcl 13153	. . . . . . . . . . . 12 |- ( ( R e. SRing /\ z e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( z ( .r ` R ) x ) e. ( Base ` R ) )
24	18, 19, 20, 23	syl3anc 1238	. . . . . . . . . . 11 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> ( z ( .r ` R ) x ) e. ( Base ` R ) )
25	17, 24	eqeltrrd 2255	. . . . . . . . . 10 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> y e. ( Base ` R ) )
26	25	rexlimdva2 2597	. . . . . . . . 9 |- ( ( R e. SRing /\ x e. ( Base ` R ) ) -> ( E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y -> y e. ( Base ` R ) ) )
27	26	imdistanda 448	. . . . . . . 8 |- ( R e. SRing -> ( ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) -> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) )
28	27	ssopab2dv 4279	. . . . . . 7 |- ( R e. SRing -> { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } C_ { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) } )
29		df-xp 4633	. . . . . . 7 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
30	28, 29	sseqtrrdi 3205	. . . . . 6 |- ( R e. SRing -> { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } C_ ( ( Base ` R ) X. ( Base ` R ) ) )
31	16, 30	ssexd 4144	. . . . 5 |- ( R e. SRing -> { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } e. _V )
32	1, 9, 10, 31	fvmptd3 5610	. . . 4 |- ( R e. SRing -> ( ||r ` R ) = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
33	32, 30	eqsstrd 3192	. . 3 |- ( R e. SRing -> ( ||r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) )
34		xpss 4735	. . 3 |- ( ( Base ` R ) X. ( Base ` R ) ) C_ ( _V X. _V )
35	33, 34	sstrdi 3168	. 2 |- ( R e. SRing -> ( ||r ` R ) C_ ( _V X. _V ) )
36		df-rel 4634	. 2 |- ( Rel ( ||r ` R ) <-> ( ||r ` R ) C_ ( _V X. _V ) )
37	35, 36	sylibr 134	1 |- ( R e. SRing -> Rel ( ||r ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2738   C_ wss 3130   {copab 4064   X. cxp 4625   Rel wrel 4632   Fn wfn 5212   ` cfv 5217  (class class class)co 5875   Basecbs 12462   .rcmulr 12537  SRingcsrg 13146   ||_rcdsr 13255

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-mgp 13131  df-srg 13147  df-dvdsr 13258

This theorem is referenced by:  dvdsrd  13263  isunitd  13275  subrgdvds  13356