Theorem reldvdsrsrg 13648

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 13645	. . . . 5 |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
2		fveq2 5558	. . . . . . . 8 |- ( w = R -> ( Base ` w ) = ( Base ` R ) )
3	2	eleq2d 2266	. . . . . . 7 |- ( w = R -> ( x e. ( Base ` w ) <-> x e. ( Base ` R ) ) )
4		fveq2 5558	. . . . . . . . . 10 |- ( w = R -> ( .r ` w ) = ( .r ` R ) )
5	4	oveqd 5939	. . . . . . . . 9 |- ( w = R -> ( z ( .r ` w ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2205	. . . . . . . 8 |- ( w = R -> ( ( z ( .r ` w ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2710	. . . . . . 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 4099	. . . . 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 2774	. . . . 5 |- ( R e. SRing -> R e. _V )
11		basfn 12736	. . . . . . . 8 |- Base Fn _V
12		funfvex 5575	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
13	12	funfni 5358	. . . . . . . 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 4777	. . . . . . 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 2196	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
22		eqid 2196	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
23	21, 22	srgcl 13526	. . . . . . . . . . . 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 1249	. . . . . . . . . . 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 2274	. . . . . . . . . 10 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> y e. ( Base ` R ) )
26	25	rexlimdva2 2617	. . . . . . . . 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 4313	. . . . . . 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 4669	. . . . . . 7 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
30	28, 29	sseqtrrdi 3232	. . . . . 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 4173	. . . . 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 5655	. . . 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 3219	. . 3 |- ( R e. SRing -> ( ||r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) )
34		xpss 4771	. . 3 |- ( ( Base ` R ) X. ( Base ` R ) ) C_ ( _V X. _V )
35	33, 34	sstrdi 3195	. 2 |- ( R e. SRing -> ( ||r ` R ) C_ ( _V X. _V ) )
36		df-rel 4670	. 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 1364   e. wcel 2167   E.wrex 2476   _Vcvv 2763   C_ wss 3157   {copab 4093   X. cxp 4661   Rel wrel 4668   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   .rcmulr 12756  SRingcsrg 13519   ||_rcdsr 13642

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mgp 13477  df-srg 13520  df-dvdsr 13645

This theorem is referenced by:  dvdsrd  13650  isunitd  13662  subrgdvds  13791