Theorem reldvdsrsrg 13939

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 13936	. . . . 5 |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
2		fveq2 5594	. . . . . . . 8 |- ( w = R -> ( Base ` w ) = ( Base ` R ) )
3	2	eleq2d 2276	. . . . . . 7 |- ( w = R -> ( x e. ( Base ` w ) <-> x e. ( Base ` R ) ) )
4		fveq2 5594	. . . . . . . . . 10 |- ( w = R -> ( .r ` w ) = ( .r ` R ) )
5	4	oveqd 5979	. . . . . . . . 9 |- ( w = R -> ( z ( .r ` w ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2215	. . . . . . . 8 |- ( w = R -> ( ( z ( .r ` w ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2720	. . . . . . 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 4121	. . . . 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 2785	. . . . 5 |- ( R e. SRing -> R e. _V )
11		basfn 12975	. . . . . . . 8 |- Base Fn _V
12		funfvex 5611	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
13	12	funfni 5390	. . . . . . . 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 4802	. . . . . . 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 2206	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
22		eqid 2206	. . . . . . . . . . . . 13 |- ( .r ` R ) = ( .r ` R )
23	21, 22	srgcl 13817	. . . . . . . . . . . 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 1250	. . . . . . . . . . 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 2284	. . . . . . . . . 10 |- ( ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ z e. ( Base ` R ) ) /\ ( z ( .r ` R ) x ) = y ) -> y e. ( Base ` R ) )
26	25	rexlimdva2 2627	. . . . . . . . 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 4338	. . . . . . 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 4694	. . . . . . 7 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
30	28, 29	sseqtrrdi 3246	. . . . . 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 4195	. . . . 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 5691	. . . 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 3233	. . 3 |- ( R e. SRing -> ( ||r ` R ) C_ ( ( Base ` R ) X. ( Base ` R ) ) )
34		xpss 4796	. . 3 |- ( ( Base ` R ) X. ( Base ` R ) ) C_ ( _V X. _V )
35	33, 34	sstrdi 3209	. 2 |- ( R e. SRing -> ( ||r ` R ) C_ ( _V X. _V ) )
36		df-rel 4695	. 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 1373   e. wcel 2177   E.wrex 2486   _Vcvv 2773   C_ wss 3170   {copab 4115   X. cxp 4686   Rel wrel 4693   Fn wfn 5280   ` cfv 5285  (class class class)co 5962   Basecbs 12917   .rcmulr 12995  SRingcsrg 13810   ||_rcdsr 13933

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-sep 4173  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-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-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-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-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  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-plusg 13007  df-mulr 13008  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-mgp 13768  df-srg 13811  df-dvdsr 13936

This theorem is referenced by:  dvdsrd  13941  isunitd  13953  subrgdvds  14082