Theorem dvdsrex 14099

Description: Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)

Assertion

Ref	Expression
dvdsrex	|- ( R e. SRing -> ( ||r ` R ) e. _V )

Proof of Theorem dvdsrex

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

Step	Hyp	Ref	Expression
1		eqidd 2230	. . 3 |- ( R e. SRing -> ( Base ` R ) = ( Base ` R ) )
2		eqidd 2230	. . 3 |- ( R e. SRing -> ( ||r ` R ) = ( ||r ` R ) )
3		id 19	. . 3 |- ( R e. SRing -> R e. SRing )
4		eqidd 2230	. . 3 |- ( R e. SRing -> ( .r ` R ) = ( .r ` R ) )
5	1, 2, 3, 4	dvdsrvald 14094	. 2 |- ( R e. SRing -> ( ||r ` R ) = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
6		basfn 13128	. . . . 5 |- Base Fn _V
7		elex 2812	. . . . 5 |- ( R e. SRing -> R e. _V )
8		funfvex 5650	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5427	. . . . 5 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	6, 7, 9	sylancr 414	. . . 4 |- ( R e. SRing -> ( Base ` R ) e. _V )
11		xpexg 4836	. . . 4 |- ( ( ( Base ` R ) e. _V /\ ( Base ` R ) e. _V ) -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
12	10, 10, 11	syl2anc 411	. . 3 |- ( R e. SRing -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
13		simprr 531	. . . . . . . 8 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> ( z ( .r ` R ) x ) = y )
14		simpll 527	. . . . . . . . 9 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> R e. SRing )
15		simprl 529	. . . . . . . . 9 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> z e. ( Base ` R ) )
16		simplr 528	. . . . . . . . 9 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> x e. ( Base ` R ) )
17		eqid 2229	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
18		eqid 2229	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
19	17, 18	srgcl 13970	. . . . . . . . 9 |- ( ( R e. SRing /\ z e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( z ( .r ` R ) x ) e. ( Base ` R ) )
20	14, 15, 16, 19	syl3anc 1271	. . . . . . . 8 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> ( z ( .r ` R ) x ) e. ( Base ` R ) )
21	13, 20	eqeltrrd 2307	. . . . . . 7 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
22	21	rexlimdvaa 2649	. . . . . 6 |- ( ( R e. SRing /\ x e. ( Base ` R ) ) -> ( E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y -> y e. ( Base ` R ) ) )
23	22	imdistanda 448	. . . . 5 |- ( 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 ) ) ) )
24	23	ssopab2dv 4369	. . . 4 |- ( 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 ) ) } )
25		df-xp 4727	. . . 4 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
26	24, 25	sseqtrrdi 3274	. . 3 |- ( 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 ) ) )
27	12, 26	ssexd 4225	. 2 |- ( R e. SRing -> { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } e. _V )
28	5, 27	eqeltrd 2306	1 |- ( R e. SRing -> ( ||r ` R ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2800   {copab 4145   X. cxp 4719   Fn wfn 5317   ` cfv 5322  (class class class)co 6011   Basecbs 13069   .rcmulr 13148  SRingcsrg 13963   ||_rcdsr 14086

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-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-pre-ltirr 8132  ax-pre-ltadd 8136

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-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-iota 5282  df-fun 5324  df-fn 5325  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-pnf 8204  df-mnf 8205  df-ltxr 8207  df-inn 9132  df-2 9190  df-3 9191  df-ndx 13072  df-slot 13073  df-base 13075  df-sets 13076  df-plusg 13160  df-mulr 13161  df-0g 13328  df-mgm 13426  df-sgrp 13472  df-mnd 13487  df-mgp 13921  df-srg 13964  df-dvdsr 14089

This theorem is referenced by:  isunitd  14107