Theorem dvdsrex 13260

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 2178	. . 3 |- ( R e. SRing -> ( Base ` R ) = ( Base ` R ) )
2		eqidd 2178	. . 3 |- ( R e. SRing -> ( ||r ` R ) = ( ||r ` R ) )
3		id 19	. . 3 |- ( R e. SRing -> R e. SRing )
4		eqidd 2178	. . 3 |- ( R e. SRing -> ( .r ` R ) = ( .r ` R ) )
5	1, 2, 3, 4	dvdsrvald 13255	. 2 |- ( R e. SRing -> ( ||r ` R ) = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
6		basfn 12514	. . . . 5 |- Base Fn _V
7		elex 2748	. . . . 5 |- ( R e. SRing -> R e. _V )
8		funfvex 5532	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5316	. . . . 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 4740	. . . 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 2177	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
18		eqid 2177	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
19	17, 18	srgcl 13146	. . . . . . . . 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 1238	. . . . . . . 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 2255	. . . . . . 7 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
22	21	rexlimdvaa 2595	. . . . . 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 4278	. . . 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 4632	. . . 4 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
26	24, 25	sseqtrrdi 3204	. . 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 4143	. 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 2254	1 |- ( R e. SRing -> ( ||r ` R ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2737   {copab 4063   X. cxp 4624   Fn wfn 5211   ` cfv 5216  (class class class)co 5874   Basecbs 12456   .rcmulr 12531  SRingcsrg 13139   ||_rcdsr 13248

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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12701  df-mgm 12769  df-sgrp 12802  df-mnd 12812  df-mgp 13124  df-srg 13140  df-dvdsr 13251

This theorem is referenced by:  isunitd  13268