Theorem dvdsrex 14118

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 2232	. . 3 |- ( R e. SRing -> ( Base ` R ) = ( Base ` R ) )
2		eqidd 2232	. . 3 |- ( R e. SRing -> ( ||r ` R ) = ( ||r ` R ) )
3		id 19	. . 3 |- ( R e. SRing -> R e. SRing )
4		eqidd 2232	. . 3 |- ( R e. SRing -> ( .r ` R ) = ( .r ` R ) )
5	1, 2, 3, 4	dvdsrvald 14113	. 2 |- ( R e. SRing -> ( ||r ` R ) = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
6		basfn 13146	. . . . 5 |- Base Fn _V
7		elex 2814	. . . . 5 |- ( R e. SRing -> R e. _V )
8		funfvex 5656	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5432	. . . . 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 4840	. . . 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 533	. . . . . . . 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 531	. . . . . . . . 9 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> z e. ( Base ` R ) )
16		simplr 529	. . . . . . . . 9 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> x e. ( Base ` R ) )
17		eqid 2231	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
18		eqid 2231	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
19	17, 18	srgcl 13989	. . . . . . . . 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 1273	. . . . . . . 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 2309	. . . . . . 7 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
22	21	rexlimdvaa 2651	. . . . . 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 4373	. . . 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 4731	. . . 4 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
26	24, 25	sseqtrrdi 3276	. . 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 4229	. 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 2308	1 |- ( R e. SRing -> ( ||r ` R ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   {copab 4149   X. cxp 4723   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   Basecbs 13087   .rcmulr 13166  SRingcsrg 13982   ||_rcdsr 14105

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-plusg 13178  df-mulr 13179  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-mgp 13940  df-srg 13983  df-dvdsr 14108

This theorem is referenced by:  isunitd  14126