Theorem dvdsrex 13910

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 2207	. . 3 |- ( R e. SRing -> ( Base ` R ) = ( Base ` R ) )
2		eqidd 2207	. . 3 |- ( R e. SRing -> ( ||r ` R ) = ( ||r ` R ) )
3		id 19	. . 3 |- ( R e. SRing -> R e. SRing )
4		eqidd 2207	. . 3 |- ( R e. SRing -> ( .r ` R ) = ( .r ` R ) )
5	1, 2, 3, 4	dvdsrvald 13905	. 2 |- ( R e. SRing -> ( ||r ` R ) = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
6		basfn 12940	. . . . 5 |- Base Fn _V
7		elex 2785	. . . . 5 |- ( R e. SRing -> R e. _V )
8		funfvex 5603	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5382	. . . . 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 4794	. . . 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 2206	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
18		eqid 2206	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
19	17, 18	srgcl 13782	. . . . . . . . 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 1250	. . . . . . . 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 2284	. . . . . . 7 |- ( ( ( R e. SRing /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
22	21	rexlimdvaa 2625	. . . . . 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 4330	. . . 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 4686	. . . 4 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
26	24, 25	sseqtrrdi 3244	. . 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 4189	. 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 2283	1 |- ( R e. SRing -> ( ||r ` R ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   E.wrex 2486   _Vcvv 2773   {copab 4109   X. cxp 4678   Fn wfn 5272   ` cfv 5277  (class class class)co 5954   Basecbs 12882   .rcmulr 12960  SRingcsrg 13775   ||_rcdsr 13898

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 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-addass 8040  ax-i2m1 8043  ax-0lt1 8044  ax-0id 8046  ax-rnegex 8047  ax-pre-ltirr 8050  ax-pre-ltadd 8054

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 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fn 5280  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-pnf 8122  df-mnf 8123  df-ltxr 8125  df-inn 9050  df-2 9108  df-3 9109  df-ndx 12885  df-slot 12886  df-base 12888  df-sets 12889  df-plusg 12972  df-mulr 12973  df-0g 13140  df-mgm 13238  df-sgrp 13284  df-mnd 13299  df-mgp 13733  df-srg 13776  df-dvdsr 13901

This theorem is referenced by:  isunitd  13918