Theorem dvdsrvald 13826

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
dvdsrvald.1	|- ( ph -> B = ( Base ` R ) )
dvdsrvald.2	|- ( ph -> .|| = ( ||r ` R ) )
dvdsrvald.r	|- ( ph -> R e. SRing )
dvdsrvald.3	|- ( ph -> .x. = ( .r ` R ) )

Assertion

Ref	Expression
dvdsrvald	|- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )

Distinct variable groups:   x, y, .||   x, z, B, y   x, R, y, z   x, .x. , y, z   ph, x, y, z

Allowed substitution hint:   .|| ( z)

Proof of Theorem dvdsrvald

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvdsr 13822	. . 3 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
2		fveq2 5575	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2274	. . . . 5 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4		fveq2 5575	. . . . . . . 8 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5	4	oveqd 5960	. . . . . . 7 |- ( r = R -> ( z ( .r ` r ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2213	. . . . . 6 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2718	. . . . 5 |- ( r = R -> ( E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y <-> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
8	3, 7	anbi12d 473	. . . 4 |- ( r = R -> ( ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) <-> ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) ) )
9	8	opabbidv 4109	. . 3 |- ( r = R -> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
10		dvdsrvald.r	. . . 4 |- ( ph -> R e. SRing )
11	10	elexd 2784	. . 3 |- ( ph -> R e. _V )
12		basfn 12861	. . . . . 6 |- Base Fn _V
13		funfvex 5592	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5375	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
15	12, 11, 14	sylancr 414	. . . . 5 |- ( ph -> ( Base ` R ) e. _V )
16		xpexg 4788	. . . . 5 |- ( ( ( Base ` R ) e. _V /\ ( Base ` R ) e. _V ) -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
17	15, 15, 16	syl2anc 411	. . . 4 |- ( ph -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
18		simprr 531	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> ( z ( .r ` R ) x ) = y )
19	10	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> R e. SRing )
20		simprl 529	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> z e. ( Base ` R ) )
21		simplr 528	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> x e. ( Base ` R ) )
22		eqid 2204	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
23		eqid 2204	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
24	22, 23	srgcl 13703	. . . . . . . . . 10 |- ( ( R e. SRing /\ z e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( z ( .r ` R ) x ) e. ( Base ` R ) )
25	19, 20, 21, 24	syl3anc 1249	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> ( z ( .r ` R ) x ) e. ( Base ` R ) )
26	18, 25	eqeltrrd 2282	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
27	26	rexlimdvaa 2623	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` R ) ) -> ( E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y -> y e. ( Base ` R ) ) )
28	27	imdistanda 448	. . . . . 6 |- ( ph -> ( ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) -> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) )
29	28	ssopab2dv 4324	. . . . 5 |- ( ph -> { <. 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 ) ) } )
30		df-xp 4680	. . . . 5 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
31	29, 30	sseqtrrdi 3241	. . . 4 |- ( ph -> { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } C_ ( ( Base ` R ) X. ( Base ` R ) ) )
32	17, 31	ssexd 4183	. . 3 |- ( ph -> { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } e. _V )
33	1, 9, 11, 32	fvmptd3 5672	. 2 |- ( ph -> ( ||r ` R ) = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
34		dvdsrvald.2	. 2 |- ( ph -> .|| = ( ||r ` R ) )
35		dvdsrvald.1	. . . . 5 |- ( ph -> B = ( Base ` R ) )
36	35	eleq2d 2274	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` R ) ) )
37		dvdsrvald.3	. . . . . . 7 |- ( ph -> .x. = ( .r ` R ) )
38	37	oveqd 5960	. . . . . 6 |- ( ph -> ( z .x. x ) = ( z ( .r ` R ) x ) )
39	38	eqeq1d 2213	. . . . 5 |- ( ph -> ( ( z .x. x ) = y <-> ( z ( .r ` R ) x ) = y ) )
40	35, 39	rexeqbidv 2718	. . . 4 |- ( ph -> ( E. z e. B ( z .x. x ) = y <-> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
41	36, 40	anbi12d 473	. . 3 |- ( ph -> ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) <-> ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) ) )
42	41	opabbidv 4109	. 2 |- ( ph -> { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } = { <. x , y >. | ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) } )
43	33, 34, 42	3eqtr4d 2247	1 |- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   E.wrex 2484   _Vcvv 2771   {copab 4103   X. cxp 4672   Fn wfn 5265   ` cfv 5270  (class class class)co 5943   Basecbs 12803   .rcmulr 12881  SRingcsrg 13696   ||_rcdsr 13819

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-plusg 12893  df-mulr 12894  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-mgp 13654  df-srg 13697  df-dvdsr 13822

This theorem is referenced by:  dvdsrd  13827  dvdsrex  13831  dvdsrpropdg  13880  dvdsrzring  14336