Theorem dvdsrvald 14051

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 14047	. . 3 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
2		fveq2 5626	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2299	. . . . 5 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4		fveq2 5626	. . . . . . . 8 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5	4	oveqd 6017	. . . . . . 7 |- ( r = R -> ( z ( .r ` r ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2238	. . . . . 6 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2745	. . . . 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 4149	. . 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 2813	. . 3 |- ( ph -> R e. _V )
12		basfn 13086	. . . . . 6 |- Base Fn _V
13		funfvex 5643	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5422	. . . . . 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 4832	. . . . 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 2229	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
23		eqid 2229	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
24	22, 23	srgcl 13928	. . . . . . . . . 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 1271	. . . . . . . . 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 2307	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
27	26	rexlimdvaa 2649	. . . . . . 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 4366	. . . . 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 4724	. . . . 5 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
31	29, 30	sseqtrrdi 3273	. . . 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 4223	. . 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 5727	. 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 2299	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` R ) ) )
37		dvdsrvald.3	. . . . . . 7 |- ( ph -> .x. = ( .r ` R ) )
38	37	oveqd 6017	. . . . . 6 |- ( ph -> ( z .x. x ) = ( z ( .r ` R ) x ) )
39	38	eqeq1d 2238	. . . . 5 |- ( ph -> ( ( z .x. x ) = y <-> ( z ( .r ` R ) x ) = y ) )
40	35, 39	rexeqbidv 2745	. . . 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 4149	. 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 2272	1 |- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   {copab 4143   X. cxp 4716   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   Basecbs 13027   .rcmulr 13106  SRingcsrg 13921   ||_rcdsr 14044

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-mgp 13879  df-srg 13922  df-dvdsr 14047

This theorem is referenced by:  dvdsrd  14052  dvdsrex  14056  dvdsrpropdg  14105  dvdsrzring  14561