Theorem dvdsrvald 14238

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 14233	. . 3 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
2		fveq2 5670	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2302	. . . . 5 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4		fveq2 5670	. . . . . . . 8 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5	4	oveqd 6067	. . . . . . 7 |- ( r = R -> ( z ( .r ` r ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2241	. . . . . 6 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2758	. . . . 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 4176	. . 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 2827	. . 3 |- ( ph -> R e. _V )
12		basfn 13271	. . . . . 6 |- Base Fn _V
13		funfvex 5687	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5458	. . . . . 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 4864	. . . . 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 533	. . . . . . . . 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 531	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> z e. ( Base ` R ) )
21		simplr 529	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> x e. ( Base ` R ) )
22		eqid 2232	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
23		eqid 2232	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
24	22, 23	srgcl 14114	. . . . . . . . . 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 1274	. . . . . . . . 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 2310	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
27	26	rexlimdvaa 2661	. . . . . . 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 4397	. . . . 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 4755	. . . . 5 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
31	29, 30	sseqtrrdi 3287	. . . 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 4250	. . 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 5771	. 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 2302	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` R ) ) )
37		dvdsrvald.3	. . . . . . 7 |- ( ph -> .x. = ( .r ` R ) )
38	37	oveqd 6067	. . . . . 6 |- ( ph -> ( z .x. x ) = ( z ( .r ` R ) x ) )
39	38	eqeq1d 2241	. . . . 5 |- ( ph -> ( ( z .x. x ) = y <-> ( z ( .r ` R ) x ) = y ) )
40	35, 39	rexeqbidv 2758	. . . 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 4176	. 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 2275	1 |- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   E.wrex 2521   _Vcvv 2813   {copab 4170   X. cxp 4747   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Basecbs 13212   .rcmulr 13291  SRingcsrg 14107   ||_rcdsr 14230

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mgp 14065  df-srg 14108  df-dvdsr 14233

This theorem is referenced by:  dvdsrd  14239  dvdsrex  14243  dvdsrpropdg  14292  dvdsrzring  14751