Theorem dvdsrvald 14373

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 14368	. . 3 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
2		fveq2 5690	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3	2	eleq2d 2308	. . . . 5 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. ( Base ` R ) ) )
4		fveq2 5690	. . . . . . . 8 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5	4	oveqd 6092	. . . . . . 7 |- ( r = R -> ( z ( .r ` r ) x ) = ( z ( .r ` R ) x ) )
6	5	eqeq1d 2247	. . . . . 6 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z ( .r ` R ) x ) = y ) )
7	2, 6	rexeqbidv 2766	. . . . 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 477	. . . 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 4192	. . 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 2835	. . 3 |- ( ph -> R e. _V )
12		basfn 13389	. . . . . 6 |- Base Fn _V
13		funfvex 5707	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5478	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
15	12, 11, 14	sylancr 418	. . . . 5 |- ( ph -> ( Base ` R ) e. _V )
16		xpexg 4884	. . . . 5 |- ( ( ( Base ` R ) e. _V /\ ( Base ` R ) e. _V ) -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
17	15, 15, 16	syl2anc 415	. . . 4 |- ( ph -> ( ( Base ` R ) X. ( Base ` R ) ) e. _V )
18		simprr 537	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> ( z ( .r ` R ) x ) = y )
19	10	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> R e. SRing )
20		simprl 535	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> z e. ( Base ` R ) )
21		simplr 533	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> x e. ( Base ` R ) )
22		eqid 2238	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
23		eqid 2238	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
24	22, 23	srgcl 14248	. . . . . . . . . 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 1278	. . . . . . . . 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 2316	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` R ) ) /\ ( z e. ( Base ` R ) /\ ( z ( .r ` R ) x ) = y ) ) -> y e. ( Base ` R ) )
27	26	rexlimdvaa 2669	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` R ) ) -> ( E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y -> y e. ( Base ` R ) ) )
28	27	imdistanda 452	. . . . . 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 4416	. . . . 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 4775	. . . . 5 |- ( ( Base ` R ) X. ( Base ` R ) ) = { <. x , y >. | ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) }
31	29, 30	sseqtrrdi 3297	. . . 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 4268	. . 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 5793	. 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 2308	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` R ) ) )
37		dvdsrvald.3	. . . . . . 7 |- ( ph -> .x. = ( .r ` R ) )
38	37	oveqd 6092	. . . . . 6 |- ( ph -> ( z .x. x ) = ( z ( .r ` R ) x ) )
39	38	eqeq1d 2247	. . . . 5 |- ( ph -> ( ( z .x. x ) = y <-> ( z ( .r ` R ) x ) = y ) )
40	35, 39	rexeqbidv 2766	. . . 4 |- ( ph -> ( E. z e. B ( z .x. x ) = y <-> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
41	36, 40	anbi12d 477	. . 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 4192	. 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 2281	1 |- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   _Vcvv 2821   {copab 4186   X. cxp 4767   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   .rcmulr 13409  SRingcsrg 14241   ||_rcdsr 14365

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mgp 14195  df-srg 14242  df-dvdsr 14368

This theorem is referenced by:  dvdsrd  14374  dvdsrex  14378  dvdsrpropdg  14427  dvdsrzring  14910