Theorem dvrfval 18684

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
dvrval.b	|- B = ( Base ` R )
dvrval.t	|- .x. = ( .r ` R )
dvrval.u	|- U = (Unit ` R )
dvrval.i	|- I = ( invr ` R )
dvrval.d	|- ./ = (/r ` R )

Assertion

Ref	Expression
dvrfval	|- ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) )

Distinct variable groups:   x, y, B   x, I, y   x, R, y   x, .x. , y   x, U, y

Allowed substitution hints:   ./ ( x, y)

Proof of Theorem dvrfval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrval.d	. 2 |- ./ = (/r ` R )
2		fveq2 6191	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		dvrval.b	. . . . . 6 |- B = ( Base ` R )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( r = R -> ( Base ` r ) = B )
5		fveq2 6191	. . . . . 6 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
6		dvrval.u	. . . . . 6 |- U = (Unit ` R )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( r = R -> (Unit ` r ) = U )
8		fveq2 6191	. . . . . . 7 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
9		dvrval.t	. . . . . . 7 |- .x. = ( .r ` R )
10	8, 9	syl6eqr 2674	. . . . . 6 |- ( r = R -> ( .r ` r ) = .x. )
11		eqidd 2623	. . . . . 6 |- ( r = R -> x = x )
12		fveq2 6191	. . . . . . . 8 |- ( r = R -> ( invr ` r ) = ( invr ` R ) )
13		dvrval.i	. . . . . . . 8 |- I = ( invr ` R )
14	12, 13	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( invr ` r ) = I )
15	14	fveq1d 6193	. . . . . 6 |- ( r = R -> ( ( invr ` r ) ` y ) = ( I ` y ) )
16	10, 11, 15	oveq123d 6671	. . . . 5 |- ( r = R -> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) = ( x .x. ( I ` y ) ) )
17	4, 7, 16	mpt2eq123dv 6717	. . . 4 |- ( r = R -> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
18		df-dvr 18683	. . . 4 |- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
19		fvex 6201	. . . . . 6 |- ( Base ` R ) e. _V
20	3, 19	eqeltri 2697	. . . . 5 |- B e. _V
21		fvex 6201	. . . . . 6 |- (Unit ` R ) e. _V
22	6, 21	eqeltri 2697	. . . . 5 |- U e. _V
23	20, 22	mpt2ex 7247	. . . 4 |- ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) e. _V
24	17, 18, 23	fvmpt 6282	. . 3 |- ( R e. _V -> (/r ` R ) = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
25		fvprc 6185	. . . 4 |- ( -. R e. _V -> (/r ` R ) = (/) )
26		fvprc 6185	. . . . . . 7 |- ( -. R e. _V -> ( Base ` R ) = (/) )
27	3, 26	syl5eq 2668	. . . . . 6 |- ( -. R e. _V -> B = (/) )
28		eqid 2622	. . . . . 6 |- U = U
29		mpt2eq12 6715	. . . . . 6 |- ( ( B = (/) /\ U = U ) -> ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) = ( x e. (/) , y e. U |-> ( x .x. ( I ` y ) ) ) )
30	27, 28, 29	sylancl 694	. . . . 5 |- ( -. R e. _V -> ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) = ( x e. (/) , y e. U |-> ( x .x. ( I ` y ) ) ) )
31		mpt20 6725	. . . . 5 |- ( x e. (/) , y e. U |-> ( x .x. ( I ` y ) ) ) = (/)
32	30, 31	syl6eq 2672	. . . 4 |- ( -. R e. _V -> ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) = (/) )
33	25, 32	eqtr4d 2659	. . 3 |- ( -. R e. _V -> (/r ` R ) = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
34	24, 33	pm2.61i 176	. 2 |- (/r ` R ) = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) )
35	1, 34	eqtri 2644	1 |- ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   .rcmulr 15942  Unitcui 18639   invrcinvr 18671  /_rcdvr 18682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-dvr 18683

This theorem is referenced by:  dvrval  18685  cnflddiv  19776  dvrcn  21987