Theorem dvrfvald 14091

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

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

Assertion

Ref	Expression
dvrfvald	|- ( ph -> ./ = ( 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   ph, x, y

Allowed substitution hints:   ./ ( x, y)

Proof of Theorem dvrfvald

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvr 14090	. . 3 |- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
2		fveq2 5626	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		fveq2 5626	. . . 4 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
4		fveq2 5626	. . . . 5 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5		eqidd 2230	. . . . 5 |- ( r = R -> x = x )
6		fveq2 5626	. . . . . 6 |- ( r = R -> ( invr ` r ) = ( invr ` R ) )
7	6	fveq1d 5628	. . . . 5 |- ( r = R -> ( ( invr ` r ) ` y ) = ( ( invr ` R ) ` y ) )
8	4, 5, 7	oveq123d 6021	. . . 4 |- ( r = R -> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
9	2, 3, 8	mpoeq123dv 6065	. . 3 |- ( r = R -> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) = ( x e. ( Base ` R ) , y e. (Unit ` R ) |-> ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) ) )
10		dvrfvald.r	. . . 4 |- ( ph -> R e. SRing )
11	10	elexd 2813	. . 3 |- ( ph -> R e. _V )
12		basfn 13086	. . . . 5 |- Base Fn _V
13		funfvex 5643	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5422	. . . . 5 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
15	12, 11, 14	sylancr 414	. . . 4 |- ( ph -> ( Base ` R ) e. _V )
16		eqidd 2230	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
17		eqidd 2230	. . . . . 6 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
18	16, 17, 10	unitssd 14067	. . . . 5 |- ( ph -> (Unit ` R ) C_ ( Base ` R ) )
19	15, 18	ssexd 4223	. . . 4 |- ( ph -> (Unit ` R ) e. _V )
20		mpoexga 6356	. . . 4 |- ( ( ( Base ` R ) e. _V /\ (Unit ` R ) e. _V ) -> ( x e. ( Base ` R ) , y e. (Unit ` R ) |-> ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) ) e. _V )
21	15, 19, 20	syl2anc 411	. . 3 |- ( ph -> ( x e. ( Base ` R ) , y e. (Unit ` R ) |-> ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) ) e. _V )
22	1, 9, 11, 21	fvmptd3 5727	. 2 |- ( ph -> (/r ` R ) = ( x e. ( Base ` R ) , y e. (Unit ` R ) |-> ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) ) )
23		dvrfvald.d	. 2 |- ( ph -> ./ = (/r ` R ) )
24		dvrfvald.b	. . 3 |- ( ph -> B = ( Base ` R ) )
25		dvrfvald.u	. . 3 |- ( ph -> U = (Unit ` R ) )
26		dvrfvald.t	. . . 4 |- ( ph -> .x. = ( .r ` R ) )
27		eqidd 2230	. . . 4 |- ( ph -> x = x )
28		dvrfvald.i	. . . . 5 |- ( ph -> I = ( invr ` R ) )
29	28	fveq1d 5628	. . . 4 |- ( ph -> ( I ` y ) = ( ( invr ` R ) ` y ) )
30	26, 27, 29	oveq123d 6021	. . 3 |- ( ph -> ( x .x. ( I ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
31	24, 25, 30	mpoeq123dv 6065	. 2 |- ( ph -> ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) = ( x e. ( Base ` R ) , y e. (Unit ` R ) |-> ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) ) )
32	22, 23, 31	3eqtr4d 2272	1 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   Basecbs 13027   .rcmulr 13106  SRingcsrg 13921  Unitcui 14045   invrcinvr 14078  /_rcdvr 14089

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-coll 4198  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-reu 2515  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-iun 3966  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-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  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  df-unit 14048  df-dvr 14090

This theorem is referenced by:  dvrvald  14092