Theorem dvrfvald 13632

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 13631	. . 3 |- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
2		fveq2 5555	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		fveq2 5555	. . . 4 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
4		fveq2 5555	. . . . 5 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5		eqidd 2194	. . . . 5 |- ( r = R -> x = x )
6		fveq2 5555	. . . . . 6 |- ( r = R -> ( invr ` r ) = ( invr ` R ) )
7	6	fveq1d 5557	. . . . 5 |- ( r = R -> ( ( invr ` r ) ` y ) = ( ( invr ` R ) ` y ) )
8	4, 5, 7	oveq123d 5940	. . . 4 |- ( r = R -> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
9	2, 3, 8	mpoeq123dv 5981	. . 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 2773	. . 3 |- ( ph -> R e. _V )
12		basfn 12679	. . . . 5 |- Base Fn _V
13		funfvex 5572	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5355	. . . . 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 2194	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
17		eqidd 2194	. . . . . 6 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
18	16, 17, 10	unitssd 13608	. . . . 5 |- ( ph -> (Unit ` R ) C_ ( Base ` R ) )
19	15, 18	ssexd 4170	. . . 4 |- ( ph -> (Unit ` R ) e. _V )
20		mpoexga 6267	. . . 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 5652	. 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 2194	. . . 4 |- ( ph -> x = x )
28		dvrfvald.i	. . . . 5 |- ( ph -> I = ( invr ` R ) )
29	28	fveq1d 5557	. . . 4 |- ( ph -> ( I ` y ) = ( ( invr ` R ) ` y ) )
30	26, 27, 29	oveq123d 5940	. . 3 |- ( ph -> ( x .x. ( I ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
31	24, 25, 30	mpoeq123dv 5981	. 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 2236	1 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   e. cmpo 5921   Basecbs 12621   .rcmulr 12699  SRingcsrg 13462  Unitcui 13586   invrcinvr 13619  /_rcdvr 13630

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mgp 13420  df-srg 13463  df-dvdsr 13588  df-unit 13589  df-dvr 13631

This theorem is referenced by:  dvrvald  13633