Theorem dvrfvald 14211

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 14210	. . 3 |- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
2		fveq2 5648	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		fveq2 5648	. . . 4 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
4		fveq2 5648	. . . . 5 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5		eqidd 2232	. . . . 5 |- ( r = R -> x = x )
6		fveq2 5648	. . . . . 6 |- ( r = R -> ( invr ` r ) = ( invr ` R ) )
7	6	fveq1d 5650	. . . . 5 |- ( r = R -> ( ( invr ` r ) ` y ) = ( ( invr ` R ) ` y ) )
8	4, 5, 7	oveq123d 6049	. . . 4 |- ( r = R -> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
9	2, 3, 8	mpoeq123dv 6093	. . 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 2817	. . 3 |- ( ph -> R e. _V )
12		basfn 13204	. . . . 5 |- Base Fn _V
13		funfvex 5665	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5439	. . . . 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 2232	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
17		eqidd 2232	. . . . . 6 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
18	16, 17, 10	unitssd 14187	. . . . 5 |- ( ph -> (Unit ` R ) C_ ( Base ` R ) )
19	15, 18	ssexd 4234	. . . 4 |- ( ph -> (Unit ` R ) e. _V )
20		mpoexga 6386	. . . 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 5749	. 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 2232	. . . 4 |- ( ph -> x = x )
28		dvrfvald.i	. . . . 5 |- ( ph -> I = ( invr ` R ) )
29	28	fveq1d 5650	. . . 4 |- ( ph -> ( I ` y ) = ( ( invr ` R ) ` y ) )
30	26, 27, 29	oveq123d 6049	. . 3 |- ( ph -> ( x .x. ( I ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
31	24, 25, 30	mpoeq123dv 6093	. 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 2274	1 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   Basecbs 13145   .rcmulr 13224  SRingcsrg 14040  Unitcui 14164   invrcinvr 14198  /_rcdvr 14209

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-plusg 13236  df-mulr 13237  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-mgp 13998  df-srg 14041  df-dvdsr 14166  df-unit 14167  df-dvr 14210

This theorem is referenced by:  dvrvald  14212