Theorem dvrfvald 13438

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 13437	. . 3 |- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
2		fveq2 5527	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		fveq2 5527	. . . 4 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
4		fveq2 5527	. . . . 5 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5		eqidd 2188	. . . . 5 |- ( r = R -> x = x )
6		fveq2 5527	. . . . . 6 |- ( r = R -> ( invr ` r ) = ( invr ` R ) )
7	6	fveq1d 5529	. . . . 5 |- ( r = R -> ( ( invr ` r ) ` y ) = ( ( invr ` R ) ` y ) )
8	4, 5, 7	oveq123d 5909	. . . 4 |- ( r = R -> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
9	2, 3, 8	mpoeq123dv 5950	. . 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 2762	. . 3 |- ( ph -> R e. _V )
12		basfn 12534	. . . . 5 |- Base Fn _V
13		funfvex 5544	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5328	. . . . 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 2188	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
17		eqidd 2188	. . . . . 6 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
18	16, 17, 10	unitssd 13414	. . . . 5 |- ( ph -> (Unit ` R ) C_ ( Base ` R ) )
19	15, 18	ssexd 4155	. . . 4 |- ( ph -> (Unit ` R ) e. _V )
20		mpoexga 6227	. . . 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 5622	. 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 2188	. . . 4 |- ( ph -> x = x )
28		dvrfvald.i	. . . . 5 |- ( ph -> I = ( invr ` R ) )
29	28	fveq1d 5529	. . . 4 |- ( ph -> ( I ` y ) = ( ( invr ` R ) ` y ) )
30	26, 27, 29	oveq123d 5909	. . 3 |- ( ph -> ( x .x. ( I ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
31	24, 25, 30	mpoeq123dv 5950	. 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 2230	1 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2158   _Vcvv 2749   Fn wfn 5223   ` cfv 5228  (class class class)co 5888   e. cmpo 5890   Basecbs 12476   .rcmulr 12552  SRingcsrg 13272  Unitcui 13392   invrcinvr 13425  /_rcdvr 13436

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-plusg 12564  df-mulr 12565  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-mgp 13230  df-srg 13273  df-dvdsr 13394  df-unit 13395  df-dvr 13437

This theorem is referenced by:  dvrvald  13439