Theorem dvrfvald 14378

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 14377	. . 3 |- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
2		fveq2 5675	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		fveq2 5675	. . . 4 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
4		fveq2 5675	. . . . 5 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
5		eqidd 2235	. . . . 5 |- ( r = R -> x = x )
6		fveq2 5675	. . . . . 6 |- ( r = R -> ( invr ` r ) = ( invr ` R ) )
7	6	fveq1d 5677	. . . . 5 |- ( r = R -> ( ( invr ` r ) ` y ) = ( ( invr ` R ) ` y ) )
8	4, 5, 7	oveq123d 6079	. . . 4 |- ( r = R -> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
9	2, 3, 8	mpoeq123dv 6123	. . 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 2829	. . 3 |- ( ph -> R e. _V )
12		basfn 13355	. . . . 5 |- Base Fn _V
13		funfvex 5692	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5463	. . . . 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 2235	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
17		eqidd 2235	. . . . . 6 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
18	16, 17, 10	unitssd 14354	. . . . 5 |- ( ph -> (Unit ` R ) C_ ( Base ` R ) )
19	15, 18	ssexd 4255	. . . 4 |- ( ph -> (Unit ` R ) e. _V )
20		mpoexga 6421	. . . 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 5776	. 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 2235	. . . 4 |- ( ph -> x = x )
28		dvrfvald.i	. . . . 5 |- ( ph -> I = ( invr ` R ) )
29	28	fveq1d 5677	. . . 4 |- ( ph -> ( I ` y ) = ( ( invr ` R ) ` y ) )
30	26, 27, 29	oveq123d 6079	. . 3 |- ( ph -> ( x .x. ( I ` y ) ) = ( x ( .r ` R ) ( ( invr ` R ) ` y ) ) )
31	24, 25, 30	mpoeq123dv 6123	. 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 2277	1 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   e. cmpo 6060   Basecbs 13296   .rcmulr 13375  SRingcsrg 14206  Unitcui 14331   invrcinvr 14365  /_rcdvr 14376

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mgp 14160  df-srg 14207  df-dvdsr 14333  df-unit 14334  df-dvr 14377

This theorem is referenced by:  dvrvald  14379