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Theorem rngdir 38682
Description: Distributive law for the multiplication operation of a nonunital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
rngdi.b  |-  B  =  ( Base `  R
)
rngdi.p  |-  .+  =  ( +g  `  R )
rngdi.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
rngdir  |-  ( ( R  e. Rng  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) )

Proof of Theorem rngdir
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngdi.b . . . 4  |-  B  =  ( Base `  R
)
2 eqid 2420 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 rngdi.p . . . 4  |-  .+  =  ( +g  `  R )
4 rngdi.t . . . 4  |-  .x.  =  ( .r `  R )
51, 2, 3, 4isrng 38676 . . 3  |-  ( R  e. Rng 
<->  ( R  e.  Abel  /\  (mulGrp `  R )  e. SGrp  /\  A. a  e.  B  A. b  e.  B  A. c  e.  B  ( ( a 
.x.  ( b  .+  c ) )  =  ( ( a  .x.  b )  .+  (
a  .x.  c )
)  /\  ( (
a  .+  b )  .x.  c )  =  ( ( a  .x.  c
)  .+  ( b  .x.  c ) ) ) ) )
6 oveq1 6303 . . . . . . . 8  |-  ( a  =  X  ->  (
a  .x.  ( b  .+  c ) )  =  ( X  .x.  (
b  .+  c )
) )
7 oveq1 6303 . . . . . . . . 9  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
8 oveq1 6303 . . . . . . . . 9  |-  ( a  =  X  ->  (
a  .x.  c )  =  ( X  .x.  c ) )
97, 8oveq12d 6314 . . . . . . . 8  |-  ( a  =  X  ->  (
( a  .x.  b
)  .+  ( a  .x.  c ) )  =  ( ( X  .x.  b )  .+  ( X  .x.  c ) ) )
106, 9eqeq12d 2442 . . . . . . 7  |-  ( a  =  X  ->  (
( a  .x.  (
b  .+  c )
)  =  ( ( a  .x.  b ) 
.+  ( a  .x.  c ) )  <->  ( X  .x.  ( b  .+  c
) )  =  ( ( X  .x.  b
)  .+  ( X  .x.  c ) ) ) )
11 oveq1 6303 . . . . . . . . 9  |-  ( a  =  X  ->  (
a  .+  b )  =  ( X  .+  b ) )
1211oveq1d 6311 . . . . . . . 8  |-  ( a  =  X  ->  (
( a  .+  b
)  .x.  c )  =  ( ( X 
.+  b )  .x.  c ) )
138oveq1d 6311 . . . . . . . 8  |-  ( a  =  X  ->  (
( a  .x.  c
)  .+  ( b  .x.  c ) )  =  ( ( X  .x.  c )  .+  (
b  .x.  c )
) )
1412, 13eqeq12d 2442 . . . . . . 7  |-  ( a  =  X  ->  (
( ( a  .+  b )  .x.  c
)  =  ( ( a  .x.  c ) 
.+  ( b  .x.  c ) )  <->  ( ( X  .+  b )  .x.  c )  =  ( ( X  .x.  c
)  .+  ( b  .x.  c ) ) ) )
1510, 14anbi12d 715 . . . . . 6  |-  ( a  =  X  ->  (
( ( a  .x.  ( b  .+  c
) )  =  ( ( a  .x.  b
)  .+  ( a  .x.  c ) )  /\  ( ( a  .+  b )  .x.  c
)  =  ( ( a  .x.  c ) 
.+  ( b  .x.  c ) ) )  <-> 
( ( X  .x.  ( b  .+  c
) )  =  ( ( X  .x.  b
)  .+  ( X  .x.  c ) )  /\  ( ( X  .+  b )  .x.  c
)  =  ( ( X  .x.  c ) 
.+  ( b  .x.  c ) ) ) ) )
16 oveq1 6303 . . . . . . . . 9  |-  ( b  =  Y  ->  (
b  .+  c )  =  ( Y  .+  c ) )
1716oveq2d 6312 . . . . . . . 8  |-  ( b  =  Y  ->  ( X  .x.  ( b  .+  c ) )  =  ( X  .x.  ( Y  .+  c ) ) )
18 oveq2 6304 . . . . . . . . 9  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
1918oveq1d 6311 . . . . . . . 8  |-  ( b  =  Y  ->  (
( X  .x.  b
)  .+  ( X  .x.  c ) )  =  ( ( X  .x.  Y )  .+  ( X  .x.  c ) ) )
2017, 19eqeq12d 2442 . . . . . . 7  |-  ( b  =  Y  ->  (
( X  .x.  (
b  .+  c )
)  =  ( ( X  .x.  b ) 
.+  ( X  .x.  c ) )  <->  ( X  .x.  ( Y  .+  c
) )  =  ( ( X  .x.  Y
)  .+  ( X  .x.  c ) ) ) )
21 oveq2 6304 . . . . . . . . 9  |-  ( b  =  Y  ->  ( X  .+  b )  =  ( X  .+  Y
) )
2221oveq1d 6311 . . . . . . . 8  |-  ( b  =  Y  ->  (
( X  .+  b
)  .x.  c )  =  ( ( X 
.+  Y )  .x.  c ) )
23 oveq1 6303 . . . . . . . . 9  |-  ( b  =  Y  ->  (
b  .x.  c )  =  ( Y  .x.  c ) )
2423oveq2d 6312 . . . . . . . 8  |-  ( b  =  Y  ->  (
( X  .x.  c
)  .+  ( b  .x.  c ) )  =  ( ( X  .x.  c )  .+  ( Y  .x.  c ) ) )
2522, 24eqeq12d 2442 . . . . . . 7  |-  ( b  =  Y  ->  (
( ( X  .+  b )  .x.  c
)  =  ( ( X  .x.  c ) 
.+  ( b  .x.  c ) )  <->  ( ( X  .+  Y )  .x.  c )  =  ( ( X  .x.  c
)  .+  ( Y  .x.  c ) ) ) )
2620, 25anbi12d 715 . . . . . 6  |-  ( b  =  Y  ->  (
( ( X  .x.  ( b  .+  c
) )  =  ( ( X  .x.  b
)  .+  ( X  .x.  c ) )  /\  ( ( X  .+  b )  .x.  c
)  =  ( ( X  .x.  c ) 
.+  ( b  .x.  c ) ) )  <-> 
( ( X  .x.  ( Y  .+  c ) )  =  ( ( X  .x.  Y ) 
.+  ( X  .x.  c ) )  /\  ( ( X  .+  Y )  .x.  c
)  =  ( ( X  .x.  c ) 
.+  ( Y  .x.  c ) ) ) ) )
27 oveq2 6304 . . . . . . . . 9  |-  ( c  =  Z  ->  ( Y  .+  c )  =  ( Y  .+  Z
) )
2827oveq2d 6312 . . . . . . . 8  |-  ( c  =  Z  ->  ( X  .x.  ( Y  .+  c ) )  =  ( X  .x.  ( Y  .+  Z ) ) )
29 oveq2 6304 . . . . . . . . 9  |-  ( c  =  Z  ->  ( X  .x.  c )  =  ( X  .x.  Z
) )
3029oveq2d 6312 . . . . . . . 8  |-  ( c  =  Z  ->  (
( X  .x.  Y
)  .+  ( X  .x.  c ) )  =  ( ( X  .x.  Y )  .+  ( X  .x.  Z ) ) )
3128, 30eqeq12d 2442 . . . . . . 7  |-  ( c  =  Z  ->  (
( X  .x.  ( Y  .+  c ) )  =  ( ( X 
.x.  Y )  .+  ( X  .x.  c ) )  <->  ( X  .x.  ( Y  .+  Z ) )  =  ( ( X  .x.  Y ) 
.+  ( X  .x.  Z ) ) ) )
32 oveq2 6304 . . . . . . . 8  |-  ( c  =  Z  ->  (
( X  .+  Y
)  .x.  c )  =  ( ( X 
.+  Y )  .x.  Z ) )
33 oveq2 6304 . . . . . . . . 9  |-  ( c  =  Z  ->  ( Y  .x.  c )  =  ( Y  .x.  Z
) )
3429, 33oveq12d 6314 . . . . . . . 8  |-  ( c  =  Z  ->  (
( X  .x.  c
)  .+  ( Y  .x.  c ) )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )
3532, 34eqeq12d 2442 . . . . . . 7  |-  ( c  =  Z  ->  (
( ( X  .+  Y )  .x.  c
)  =  ( ( X  .x.  c ) 
.+  ( Y  .x.  c ) )  <->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) ) )
3631, 35anbi12d 715 . . . . . 6  |-  ( c  =  Z  ->  (
( ( X  .x.  ( Y  .+  c ) )  =  ( ( X  .x.  Y ) 
.+  ( X  .x.  c ) )  /\  ( ( X  .+  Y )  .x.  c
)  =  ( ( X  .x.  c ) 
.+  ( Y  .x.  c ) ) )  <-> 
( ( X  .x.  ( Y  .+  Z ) )  =  ( ( X  .x.  Y ) 
.+  ( X  .x.  Z ) )  /\  ( ( X  .+  Y )  .x.  Z
)  =  ( ( X  .x.  Z ) 
.+  ( Y  .x.  Z ) ) ) ) )
3715, 26, 36rspc3v 3191 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( A. a  e.  B  A. b  e.  B  A. c  e.  B  ( ( a 
.x.  ( b  .+  c ) )  =  ( ( a  .x.  b )  .+  (
a  .x.  c )
)  /\  ( (
a  .+  b )  .x.  c )  =  ( ( a  .x.  c
)  .+  ( b  .x.  c ) ) )  ->  ( ( X 
.x.  ( Y  .+  Z ) )  =  ( ( X  .x.  Y )  .+  ( X  .x.  Z ) )  /\  ( ( X 
.+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) ) ) )
38 simpr 462 . . . . 5  |-  ( ( ( X  .x.  ( Y  .+  Z ) )  =  ( ( X 
.x.  Y )  .+  ( X  .x.  Z ) )  /\  ( ( X  .+  Y ) 
.x.  Z )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) )
3937, 38syl6com 36 . . . 4  |-  ( A. a  e.  B  A. b  e.  B  A. c  e.  B  (
( a  .x.  (
b  .+  c )
)  =  ( ( a  .x.  b ) 
.+  ( a  .x.  c ) )  /\  ( ( a  .+  b )  .x.  c
)  =  ( ( a  .x.  c ) 
.+  ( b  .x.  c ) ) )  ->  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  (
( X  .+  Y
)  .x.  Z )  =  ( ( X 
.x.  Z )  .+  ( Y  .x.  Z ) ) ) )
40393ad2ant3 1028 . . 3  |-  ( ( R  e.  Abel  /\  (mulGrp `  R )  e. SGrp  /\  A. a  e.  B  A. b  e.  B  A. c  e.  B  (
( a  .x.  (
b  .+  c )
)  =  ( ( a  .x.  b ) 
.+  ( a  .x.  c ) )  /\  ( ( a  .+  b )  .x.  c
)  =  ( ( a  .x.  c ) 
.+  ( b  .x.  c ) ) ) )  ->  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Y )  .x.  Z
)  =  ( ( X  .x.  Z ) 
.+  ( Y  .x.  Z ) ) ) )
415, 40sylbi 198 . 2  |-  ( R  e. Rng  ->  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  (
( X  .+  Y
)  .x.  Z )  =  ( ( X 
.x.  Z )  .+  ( Y  .x.  Z ) ) ) )
4241imp 430 1  |-  ( ( R  e. Rng  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   ` cfv 5592  (class class class)co 6296   Basecbs 15073   +g cplusg 15142   .rcmulr 15143  SGrpcsgrp 16470   Abelcabl 17359  mulGrpcmgp 17651  Rngcrng 38674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-nul 4547
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-rng0 38675
This theorem is referenced by:  rnglz  38684
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