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Theorem muldivdirap 8611
Description: Distribution of division over addition with a multiplication. (Contributed by Jim Kingdon, 11-Nov-2021.)
Assertion
Ref Expression
muldivdirap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( C  x.  A )  +  B )  /  C
)  =  ( A  +  ( B  /  C ) ) )

Proof of Theorem muldivdirap
StepHypRef Expression
1 simp3l 1020 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C  e.  CC )
2 simp1 992 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  A  e.  CC )
31, 2mulcld 7927 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( C  x.  A
)  e.  CC )
4 divdirap 8601 . . 3  |-  ( ( ( C  x.  A
)  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( C  x.  A )  +  B )  /  C
)  =  ( ( ( C  x.  A
)  /  C )  +  ( B  /  C ) ) )
53, 4syld3an1 1279 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( C  x.  A )  +  B )  /  C
)  =  ( ( ( C  x.  A
)  /  C )  +  ( B  /  C ) ) )
6 3anass 977 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )
76biimpri 132 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  e.  CC  /\  C  e.  CC  /\  C #  0 ) )
873adant2 1011 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  e.  CC  /\  C  e.  CC  /\  C #  0 ) )
9 divcanap3 8602 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  (
( C  x.  A
)  /  C )  =  A )
108, 9syl 14 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( C  x.  A )  /  C
)  =  A )
1110oveq1d 5865 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( C  x.  A )  /  C )  +  ( B  /  C ) )  =  ( A  +  ( B  /  C ) ) )
125, 11eqtrd 2203 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( C  x.  A )  +  B )  /  C
)  =  ( A  +  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987  (class class class)co 5850   CCcc 7759   0cc0 7761    + caddc 7764    x. cmul 7766   # cap 8487    / cdiv 8576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577
This theorem is referenced by:  ltoddhalfle  11839  halfleoddlt  11840
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