ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  distrprg Unicode version

Theorem distrprg 6744
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrprg  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) ) )

Proof of Theorem distrprg
StepHypRef Expression
1 distrlem1prl 6738 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) ) )
2 distrlem5prl 6742 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )  C_  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
31, 2eqssd 2990 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  =  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
4 distrlem1pru 6739 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) ) )
5 distrlem5pru 6743 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )  C_  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
64, 5eqssd 2990 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  =  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
7 simp1 915 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
8 simp2 916 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  B  e.  P. )
9 simp3 917 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  C  e.  P. )
10 addclpr 6693 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
118, 9, 10syl2anc 397 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
12 mulclpr 6728 . . . 4  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
137, 11, 12syl2anc 397 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  e.  P. )
14 mulclpr 6728 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
157, 8, 14syl2anc 397 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
16 mulclpr 6728 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
177, 9, 16syl2anc 397 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
18 addclpr 6693 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( ( A  .P.  B )  +P.  ( A  .P.  C ) )  e.  P. )
1915, 17, 18syl2anc 397 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  e. 
P. )
20 preqlu 6628 . . 3  |-  ( ( ( A  .P.  ( B  +P.  C ) )  e.  P.  /\  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  e. 
P. )  ->  (
( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B )  +P.  ( A  .P.  C
) )  <->  ( ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  =  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  /\  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  =  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) ) )
2113, 19, 20syl2anc 397 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B )  +P.  ( A  .P.  C
) )  <->  ( ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  =  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  /\  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  =  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) ) )
223, 6, 21mpbir2and 862 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   P.cnp 6447    +P. cpp 6449    .P. cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-imp 6625
This theorem is referenced by:  ltmprr  6798  mulcmpblnrlemg  6883  mulasssrg  6901  distrsrg  6902  m1m1sr  6904  1idsr  6911  recexgt0sr  6916  mulgt0sr  6920  mulextsr1lem  6922  recidpirqlemcalc  6991
  Copyright terms: Public domain W3C validator