MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  distrpr Unicode version

Theorem distrpr 8532
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
distrpr  |-  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) )

Proof of Theorem distrpr
StepHypRef Expression
1 distrlem1pr 8529 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  C_  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) )
2 distrlem5pr 8531 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )
31, 2eqssd 3117 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) ) )
4 dmplp 8516 . . 3  |-  dom  +P.  =  ( P.  X.  P. )
5 0npr 8496 . . 3  |-  -.  (/)  e.  P.
6 dmmp 8517 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
74, 5, 6ndmovdistr 5861 . 2  |-  ( -.  ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )
83, 7pm2.61i 158 1  |-  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5710   P.cnp 8361    +P. cpp 8363    .P. cmp 8364
This theorem is referenced by:  mulcmpblnrlem  8575  mulasssr  8592  distrsr  8593  m1m1sr  8595  1idsr  8600  recexsrlem  8605  mulgt0sr  8607
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6546  df-ni 8376  df-pli 8377  df-mi 8378  df-lti 8379  df-plpq 8412  df-mpq 8413  df-ltpq 8414  df-enq 8415  df-nq 8416  df-erq 8417  df-plq 8418  df-mq 8419  df-1nq 8420  df-rq 8421  df-ltnq 8422  df-np 8485  df-plp 8487  df-mp 8488
  Copyright terms: Public domain W3C validator