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Theorem distrpr 9442
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 9439 . . 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 9441 . . 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 3478 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) ) )
4 dmplp 9426 . . 3  |-  dom  +P.  =  ( P.  X.  P. )
5 0npr 9406 . . 3  |-  -.  (/)  e.  P.
6 dmmp 9427 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
74, 5, 6ndmovdistr 6463 . 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 167 1  |-  ( A  .P.  ( B  +P.  C ) )  =  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 982    = wceq 1437    e. wcel 1867  (class class class)co 6296   P.cnp 9273    +P. cpp 9275    .P. cmp 9276
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-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-omul 7186  df-er 7362  df-ni 9286  df-pli 9287  df-mi 9288  df-lti 9289  df-plpq 9322  df-mpq 9323  df-ltpq 9324  df-enq 9325  df-nq 9326  df-erq 9327  df-plq 9328  df-mq 9329  df-1nq 9330  df-rq 9331  df-ltnq 9332  df-np 9395  df-plp 9397  df-mp 9398
This theorem is referenced by:  mulcmpblnrlem  9483  mulasssr  9503  distrsr  9504  m1m1sr  9506  1idsr  9511  recexsrlem  9516  mulgt0sr  9518
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