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Theorem distrprg 7396
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 7390 . . 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 7394 . . 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 3114 . 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 7391 . . 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 7395 . . 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 3114 . 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 981 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
8 simp2 982 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  B  e.  P. )
9 simp3 983 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  C  e.  P. )
10 addclpr 7345 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
118, 9, 10syl2anc 408 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
12 mulclpr 7380 . . . 4  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
137, 11, 12syl2anc 408 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  e.  P. )
14 mulclpr 7380 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
157, 8, 14syl2anc 408 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
16 mulclpr 7380 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
177, 9, 16syl2anc 408 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
18 addclpr 7345 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( ( A  .P.  B )  +P.  ( A  .P.  C ) )  e.  P. )
1915, 17, 18syl2anc 408 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  e. 
P. )
20 preqlu 7280 . . 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 408 . 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 928 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   P.cnp 7099    +P. cpp 7101    .P. cmp 7102
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-imp 7277
This theorem is referenced by:  ltmprr  7450  mulcmpblnrlemg  7548  mulasssrg  7566  distrsrg  7567  m1m1sr  7569  1idsr  7576  recexgt0sr  7581  mulgt0sr  7586  mulextsr1lem  7588  recidpirqlemcalc  7665
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