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Theorem distrprg 7126
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 7120 . . 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 7124 . . 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 3040 . 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 7121 . . 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 7125 . . 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 3040 . 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 943 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
8 simp2 944 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  B  e.  P. )
9 simp3 945 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  C  e.  P. )
10 addclpr 7075 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
118, 9, 10syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
12 mulclpr 7110 . . . 4  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
137, 11, 12syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  e.  P. )
14 mulclpr 7110 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
157, 8, 14syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
16 mulclpr 7110 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
177, 9, 16syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
18 addclpr 7075 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( ( A  .P.  B )  +P.  ( A  .P.  C ) )  e.  P. )
1915, 17, 18syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  e. 
P. )
20 preqlu 7010 . . 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 403 . 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 890 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   P.cnp 6829    +P. cpp 6831    .P. cmp 6832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-imp 7007
This theorem is referenced by:  ltmprr  7180  mulcmpblnrlemg  7265  mulasssrg  7283  distrsrg  7284  m1m1sr  7286  1idsr  7293  recexgt0sr  7298  mulgt0sr  7302  mulextsr1lem  7304  recidpirqlemcalc  7373
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