Theorem distrprg 7955

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

Step	Hyp	Ref	Expression
1		distrlem1prl 7949	. . 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 7953	. . 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 ) ) ) )
3	1, 2	eqssd 3265	. 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 7950	. . 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 7954	. . 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 ) ) ) )
6	4, 5	eqssd 3265	. 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 1028	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
8		simp2 1029	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> B e. P. )
9		simp3 1030	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> C e. P. )
10		addclpr 7904	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
11	8, 9, 10	syl2anc 415	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
12		mulclpr 7939	. . . 4 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
13	7, 11, 12	syl2anc 415	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
14		mulclpr 7939	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
15	7, 8, 14	syl2anc 415	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
16		mulclpr 7939	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
17	7, 9, 16	syl2anc 415	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
18		addclpr 7904	. . . 4 |- ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) e. P. )
19	15, 17, 18	syl2anc 415	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) e. P. )
20		preqlu 7839	. . 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 ) ) ) ) ) )
21	13, 19, 20	syl2anc 415	. 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 ) ) ) ) ) )
22	3, 6, 21	mpbir2and 957	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. ( B +P. C ) ) = ( ( A .P. B ) +P. ( A .P. C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   P.cnp 7658   +P. cpp 7660   .P. cmp 7661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835  df-imp 7836

This theorem is used by:  ltmprr  8009  mulcmpblnrlemg  8107  mulasssrg  8125  distrsrg  8126  m1m1sr  8128  1idsr  8135  recexgt0sr  8140  mulgt0sr  8145  mulextsr1lem  8147  recidpirqlemcalc  8224