Theorem distrprg 7945

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 7939	. . 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 7943	. . 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 7940	. . 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 7944	. . 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 7894	. . . . 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 7929	. . . 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 7929	. . . . 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 7929	. . . . 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 7894	. . . 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 7829	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   P.cnp 7648   +P. cpp 7650   .P. cmp 7651

This theorem was proved from 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-imp 7826

This theorem is referenced by:  ltmprr  7999  mulcmpblnrlemg  8097  mulasssrg  8115  distrsrg  8116  m1m1sr  8118  1idsr  8125  recexgt0sr  8130  mulgt0sr  8135  mulextsr1lem  8137  recidpirqlemcalc  8214