Theorem distrprg 7771

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 7765	. . 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 7769	. . 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 3241	. 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 7766	. . 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 7770	. . 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 3241	. 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 1021	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
8		simp2 1022	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> B e. P. )
9		simp3 1023	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> C e. P. )
10		addclpr 7720	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
11	8, 9, 10	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
12		mulclpr 7755	. . . 4 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
13	7, 11, 12	syl2anc 411	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
14		mulclpr 7755	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
15	7, 8, 14	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
16		mulclpr 7755	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
17	7, 9, 16	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
18		addclpr 7720	. . . 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 411	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) e. P. )
20		preqlu 7655	. . 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 411	. 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 950	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 1002   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   1stc1st 6282   2ndc2nd 6283   P.cnp 7474   +P. cpp 7476   .P. cmp 7477

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-iplp 7651  df-imp 7652

This theorem is referenced by:  ltmprr  7825  mulcmpblnrlemg  7923  mulasssrg  7941  distrsrg  7942  m1m1sr  7944  1idsr  7951  recexgt0sr  7956  mulgt0sr  7961  mulextsr1lem  7963  recidpirqlemcalc  8040