Theorem distrprg 7550

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 7544	. . 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 7548	. . 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 3164	. 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 7545	. . 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 7549	. . 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 3164	. 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 992	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
8		simp2 993	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> B e. P. )
9		simp3 994	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> C e. P. )
10		addclpr 7499	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
11	8, 9, 10	syl2anc 409	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
12		mulclpr 7534	. . . 4 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
13	7, 11, 12	syl2anc 409	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
14		mulclpr 7534	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
15	7, 8, 14	syl2anc 409	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
16		mulclpr 7534	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
17	7, 9, 16	syl2anc 409	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
18		addclpr 7499	. . . 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 409	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) e. P. )
20		preqlu 7434	. . 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 409	. 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 939	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   P.cnp 7253   +P. cpp 7255   .P. cmp 7256

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431

This theorem is referenced by:  ltmprr  7604  mulcmpblnrlemg  7702  mulasssrg  7720  distrsrg  7721  m1m1sr  7723  1idsr  7730  recexgt0sr  7735  mulgt0sr  7740  mulextsr1lem  7742  recidpirqlemcalc  7819