Theorem distrprg 7736

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 7730	. . 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 7734	. . 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 3218	. 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 7731	. . 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 7735	. . 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 3218	. 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 1000	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
8		simp2 1001	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> B e. P. )
9		simp3 1002	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> C e. P. )
10		addclpr 7685	. . . . 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 7720	. . . 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 7720	. . . . 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 7720	. . . . 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 7685	. . . 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 7620	. . 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 947	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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   1stc1st 6247   2ndc2nd 6248   P.cnp 7439   +P. cpp 7441   .P. cmp 7442

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iplp 7616  df-imp 7617

This theorem is referenced by:  ltmprr  7790  mulcmpblnrlemg  7888  mulasssrg  7906  distrsrg  7907  m1m1sr  7909  1idsr  7916  recexgt0sr  7921  mulgt0sr  7926  mulextsr1lem  7928  recidpirqlemcalc  8005