Theorem distrpr 5132

Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124.

Hypotheses

Ref	Expression
distrpr.1	|- B e. V
distrpr.2	|- C e. V

Assertion

Ref	Expression
distrpr	|- (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C))

Proof of Theorem distrpr

Step	Hyp	Ref	Expression
1		distrlem1pr 5127	. . . 4 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> (A .P. (B +P. C)) (_ ((A .P. B) +P. (A .P. C)))
2		distrlem5pr 5131	. . . 4 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> ((A .P. B) +P. (A .P. C)) (_ (A .P. (B +P. C)))
3	1, 2	eqssd 2079	. . 3 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C)))
4	3	3impb 829	. 2 |- ((A e. P. /\ B e. P. /\ C e. P.) -> (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C)))
5		distrpr.1	. . 3 |- B e. V
6		dmplp 5115	. . 3 |- dom +P. = (P. X. P.)
7		distrpr.2	. . 3 |- C e. V
8		0npr 5096	. . 3 |- -. (/) e. P.
9		dmmp 5116	. . 3 |- dom .P. = (P. X. P.)
10	5, 6, 7, 8, 9	ndmoprdistr 4049	. 2 |- (-. (A e. P. /\ B e. P. /\ C e. P.) -> (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C)))
11	4, 10	pm2.61i 126	1 |- (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C))

Syntax hints:   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  Vcvv 1811  (class class class)co 3963  P.cnp 4985   +P. cpp 4987   .P. cmp 4988

This theorem is referenced by:  mulcmpblnrlem 5182  mulasssr 5199  distrsr 5200  m1m1sr 5202  1idsr 5207  recexsrlem 5212  mulgt0sr 5214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-plp 5088  df-mp 5089

Copyright terms: Public domain