![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > distrpr | Structured version Visualization version GIF version |
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
distrpr | โข (๐ด ยทP (๐ต +P ๐ถ)) = ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrlem1pr 10968 | . . 3 โข ((๐ด โ P โง ๐ต โ P โง ๐ถ โ P) โ (๐ด ยทP (๐ต +P ๐ถ)) โ ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ))) | |
2 | distrlem5pr 10970 | . . 3 โข ((๐ด โ P โง ๐ต โ P โง ๐ถ โ P) โ ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)) โ (๐ด ยทP (๐ต +P ๐ถ))) | |
3 | 1, 2 | eqssd 3966 | . 2 โข ((๐ด โ P โง ๐ต โ P โง ๐ถ โ P) โ (๐ด ยทP (๐ต +P ๐ถ)) = ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ))) |
4 | dmplp 10955 | . . 3 โข dom +P = (P ร P) | |
5 | 0npr 10935 | . . 3 โข ยฌ โ โ P | |
6 | dmmp 10956 | . . 3 โข dom ยทP = (P ร P) | |
7 | 4, 5, 6 | ndmovdistr 7548 | . 2 โข (ยฌ (๐ด โ P โง ๐ต โ P โง ๐ถ โ P) โ (๐ด ยทP (๐ต +P ๐ถ)) = ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ))) |
8 | 3, 7 | pm2.61i 182 | 1 โข (๐ด ยทP (๐ต +P ๐ถ)) = ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: โง w3a 1088 = wceq 1542 โ wcel 2107 (class class class)co 7362 Pcnp 10802 +P cpp 10804 ยทP cmp 10805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-mpq 10852 df-ltpq 10853 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-mq 10858 df-1nq 10859 df-rq 10860 df-ltnq 10861 df-np 10924 df-plp 10926 df-mp 10927 |
This theorem is referenced by: mulcmpblnrlem 11013 mulasssr 11033 distrsr 11034 m1m1sr 11036 1idsr 11041 recexsrlem 11046 mulgt0sr 11048 |
Copyright terms: Public domain | W3C validator |