Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > distrprg | Unicode version |
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
distrprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrlem1prl 7358 | . . 3 | |
2 | distrlem5prl 7362 | . . 3 | |
3 | 1, 2 | eqssd 3084 | . 2 |
4 | distrlem1pru 7359 | . . 3 | |
5 | distrlem5pru 7363 | . . 3 | |
6 | 4, 5 | eqssd 3084 | . 2 |
7 | simp1 966 | . . . 4 | |
8 | simp2 967 | . . . . 5 | |
9 | simp3 968 | . . . . 5 | |
10 | addclpr 7313 | . . . . 5 | |
11 | 8, 9, 10 | syl2anc 408 | . . . 4 |
12 | mulclpr 7348 | . . . 4 | |
13 | 7, 11, 12 | syl2anc 408 | . . 3 |
14 | mulclpr 7348 | . . . . 5 | |
15 | 7, 8, 14 | syl2anc 408 | . . . 4 |
16 | mulclpr 7348 | . . . . 5 | |
17 | 7, 9, 16 | syl2anc 408 | . . . 4 |
18 | addclpr 7313 | . . . 4 | |
19 | 15, 17, 18 | syl2anc 408 | . . 3 |
20 | preqlu 7248 | . . 3 | |
21 | 13, 19, 20 | syl2anc 408 | . 2 |
22 | 3, 6, 21 | mpbir2and 913 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 cfv 5093 (class class class)co 5742 c1st 6004 c2nd 6005 cnp 7067 cpp 7069 cmp 7070 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-iplp 7244 df-imp 7245 |
This theorem is referenced by: ltmprr 7418 mulcmpblnrlemg 7516 mulasssrg 7534 distrsrg 7535 m1m1sr 7537 1idsr 7544 recexgt0sr 7549 mulgt0sr 7554 mulextsr1lem 7556 recidpirqlemcalc 7633 |
Copyright terms: Public domain | W3C validator |