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Mirrors > Home > ILE Home > Th. List > mulcomprg | Unicode version |
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
mulcomprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 6573 | . . . . . . . . 9 | |
2 | elprnql 6579 | . . . . . . . . 9 | |
3 | 1, 2 | sylan 267 | . . . . . . . 8 |
4 | prop 6573 | . . . . . . . . . . . . 13 | |
5 | elprnql 6579 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | sylan 267 | . . . . . . . . . . . 12 |
7 | mulcomnqg 6481 | . . . . . . . . . . . . 13 | |
8 | 7 | eqeq2d 2051 | . . . . . . . . . . . 12 |
9 | 6, 8 | sylan2 270 | . . . . . . . . . . 11 |
10 | 9 | anassrs 380 | . . . . . . . . . 10 |
11 | 10 | rexbidva 2323 | . . . . . . . . 9 |
12 | 11 | ancoms 255 | . . . . . . . 8 |
13 | 3, 12 | sylan2 270 | . . . . . . 7 |
14 | 13 | anassrs 380 | . . . . . 6 |
15 | 14 | rexbidva 2323 | . . . . 5 |
16 | rexcom 2474 | . . . . 5 | |
17 | 15, 16 | syl6bb 185 | . . . 4 |
18 | 17 | rabbidv 2549 | . . 3 |
19 | elprnqu 6580 | . . . . . . . . 9 | |
20 | 1, 19 | sylan 267 | . . . . . . . 8 |
21 | elprnqu 6580 | . . . . . . . . . . . . 13 | |
22 | 4, 21 | sylan 267 | . . . . . . . . . . . 12 |
23 | 22, 8 | sylan2 270 | . . . . . . . . . . 11 |
24 | 23 | anassrs 380 | . . . . . . . . . 10 |
25 | 24 | rexbidva 2323 | . . . . . . . . 9 |
26 | 25 | ancoms 255 | . . . . . . . 8 |
27 | 20, 26 | sylan2 270 | . . . . . . 7 |
28 | 27 | anassrs 380 | . . . . . 6 |
29 | 28 | rexbidva 2323 | . . . . 5 |
30 | rexcom 2474 | . . . . 5 | |
31 | 29, 30 | syl6bb 185 | . . . 4 |
32 | 31 | rabbidv 2549 | . . 3 |
33 | 18, 32 | opeq12d 3557 | . 2 |
34 | mpvlu 6637 | . . 3 | |
35 | 34 | ancoms 255 | . 2 |
36 | mpvlu 6637 | . 2 | |
37 | 33, 35, 36 | 3eqtr4rd 2083 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wrex 2307 crab 2310 cop 3378 cfv 4902 (class class class)co 5512 c1st 5765 c2nd 5766 cnq 6378 cmq 6381 cnp 6389 cmp 6392 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 df-inp 6564 df-imp 6567 |
This theorem is referenced by: ltmprr 6740 mulcmpblnrlemg 6825 mulcomsrg 6842 mulasssrg 6843 m1m1sr 6846 recexgt0sr 6858 mulgt0sr 6862 mulextsr1lem 6864 recidpirqlemcalc 6933 |
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