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Mirrors > Home > ILE Home > Th. List > mulcomnq0 | Unicode version |
Description: Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.) |
Ref | Expression |
---|---|
mulcomnq0 | Q0 Q0 ·Q0 ·Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 6615 | . 2 Q0 ~Q0 | |
2 | oveq1 5539 | . . 3 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
3 | oveq2 5540 | . . 3 ~Q0 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 | |
4 | 2, 3 | eqeq12d 2095 | . 2 ~Q0 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 |
5 | oveq2 5540 | . . 3 ~Q0 ·Q0 ~Q0 ·Q0 | |
6 | oveq1 5539 | . . 3 ~Q0 ~Q0 ·Q0 ·Q0 | |
7 | 5, 6 | eqeq12d 2095 | . 2 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 ·Q0 ·Q0 |
8 | nnmcom 6091 | . . . . 5 | |
9 | 8 | ad2ant2r 492 | . . . 4 |
10 | pinn 6499 | . . . . . 6 | |
11 | pinn 6499 | . . . . . 6 | |
12 | nnmcom 6091 | . . . . . 6 | |
13 | 10, 11, 12 | syl2an 283 | . . . . 5 |
14 | 13 | ad2ant2l 491 | . . . 4 |
15 | opeq12 3572 | . . . . 5 | |
16 | 15 | eceq1d 6165 | . . . 4 ~Q0 ~Q0 |
17 | 9, 14, 16 | syl2anc 403 | . . 3 ~Q0 ~Q0 |
18 | mulnnnq0 6640 | . . 3 ~Q0 ·Q0 ~Q0 ~Q0 | |
19 | mulnnnq0 6640 | . . . 4 ~Q0 ·Q0 ~Q0 ~Q0 | |
20 | 19 | ancoms 264 | . . 3 ~Q0 ·Q0 ~Q0 ~Q0 |
21 | 17, 18, 20 | 3eqtr4d 2123 | . 2 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 ~Q0 |
22 | 1, 4, 7, 21 | 2ecoptocl 6217 | 1 Q0 Q0 ·Q0 ·Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cop 3401 com 4331 (class class class)co 5532 comu 6022 cec 6127 cnpi 6462 ~Q0 ceq0 6476 Q0cnq0 6477 ·Q0 cmq0 6480 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-enq0 6614 df-nq0 6615 df-mq0 6618 |
This theorem is referenced by: distnq0r 6653 |
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