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Mirrors > Home > MPE Home > Th. List > mulcomnq | Structured version Visualization version GIF version |
Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcomnq | โข (๐ด ยทQ ๐ต) = (๐ต ยทQ ๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcompq 10949 | . . . 4 โข (๐ด ยทpQ ๐ต) = (๐ต ยทpQ ๐ด) | |
2 | 1 | fveq2i 6888 | . . 3 โข ([Q]โ(๐ด ยทpQ ๐ต)) = ([Q]โ(๐ต ยทpQ ๐ด)) |
3 | mulpqnq 10938 | . . 3 โข ((๐ด โ Q โง ๐ต โ Q) โ (๐ด ยทQ ๐ต) = ([Q]โ(๐ด ยทpQ ๐ต))) | |
4 | mulpqnq 10938 | . . . 4 โข ((๐ต โ Q โง ๐ด โ Q) โ (๐ต ยทQ ๐ด) = ([Q]โ(๐ต ยทpQ ๐ด))) | |
5 | 4 | ancoms 458 | . . 3 โข ((๐ด โ Q โง ๐ต โ Q) โ (๐ต ยทQ ๐ด) = ([Q]โ(๐ต ยทpQ ๐ด))) |
6 | 2, 3, 5 | 3eqtr4a 2792 | . 2 โข ((๐ด โ Q โง ๐ต โ Q) โ (๐ด ยทQ ๐ต) = (๐ต ยทQ ๐ด)) |
7 | mulnqf 10946 | . . . 4 โข ยทQ :(Q ร Q)โถQ | |
8 | 7 | fdmi 6723 | . . 3 โข dom ยทQ = (Q ร Q) |
9 | 8 | ndmovcom 7591 | . 2 โข (ยฌ (๐ด โ Q โง ๐ต โ Q) โ (๐ด ยทQ ๐ต) = (๐ต ยทQ ๐ด)) |
10 | 6, 9 | pm2.61i 182 | 1 โข (๐ด ยทQ ๐ต) = (๐ต ยทQ ๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โง wa 395 = wceq 1533 โ wcel 2098 ร cxp 5667 โcfv 6537 (class class class)co 7405 ยทpQ cmpq 10846 Qcnq 10849 [Q]cerq 10851 ยทQ cmq 10853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-omul 8472 df-er 8705 df-ni 10869 df-mi 10871 df-lti 10872 df-mpq 10906 df-enq 10908 df-nq 10909 df-erq 10910 df-mq 10912 df-1nq 10913 |
This theorem is referenced by: recmulnq 10961 recrecnq 10964 halfnq 10973 ltrnq 10976 addclprlem1 11013 addclprlem2 11014 mulclprlem 11016 mulclpr 11017 mulcompr 11020 distrlem4pr 11023 1idpr 11026 prlem934 11030 prlem936 11044 reclem3pr 11046 reclem4pr 11047 |
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