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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 10985 | . . . 4 โข (๐ด ยทpQ ๐ต) = (๐ต ยทpQ ๐ด) | |
2 | 1 | fveq2i 6905 | . . 3 โข ([Q]โ(๐ด ยทpQ ๐ต)) = ([Q]โ(๐ต ยทpQ ๐ด)) |
3 | mulpqnq 10974 | . . 3 โข ((๐ด โ Q โง ๐ต โ Q) โ (๐ด ยทQ ๐ต) = ([Q]โ(๐ด ยทpQ ๐ต))) | |
4 | mulpqnq 10974 | . . . 4 โข ((๐ต โ Q โง ๐ด โ Q) โ (๐ต ยทQ ๐ด) = ([Q]โ(๐ต ยทpQ ๐ด))) | |
5 | 4 | ancoms 457 | . . 3 โข ((๐ด โ Q โง ๐ต โ Q) โ (๐ต ยทQ ๐ด) = ([Q]โ(๐ต ยทpQ ๐ด))) |
6 | 2, 3, 5 | 3eqtr4a 2794 | . 2 โข ((๐ด โ Q โง ๐ต โ Q) โ (๐ด ยทQ ๐ต) = (๐ต ยทQ ๐ด)) |
7 | mulnqf 10982 | . . . 4 โข ยทQ :(Q ร Q)โถQ | |
8 | 7 | fdmi 6739 | . . 3 โข dom ยทQ = (Q ร Q) |
9 | 8 | ndmovcom 7615 | . 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 394 = wceq 1533 โ wcel 2098 ร cxp 5680 โcfv 6553 (class class class)co 7426 ยทpQ cmpq 10882 Qcnq 10885 [Q]cerq 10887 ยทQ cmq 10889 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-omul 8500 df-er 8733 df-ni 10905 df-mi 10907 df-lti 10908 df-mpq 10942 df-enq 10944 df-nq 10945 df-erq 10946 df-mq 10948 df-1nq 10949 |
This theorem is referenced by: recmulnq 10997 recrecnq 11000 halfnq 11009 ltrnq 11012 addclprlem1 11049 addclprlem2 11050 mulclprlem 11052 mulclpr 11053 mulcompr 11056 distrlem4pr 11059 1idpr 11062 prlem934 11066 prlem936 11080 reclem3pr 11082 reclem4pr 11083 |
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