![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mulasspr | Structured version Visualization version GIF version |
Description: Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulasspr | โข ((๐ด ยทP ๐ต) ยทP ๐ถ) = (๐ด ยทP (๐ต ยทP ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mp 10981 | . 2 โข ยทP = (๐ค โ P, ๐ฃ โ P โฆ {๐ฅ โฃ โ๐ฆ โ ๐ค โ๐ง โ ๐ฃ ๐ฅ = (๐ฆ ยทQ ๐ง)}) | |
2 | mulclnq 10944 | . 2 โข ((๐ฆ โ Q โง ๐ง โ Q) โ (๐ฆ ยทQ ๐ง) โ Q) | |
3 | dmmp 11010 | . 2 โข dom ยทP = (P ร P) | |
4 | mulclpr 11017 | . 2 โข ((๐ โ P โง ๐ โ P) โ (๐ ยทP ๐) โ P) | |
5 | mulassnq 10956 | . 2 โข ((๐ ยทQ ๐) ยทQ โ) = (๐ ยทQ (๐ ยทQ โ)) | |
6 | 1, 2, 3, 4, 5 | genpass 11006 | 1 โข ((๐ด ยทP ๐ต) ยทP ๐ถ) = (๐ด ยทP (๐ต ยทP ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7405 ยทQ cmq 10853 ยทP cmp 10859 |
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-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 |
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-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-mp 10981 |
This theorem is referenced by: mulasssr 11087 |
Copyright terms: Public domain | W3C validator |