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Mirrors > Home > MPE Home > Th. List > sltmuldiv2d | Structured version Visualization version GIF version |
Description: Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.) |
Ref | Expression |
---|---|
sltdivmuld.1 | โข (๐ โ ๐ด โ No ) |
sltdivmuld.2 | โข (๐ โ ๐ต โ No ) |
sltdivmuld.3 | โข (๐ โ ๐ถ โ No ) |
sltdivmuld.4 | โข (๐ โ 0s <s ๐ถ) |
Ref | Expression |
---|---|
sltmuldiv2d | โข (๐ โ ((๐ถ ยทs ๐ด) <s ๐ต โ ๐ด <s (๐ต /su ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltdivmuld.1 | . 2 โข (๐ โ ๐ด โ No ) | |
2 | sltdivmuld.2 | . 2 โข (๐ โ ๐ต โ No ) | |
3 | sltdivmuld.3 | . 2 โข (๐ โ ๐ถ โ No ) | |
4 | sltdivmuld.4 | . 2 โข (๐ โ 0s <s ๐ถ) | |
5 | 4 | sgt0ne0d 27894 | . . 3 โข (๐ โ ๐ถ โ 0s ) |
6 | 3, 5 | recsexd 28258 | . 2 โข (๐ โ โ๐ฅ โ No (๐ถ ยทs ๐ฅ) = 1s ) |
7 | 1, 2, 3, 4, 6 | sltmuldiv2wd 28241 | 1 โข (๐ โ ((๐ถ ยทs ๐ด) <s ๐ต โ ๐ด <s (๐ต /su ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 206 โ wcel 2105 class class class wbr 5147 (class class class)co 7430 No csur 27698 <s cslt 27699 0s c0s 27881 ยทs cmuls 28146 /su cdivs 28227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-dc 10483 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-nadd 8702 df-no 27701 df-slt 27702 df-bday 27703 df-sle 27804 df-sslt 27840 df-scut 27842 df-0s 27883 df-1s 27884 df-made 27900 df-old 27901 df-left 27903 df-right 27904 df-norec 27985 df-norec2 27996 df-adds 28007 df-negs 28067 df-subs 28068 df-muls 28147 df-divs 28228 |
This theorem is referenced by: zs12bday 28438 |
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