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Mirrors > Home > MPE Home > Th. List > sltdivmuld | 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 |
---|---|
sltdivmuld | โข (๐ โ ((๐ด /su ๐ถ) <s ๐ต โ ๐ด <s (๐ถ ยทs ๐ต))) |
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 27786 | . . 3 โข (๐ โ ๐ถ โ 0s ) |
6 | 3, 5 | recsexd 28140 | . 2 โข (๐ โ โ๐ฅ โ No (๐ถ ยทs ๐ฅ) = 1s ) |
7 | 1, 2, 3, 4, 6 | sltdivmulwd 28120 | 1 โข (๐ โ ((๐ด /su ๐ถ) <s ๐ต โ ๐ด <s (๐ถ ยทs ๐ต))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โ wcel 2098 class class class wbr 5143 (class class class)co 7416 No csur 27591 <s cslt 27592 0s c0s 27773 ยทs cmuls 28028 /su cdivs 28109 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-dc 10469 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-nadd 8685 df-no 27594 df-slt 27595 df-bday 27596 df-sle 27696 df-sslt 27732 df-scut 27734 df-0s 27775 df-1s 27776 df-made 27792 df-old 27793 df-left 27795 df-right 27796 df-norec 27873 df-norec2 27884 df-adds 27895 df-negs 27952 df-subs 27953 df-muls 28029 df-divs 28110 |
This theorem is referenced by: (None) |
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