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Mirrors > Home > MPE Home > Th. List > sleadd2im | Structured version Visualization version GIF version |
Description: Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
sleadd2im | âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðķ +s ðī) âĪs (ðķ +s ðĩ) â ðī âĪs ðĩ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addscom 27901 | . . . 4 âĒ ((ðī â No â§ ðķ â No ) â (ðī +s ðķ) = (ðķ +s ðī)) | |
2 | 1 | 3adant2 1128 | . . 3 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī +s ðķ) = (ðķ +s ðī)) |
3 | addscom 27901 | . . . 4 âĒ ((ðĩ â No â§ ðķ â No ) â (ðĩ +s ðķ) = (ðķ +s ðĩ)) | |
4 | 3 | 3adant1 1127 | . . 3 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðĩ +s ðķ) = (ðķ +s ðĩ)) |
5 | 2, 4 | breq12d 5156 | . 2 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī +s ðķ) âĪs (ðĩ +s ðķ) â (ðķ +s ðī) âĪs (ðķ +s ðĩ))) |
6 | sleadd1im 27922 | . 2 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī +s ðķ) âĪs (ðĩ +s ðķ) â ðī âĪs ðĩ)) | |
7 | 5, 6 | sylbird 259 | 1 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðķ +s ðī) âĪs (ðķ +s ðĩ) â ðī âĪs ðĩ)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ w3a 1084 = wceq 1533 â wcel 2098 class class class wbr 5143 (class class class)co 7416 No csur 27591 âĪs csle 27695 +s cadds 27894 |
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 |
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-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-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-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-1o 8485 df-2o 8486 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-made 27792 df-old 27793 df-left 27795 df-right 27796 df-norec2 27884 df-adds 27895 |
This theorem is referenced by: (None) |
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