![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sleadd1d | Structured version Visualization version GIF version |
Description: Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
addscand.1 | âĒ (ð â ðī â No ) |
addscand.2 | âĒ (ð â ðĩ â No ) |
addscand.3 | âĒ (ð â ðķ â No ) |
Ref | Expression |
---|---|
sleadd1d | âĒ (ð â (ðī âĪs ðĩ â (ðī +s ðķ) âĪs (ðĩ +s ðķ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addscand.1 | . 2 âĒ (ð â ðī â No ) | |
2 | addscand.2 | . 2 âĒ (ð â ðĩ â No ) | |
3 | addscand.3 | . 2 âĒ (ð â ðķ â No ) | |
4 | sleadd1 27810 | . 2 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī âĪs ðĩ â (ðī +s ðķ) âĪs (ðĩ +s ðķ))) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 âĒ (ð â (ðī âĪs ðĩ â (ðī +s ðķ) âĪs (ðĩ +s ðķ))) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 â wcel 2098 class class class wbr 5138 (class class class)co 7401 No csur 27477 âĪs csle 27581 +s cadds 27780 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-1o 8461 df-2o 8462 df-nadd 8660 df-no 27480 df-slt 27481 df-bday 27482 df-sle 27582 df-sslt 27618 df-scut 27620 df-0s 27661 df-made 27678 df-old 27679 df-left 27681 df-right 27682 df-norec2 27770 df-adds 27781 |
This theorem is referenced by: addsuniflem 27822 mulsuniflem 27953 |
Copyright terms: Public domain | W3C validator |