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Mirrors > Home > MPE Home > Th. List > slesubsub2bd | Structured version Visualization version GIF version |
Description: Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.) |
Ref | Expression |
---|---|
sltsubsubbd.1 | âĒ (ð â ðī â No ) |
sltsubsubbd.2 | âĒ (ð â ðĩ â No ) |
sltsubsubbd.3 | âĒ (ð â ðķ â No ) |
sltsubsubbd.4 | âĒ (ð â ð· â No ) |
Ref | Expression |
---|---|
slesubsub2bd | âĒ (ð â ((ðī -s ðĩ) âĪs (ðķ -s ð·) â (ð· -s ðķ) âĪs (ðĩ -s ðī))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltsubsubbd.3 | . . . 4 âĒ (ð â ðķ â No ) | |
2 | sltsubsubbd.4 | . . . 4 âĒ (ð â ð· â No ) | |
3 | sltsubsubbd.1 | . . . 4 âĒ (ð â ðī â No ) | |
4 | sltsubsubbd.2 | . . . 4 âĒ (ð â ðĩ â No ) | |
5 | 1, 2, 3, 4 | sltsubsub2bd 27465 | . . 3 âĒ (ð â ((ðķ -s ð·) <s (ðī -s ðĩ) â (ðĩ -s ðī) <s (ð· -s ðķ))) |
6 | 5 | notbid 317 | . 2 âĒ (ð â (ÂŽ (ðķ -s ð·) <s (ðī -s ðĩ) â ÂŽ (ðĩ -s ðī) <s (ð· -s ðķ))) |
7 | 3, 4 | subscld 27449 | . . 3 âĒ (ð â (ðī -s ðĩ) â No ) |
8 | 1, 2 | subscld 27449 | . . 3 âĒ (ð â (ðķ -s ð·) â No ) |
9 | slenlt 27182 | . . 3 âĒ (((ðī -s ðĩ) â No ⧠(ðķ -s ð·) â No ) â ((ðī -s ðĩ) âĪs (ðķ -s ð·) â ÂŽ (ðķ -s ð·) <s (ðī -s ðĩ))) | |
10 | 7, 8, 9 | syl2anc 584 | . 2 âĒ (ð â ((ðī -s ðĩ) âĪs (ðķ -s ð·) â ÂŽ (ðķ -s ð·) <s (ðī -s ðĩ))) |
11 | 2, 1 | subscld 27449 | . . 3 âĒ (ð â (ð· -s ðķ) â No ) |
12 | 4, 3 | subscld 27449 | . . 3 âĒ (ð â (ðĩ -s ðī) â No ) |
13 | slenlt 27182 | . . 3 âĒ (((ð· -s ðķ) â No ⧠(ðĩ -s ðī) â No ) â ((ð· -s ðķ) âĪs (ðĩ -s ðī) â ÂŽ (ðĩ -s ðī) <s (ð· -s ðķ))) | |
14 | 11, 12, 13 | syl2anc 584 | . 2 âĒ (ð â ((ð· -s ðķ) âĪs (ðĩ -s ðī) â ÂŽ (ðĩ -s ðī) <s (ð· -s ðķ))) |
15 | 6, 10, 14 | 3bitr4d 310 | 1 âĒ (ð â ((ðī -s ðĩ) âĪs (ðķ -s ð·) â (ð· -s ðķ) âĪs (ðĩ -s ðī))) |
Colors of variables: wff setvar class |
Syntax hints: ÂŽ wn 3 â wi 4 â wb 205 â wcel 2106 class class class wbr 5141 (class class class)co 7393 No csur 27070 <s cslt 27071 âĪs csle 27174 -s csubs 27411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4944 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-se 5625 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 6289 df-ord 6356 df-on 6357 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-1o 8448 df-2o 8449 df-nadd 8648 df-no 27073 df-slt 27074 df-bday 27075 df-sle 27175 df-sslt 27209 df-scut 27211 df-0s 27251 df-made 27265 df-old 27266 df-left 27268 df-right 27269 df-norec 27338 df-norec2 27349 df-adds 27360 df-negs 27412 df-subs 27413 |
This theorem is referenced by: mulsuniflem 27516 |
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