![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > negsex | Structured version Visualization version GIF version |
Description: Every surreal has a negative. Note that this theorem, addscl 27455, addscom 27440, addsass 27478, addsrid 27438, and sltadd1im 27458 are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025.) |
Ref | Expression |
---|---|
negsex | ⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negscl 27500 | . 2 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | |
2 | negsid 27505 | . 2 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s ) | |
3 | oveq2 7414 | . . . 4 ⊢ (𝑥 = ( -us ‘𝐴) → (𝐴 +s 𝑥) = (𝐴 +s ( -us ‘𝐴))) | |
4 | 3 | eqeq1d 2735 | . . 3 ⊢ (𝑥 = ( -us ‘𝐴) → ((𝐴 +s 𝑥) = 0s ↔ (𝐴 +s ( -us ‘𝐴)) = 0s )) |
5 | 4 | rspcev 3613 | . 2 ⊢ ((( -us ‘𝐴) ∈ No ∧ (𝐴 +s ( -us ‘𝐴)) = 0s ) → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s ) |
6 | 1, 2, 5 | syl2anc 585 | 1 ⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ‘cfv 6541 (class class class)co 7406 No csur 27133 0s c0s 27313 +s cadds 27433 -us cnegs 27484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-1o 8463 df-2o 8464 df-nadd 8662 df-no 27136 df-slt 27137 df-bday 27138 df-sle 27238 df-sslt 27273 df-scut 27275 df-0s 27315 df-made 27332 df-old 27333 df-left 27335 df-right 27336 df-norec 27412 df-norec2 27423 df-adds 27434 df-negs 27486 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |