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Mirrors > Home > MPE Home > Th. List > negnegs | Structured version Visualization version GIF version |
Description: A surreal is equal to the negative of its negative. Theorem 4(ii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.) |
Ref | Expression |
---|---|
negnegs | ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negscl 28083 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | |
2 | 1 | negsidd 28089 | . . 3 ⊢ (𝐴 ∈ No → (( -us ‘𝐴) +s ( -us ‘( -us ‘𝐴))) = 0s ) |
3 | 1 | negscld 28084 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) ∈ No ) |
4 | 3, 1 | addscomd 28015 | . . 3 ⊢ (𝐴 ∈ No → (( -us ‘( -us ‘𝐴)) +s ( -us ‘𝐴)) = (( -us ‘𝐴) +s ( -us ‘( -us ‘𝐴)))) |
5 | negsid 28088 | . . 3 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s ) | |
6 | 2, 4, 5 | 3eqtr4d 2785 | . 2 ⊢ (𝐴 ∈ No → (( -us ‘( -us ‘𝐴)) +s ( -us ‘𝐴)) = (𝐴 +s ( -us ‘𝐴))) |
7 | id 22 | . . 3 ⊢ (𝐴 ∈ No → 𝐴 ∈ No ) | |
8 | 3, 7, 1 | addscan2d 28047 | . 2 ⊢ (𝐴 ∈ No → ((( -us ‘( -us ‘𝐴)) +s ( -us ‘𝐴)) = (𝐴 +s ( -us ‘𝐴)) ↔ ( -us ‘( -us ‘𝐴)) = 𝐴)) |
9 | 6, 8 | mpbid 232 | 1 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 No csur 27699 0s c0s 27882 +s cadds 28007 -us cnegs 28066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-norec2 27997 df-adds 28008 df-negs 28068 |
This theorem is referenced by: sltneg 28092 negs11 28096 negsfo 28100 negsbday 28104 negsubsdi2d 28125 mul2negsd 28203 abssneg 28286 absslt 28288 zseo 28421 renegscl 28445 |
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