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Mirrors > Home > MPE Home > Th. List > negsval2 | Structured version Visualization version GIF version |
Description: Surreal negation in terms of subtraction. (Contributed by Scott Fenton, 15-Apr-2025.) |
Ref | Expression |
---|---|
negsval2 | ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = ( 0s -s 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 27772 | . . 3 ⊢ 0s ∈ No | |
2 | subsval 27983 | . . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s -s 𝐴) = ( 0s +s ( -us ‘𝐴))) | |
3 | 1, 2 | mpan 689 | . 2 ⊢ (𝐴 ∈ No → ( 0s -s 𝐴) = ( 0s +s ( -us ‘𝐴))) |
4 | negscl 27961 | . . 3 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | |
5 | addslid 27898 | . . 3 ⊢ (( -us ‘𝐴) ∈ No → ( 0s +s ( -us ‘𝐴)) = ( -us ‘𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ No → ( 0s +s ( -us ‘𝐴)) = ( -us ‘𝐴)) |
7 | 3, 6 | eqtr2d 2769 | 1 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = ( 0s -s 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 No csur 27586 0s c0s 27768 +s cadds 27889 -us cnegs 27945 -s csubs 27946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-1o 8487 df-2o 8488 df-no 27589 df-slt 27590 df-bday 27591 df-sslt 27727 df-scut 27729 df-0s 27770 df-made 27787 df-old 27788 df-left 27790 df-right 27791 df-norec 27868 df-norec2 27879 df-adds 27890 df-negs 27947 df-subs 27948 |
This theorem is referenced by: negsval2d 27988 |
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