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| Mirrors > Home > MPE Home > Th. List > abssneg | Structured version Visualization version GIF version | ||
| Description: Surreal absolute value of the negative. (Contributed by Scott Fenton, 16-Apr-2025.) |
| Ref | Expression |
|---|---|
| abssneg | ⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negnegs 28076 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘( -us ‘𝐴)) = 𝐴) |
| 3 | negscl 28068 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | |
| 4 | 0sno 27871 | . . . . . . . 8 ⊢ 0s ∈ No | |
| 5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ No → 0s ∈ No ) |
| 6 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ∈ No ) | |
| 7 | 5, 6 | slenegd 28080 | . . . . . 6 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s ( -us ‘ 0s ))) |
| 8 | negs0s 28058 | . . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s | |
| 9 | 8 | breq2i 5151 | . . . . . 6 ⊢ (( -us ‘𝐴) ≤s ( -us ‘ 0s ) ↔ ( -us ‘𝐴) ≤s 0s ) |
| 10 | 7, 9 | bitrdi 287 | . . . . 5 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s 0s )) |
| 11 | 10 | biimpa 476 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘𝐴) ≤s 0s ) |
| 12 | abssnid 28267 | . . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐴) ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴))) | |
| 13 | 3, 11, 12 | syl2an2r 685 | . . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴))) |
| 14 | abssid 28265 | . . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴) | |
| 15 | 2, 13, 14 | 3eqtr4d 2787 | . 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
| 16 | 6, 5 | slenegd 28080 | . . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴))) |
| 17 | 8 | breq1i 5150 | . . . . . 6 ⊢ (( -us ‘ 0s ) ≤s ( -us ‘𝐴) ↔ 0s ≤s ( -us ‘𝐴)) |
| 18 | 16, 17 | bitrdi 287 | . . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ 0s ≤s ( -us ‘𝐴))) |
| 19 | 18 | biimpa 476 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → 0s ≤s ( -us ‘𝐴)) |
| 20 | abssid 28265 | . . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴)) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴)) | |
| 21 | 3, 19, 20 | syl2an2r 685 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴)) |
| 22 | abssnid 28267 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴)) | |
| 23 | 21, 22 | eqtr4d 2780 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
| 24 | sletric 27809 | . . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s )) | |
| 25 | 4, 24 | mpan 690 | . 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s )) |
| 26 | 15, 23, 25 | mpjaodan 961 | 1 ⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 No csur 27684 ≤s csle 27789 0s c0s 27867 -us cnegs 28051 absscabss 28261 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 df-abss 28262 |
| This theorem is referenced by: absslt 28273 |
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