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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 27969 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → ( -us ‘( -us ‘𝐴)) = 𝐴) |
3 | negscl 27961 | . . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | |
4 | 0sno 27772 | . . . . . . . 8 ⊢ 0s ∈ No | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ No → 0s ∈ No ) |
6 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ∈ No ) | |
7 | 5, 6 | slenegd 27973 | . . . . . 6 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ↔ ( -us ‘𝐴) ≤s ( -us ‘ 0s ))) |
8 | negs0s 27952 | . . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s | |
9 | 8 | breq2i 5156 | . . . . . 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 28150 | . . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐴) ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴))) | |
13 | 3, 11, 12 | syl2an2r 684 | . . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = ( -us ‘( -us ‘𝐴))) |
14 | abssid 28148 | . . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴) | |
15 | 2, 13, 14 | 3eqtr4d 2778 | . 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
16 | 6, 5 | slenegd 27973 | . . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴))) |
17 | 8 | breq1i 5155 | . . . . . 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 28148 | . . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s ≤s ( -us ‘𝐴)) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴)) | |
21 | 3, 19, 20 | syl2an2r 684 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = ( -us ‘𝐴)) |
22 | abssnid 28150 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴)) | |
23 | 21, 22 | eqtr4d 2771 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
24 | sletric 27710 | . . 3 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s )) | |
25 | 4, 24 | mpan 689 | . 2 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s )) |
26 | 15, 23, 25 | mpjaodan 957 | 1 ⊢ (𝐴 ∈ No → (abss‘( -us ‘𝐴)) = (abss‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 No csur 27586 ≤s csle 27690 0s c0s 27768 -us cnegs 27945 absscabss 28144 |
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-ot 4638 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-nadd 8687 df-no 27589 df-slt 27590 df-bday 27591 df-sle 27691 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-abss 28145 |
This theorem is referenced by: absslt 28156 |
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