![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > abssge0 | Structured version Visualization version GIF version |
Description: The absolute value of a surreal number is non-negative. (Contributed by Scott Fenton, 16-Apr-2025.) |
Ref | Expression |
---|---|
abssge0 | ⊢ (𝐴 ∈ No → 0s ≤s (abss‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ ( 0s ≤s 𝐴 → 0s ≤s 𝐴) | |
2 | iftrue 4530 | . . . . 5 ⊢ ( 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = 𝐴) | |
3 | 1, 2 | breqtrrd 5170 | . . . 4 ⊢ ( 0s ≤s 𝐴 → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴))) |
4 | 3 | adantr 480 | . . 3 ⊢ (( 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴))) |
5 | negs0s 27932 | . . . . 5 ⊢ ( -us ‘ 0s ) = 0s | |
6 | 0sno 27752 | . . . . . . . . 9 ⊢ 0s ∈ No | |
7 | sletric 27690 | . . . . . . . . 9 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s )) | |
8 | 6, 7 | mpan 689 | . . . . . . . 8 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s )) |
9 | 8 | ord 863 | . . . . . . 7 ⊢ (𝐴 ∈ No → (¬ 0s ≤s 𝐴 → 𝐴 ≤s 0s )) |
10 | 9 | impcom 407 | . . . . . 6 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 𝐴 ≤s 0s ) |
11 | simpr 484 | . . . . . . 7 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 𝐴 ∈ No ) | |
12 | 6 | a1i 11 | . . . . . . 7 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ∈ No ) |
13 | 11, 12 | slenegd 27953 | . . . . . 6 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴))) |
14 | 10, 13 | mpbid 231 | . . . . 5 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴)) |
15 | 5, 14 | eqbrtrrid 5178 | . . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ≤s ( -us ‘𝐴)) |
16 | iffalse 4533 | . . . . 5 ⊢ (¬ 0s ≤s 𝐴 → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴)) | |
17 | 16 | adantr 480 | . . . 4 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴)) = ( -us ‘𝐴)) |
18 | 15, 17 | breqtrrd 5170 | . . 3 ⊢ ((¬ 0s ≤s 𝐴 ∧ 𝐴 ∈ No ) → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴))) |
19 | 4, 18 | pm2.61ian 811 | . 2 ⊢ (𝐴 ∈ No → 0s ≤s if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴))) |
20 | abssval 28126 | . 2 ⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -us ‘𝐴))) | |
21 | 19, 20 | breqtrrd 5170 | 1 ⊢ (𝐴 ∈ No → 0s ≤s (abss‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ifcif 4524 class class class wbr 5142 ‘cfv 6542 No csur 27566 ≤s csle 27670 0s c0s 27748 -us cnegs 27925 absscabss 28124 |
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 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
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 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-1o 8480 df-2o 8481 df-nadd 8680 df-no 27569 df-slt 27570 df-bday 27571 df-sle 27671 df-sslt 27707 df-scut 27709 df-0s 27750 df-made 27767 df-old 27768 df-left 27770 df-right 27771 df-norec 27848 df-norec2 27859 df-adds 27870 df-negs 27927 df-abss 28125 |
This theorem is referenced by: remulscllem2 28222 |
Copyright terms: Public domain | W3C validator |