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Mirrors > Home > MPE Home > Th. List > posdifsd | Structured version Visualization version GIF version |
Description: Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025.) |
Ref | Expression |
---|---|
posdifsd.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
posdifsd.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
Ref | Expression |
---|---|
posdifsd | ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ 0s <s (𝐵 -s 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 27909 | . . . 4 ⊢ 0s ∈ No | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 0s ∈ No ) |
3 | posdifsd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ No ) | |
4 | posdifsd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ No ) | |
5 | 3, 4 | subscld 28131 | . . 3 ⊢ (𝜑 → (𝐵 -s 𝐴) ∈ No ) |
6 | 2, 5, 4 | sltadd1d 28069 | . 2 ⊢ (𝜑 → ( 0s <s (𝐵 -s 𝐴) ↔ ( 0s +s 𝐴) <s ((𝐵 -s 𝐴) +s 𝐴))) |
7 | addslid 28039 | . . . 4 ⊢ (𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → ( 0s +s 𝐴) = 𝐴) |
9 | npcans 28143 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → ((𝐵 -s 𝐴) +s 𝐴) = 𝐵) | |
10 | 3, 4, 9 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐵 -s 𝐴) +s 𝐴) = 𝐵) |
11 | 8, 10 | breq12d 5180 | . 2 ⊢ (𝜑 → (( 0s +s 𝐴) <s ((𝐵 -s 𝐴) +s 𝐴) ↔ 𝐴 <s 𝐵)) |
12 | 6, 11 | bitr2d 280 | 1 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ 0s <s (𝐵 -s 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 class class class wbr 5167 (class class class)co 7451 No csur 27722 <s cslt 27723 0s c0s 27905 +s cadds 28030 -s csubs 28090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5304 ax-sep 5318 ax-nul 5325 ax-pow 5384 ax-pr 5448 ax-un 7773 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4933 df-int 4972 df-iun 5018 df-br 5168 df-opab 5230 df-mpt 5251 df-tr 5285 df-id 5594 df-eprel 5600 df-po 5608 df-so 5609 df-fr 5653 df-se 5654 df-we 5655 df-xp 5707 df-rel 5708 df-cnv 5709 df-co 5710 df-dm 5711 df-rn 5712 df-res 5713 df-ima 5714 df-pred 6335 df-ord 6401 df-on 6402 df-suc 6404 df-iota 6528 df-fun 6578 df-fn 6579 df-f 6580 df-f1 6581 df-fo 6582 df-f1o 6583 df-fv 6584 df-riota 7407 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8033 df-2nd 8034 df-frecs 8325 df-wrecs 8356 df-recs 8430 df-1o 8525 df-2o 8526 df-nadd 8725 df-no 27725 df-slt 27726 df-bday 27727 df-sle 27828 df-sslt 27864 df-scut 27866 df-0s 27907 df-made 27924 df-old 27925 df-left 27927 df-right 27928 df-norec 28009 df-norec2 28020 df-adds 28031 df-negs 28091 df-subs 28092 |
This theorem is referenced by: sltmul2 28235 precsexlem9 28277 |
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