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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 27674 | . . . 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 27888 | . . 3 ⊢ (𝜑 → (𝐵 -s 𝐴) ∈ No ) |
6 | 2, 5, 4 | sltadd1d 27830 | . 2 ⊢ (𝜑 → ( 0s <s (𝐵 -s 𝐴) ↔ ( 0s +s 𝐴) <s ((𝐵 -s 𝐴) +s 𝐴))) |
7 | addslid 27800 | . . . 4 ⊢ (𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → ( 0s +s 𝐴) = 𝐴) |
9 | npcans 27898 | . . . 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 5151 | . 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 205 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 No csur 27488 <s cslt 27489 0s c0s 27670 +s cadds 27791 -s csubs 27848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-1o 8461 df-2o 8462 df-nadd 8660 df-no 27491 df-slt 27492 df-bday 27493 df-sle 27593 df-sslt 27629 df-scut 27631 df-0s 27672 df-made 27689 df-old 27690 df-left 27692 df-right 27693 df-norec 27770 df-norec2 27781 df-adds 27792 df-negs 27849 df-subs 27850 |
This theorem is referenced by: sltmul2 27986 precsexlem9 28028 |
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