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Mirrors > Home > MPE Home > Th. List > sltsub1 | Structured version Visualization version GIF version |
Description: Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.) |
Ref | Expression |
---|---|
sltsub1 | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 -s 𝐶) <s (𝐵 -s 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negscl 28083 | . . 3 ⊢ (𝐶 ∈ No → ( -us ‘𝐶) ∈ No ) | |
2 | sltadd1 28040 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( -us ‘𝐶) ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 +s ( -us ‘𝐶)) <s (𝐵 +s ( -us ‘𝐶)))) | |
3 | 1, 2 | syl3an3 1164 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 +s ( -us ‘𝐶)) <s (𝐵 +s ( -us ‘𝐶)))) |
4 | subsval 28105 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 -s 𝐶) = (𝐴 +s ( -us ‘𝐶))) | |
5 | 4 | 3adant2 1130 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 -s 𝐶) = (𝐴 +s ( -us ‘𝐶))) |
6 | subsval 28105 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 -s 𝐶) = (𝐵 +s ( -us ‘𝐶))) | |
7 | 6 | 3adant1 1129 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 -s 𝐶) = (𝐵 +s ( -us ‘𝐶))) |
8 | 5, 7 | breq12d 5161 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -s 𝐶) <s (𝐵 -s 𝐶) ↔ (𝐴 +s ( -us ‘𝐶)) <s (𝐵 +s ( -us ‘𝐶)))) |
9 | 3, 8 | bitr4d 282 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 -s 𝐶) <s (𝐵 -s 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 No csur 27699 <s cslt 27700 +s cadds 28007 -us cnegs 28066 -s csubs 28067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-norec2 27997 df-adds 28008 df-negs 28068 df-subs 28069 |
This theorem is referenced by: sltsub1d 28123 |
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