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Mirrors > Home > MPE Home > Th. List > sltadd2 | Structured version Visualization version GIF version |
Description: Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
sltadd2 | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sleadd2 27453 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ≤s 𝐴 ↔ (𝐶 +s 𝐵) ≤s (𝐶 +s 𝐴))) | |
2 | 1 | 3com12 1124 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ≤s 𝐴 ↔ (𝐶 +s 𝐵) ≤s (𝐶 +s 𝐴))) |
3 | 2 | notbid 318 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (¬ 𝐵 ≤s 𝐴 ↔ ¬ (𝐶 +s 𝐵) ≤s (𝐶 +s 𝐴))) |
4 | sltnle 27236 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴)) | |
5 | 4 | 3adant3 1133 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴)) |
6 | simp3 1139 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐶 ∈ No ) | |
7 | simp1 1137 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐴 ∈ No ) | |
8 | 6, 7 | addscld 27444 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐶 +s 𝐴) ∈ No ) |
9 | simp2 1138 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No ) | |
10 | 6, 9 | addscld 27444 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐶 +s 𝐵) ∈ No ) |
11 | sltnle 27236 | . . 3 ⊢ (((𝐶 +s 𝐴) ∈ No ∧ (𝐶 +s 𝐵) ∈ No ) → ((𝐶 +s 𝐴) <s (𝐶 +s 𝐵) ↔ ¬ (𝐶 +s 𝐵) ≤s (𝐶 +s 𝐴))) | |
12 | 8, 10, 11 | syl2anc 585 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +s 𝐴) <s (𝐶 +s 𝐵) ↔ ¬ (𝐶 +s 𝐵) ≤s (𝐶 +s 𝐴))) |
13 | 3, 5, 12 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5147 (class class class)co 7404 No csur 27123 <s cslt 27124 ≤s csle 27227 +s cadds 27423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-1o 8461 df-2o 8462 df-nadd 8661 df-no 27126 df-slt 27127 df-bday 27128 df-sle 27228 df-sslt 27263 df-scut 27265 df-0s 27305 df-made 27322 df-old 27323 df-left 27325 df-right 27326 df-norec2 27413 df-adds 27424 |
This theorem is referenced by: sltadd1 27455 sltadd2d 27460 |
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