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Mirrors > Home > MPE Home > Th. List > sleadd1im | Structured version Visualization version GIF version |
Description: Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
sleadd1im | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → 𝐴 ≤s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltadd1im 27381 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐴 → (𝐵 +s 𝐶) <s (𝐴 +s 𝐶))) | |
2 | 1 | 3com12 1123 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐴 → (𝐵 +s 𝐶) <s (𝐴 +s 𝐶))) |
3 | sltnle 27180 | . . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵)) | |
4 | 3 | ancoms 459 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵)) |
5 | 4 | 3adant3 1132 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵)) |
6 | addscl 27378 | . . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) ∈ No ) | |
7 | 6 | 3adant1 1130 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) ∈ No ) |
8 | addscl 27378 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 +s 𝐶) ∈ No ) | |
9 | sltnle 27180 | . . . 4 ⊢ (((𝐵 +s 𝐶) ∈ No ∧ (𝐴 +s 𝐶) ∈ No ) → ((𝐵 +s 𝐶) <s (𝐴 +s 𝐶) ↔ ¬ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶))) | |
10 | 7, 8, 9 | 3imp3i2an 1345 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s 𝐶) <s (𝐴 +s 𝐶) ↔ ¬ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶))) |
11 | 2, 5, 10 | 3imtr3d 292 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (¬ 𝐴 ≤s 𝐵 → ¬ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶))) |
12 | 11 | con4d 115 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → 𝐴 ≤s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5140 (class class class)co 7392 No csur 27067 <s cslt 27068 ≤s csle 27171 +s cadds 27356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-se 5624 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7956 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-1o 8447 df-2o 8448 df-nadd 8647 df-no 27070 df-slt 27071 df-bday 27072 df-sle 27172 df-sslt 27206 df-scut 27208 df-0s 27248 df-made 27262 df-old 27263 df-left 27265 df-right 27266 df-norec2 27346 df-adds 27357 |
This theorem is referenced by: sleadd2im 27384 sleadd1 27385 |
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