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Theorem abslts 28640

Description: Surreal absolute value and less-than relation. (Contributed by Scott Fenton, 16-Apr-2025.)

**Assertion**
Ref	Expression
abslts	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abs_s‘𝐴) <s 𝐵 ↔ (( -_s‘𝐵) <s 𝐴 ∧ 𝐴 <s 𝐵)))

**Proof of Theorem abslts**
Step	Hyp	Ref	Expression
1		negscl 28427	. . . . . . 7 ⊢ (𝐴 ∈ No → ( -_s‘𝐴) ∈ No)
2	1	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → ( -_s‘𝐴) ∈ No)
3		absscl 28631	. . . . . . . 8 ⊢ (( -_s‘𝐴) ∈ No → (abs_s‘( -_s‘𝐴)) ∈ No)
4	1, 3	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ No → (abs_s‘( -_s‘𝐴)) ∈ No)
5	4	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → (abs_s‘( -_s‘𝐴)) ∈ No)
6		simplr 781	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → 𝐵 ∈ No)
7		leabss 28639	. . . . . . . 8 ⊢ (( -_s‘𝐴) ∈ No → ( -_s‘𝐴) ≤s (abs_s‘( -_s‘𝐴)))
8	1, 7	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ No → ( -_s‘𝐴) ≤s (abs_s‘( -_s‘𝐴)))
9	8	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → ( -_s‘𝐴) ≤s (abs_s‘( -_s‘𝐴)))
10		absnegs 28638	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (abs_s‘( -_s‘𝐴)) = (abs_s‘𝐴))
11	10	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (abs_s‘( -_s‘𝐴)) = (abs_s‘𝐴))
12	11	breq1d 5112	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abs_s‘( -_s‘𝐴)) <s 𝐵 ↔ (abs_s‘𝐴) <s 𝐵))
13	12	biimpar 483	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → (abs_s‘( -_s‘𝐴)) <s 𝐵)
14	2, 5, 6, 9, 13	leltstrd 28127	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → ( -_s‘𝐴) <s 𝐵)
15		simpll 779	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → 𝐴 ∈ No)
16		absscl 28631	. . . . . . 7 ⊢ (𝐴 ∈ No → (abs_s‘𝐴) ∈ No)
17	16	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → (abs_s‘𝐴) ∈ No)
18		leabss 28639	. . . . . . 7 ⊢ (𝐴 ∈ No → 𝐴 ≤s (abs_s‘𝐴))
19	18	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → 𝐴 ≤s (abs_s‘𝐴))
20		simpr 490	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → (abs_s‘𝐴) <s 𝐵)
21	15, 17, 6, 19, 20	leltstrd 28127	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → 𝐴 <s 𝐵)
22	14, 21	jca 521	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ (abs_s‘𝐴) <s 𝐵) → (( -_s‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵))
23	22	ex 418	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abs_s‘𝐴) <s 𝐵 → (( -_s‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵)))
24		abssor 28637	. . . . 5 ⊢ (𝐴 ∈ No → ((abs_s‘𝐴) = 𝐴 ∨ (abs_s‘𝐴) = ( -_s‘𝐴)))
25	24	adantr 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abs_s‘𝐴) = 𝐴 ∨ (abs_s‘𝐴) = ( -_s‘𝐴)))
26		breq1 5105	. . . . . . 7 ⊢ ((abs_s‘𝐴) = 𝐴 → ((abs_s‘𝐴) <s 𝐵 ↔ 𝐴 <s 𝐵))
27	26	biimprd 251	. . . . . 6 ⊢ ((abs_s‘𝐴) = 𝐴 → (𝐴 <s 𝐵 → (abs_s‘𝐴) <s 𝐵))
28		breq1 5105	. . . . . . 7 ⊢ ((abs_s‘𝐴) = ( -_s‘𝐴) → ((abs_s‘𝐴) <s 𝐵 ↔ ( -_s‘𝐴) <s 𝐵))
29	28	biimprd 251	. . . . . 6 ⊢ ((abs_s‘𝐴) = ( -_s‘𝐴) → (( -_s‘𝐴) <s 𝐵 → (abs_s‘𝐴) <s 𝐵))
30	27, 29	jaoa 970	. . . . 5 ⊢ (((abs_s‘𝐴) = 𝐴 ∨ (abs_s‘𝐴) = ( -_s‘𝐴)) → ((𝐴 <s 𝐵 ∧ ( -_s‘𝐴) <s 𝐵) → (abs_s‘𝐴) <s 𝐵))
31	30	ancomsd 471	. . . 4 ⊢ (((abs_s‘𝐴) = 𝐴 ∨ (abs_s‘𝐴) = ( -_s‘𝐴)) → ((( -_s‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵) → (abs_s‘𝐴) <s 𝐵))
32	25, 31	syl 18	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((( -_s‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵) → (abs_s‘𝐴) <s 𝐵))
33	23, 32	impbid 215	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abs_s‘𝐴) <s 𝐵 ↔ (( -_s‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵)))
34	1	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ( -_s‘𝐴) ∈ No)
35		simpr 490	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝐵 ∈ No)
36	34, 35	ltnegsd 28438	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -_s‘𝐴) <s 𝐵 ↔ ( -_s‘𝐵) <s ( -_s‘( -_s‘𝐴))))
37		negnegs 28435	. . . . . 6 ⊢ (𝐴 ∈ No → ( -_s‘( -_s‘𝐴)) = 𝐴)
38	37	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ( -_s‘( -_s‘𝐴)) = 𝐴)
39	38	breq2d 5114	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -_s‘𝐵) <s ( -_s‘( -_s‘𝐴)) ↔ ( -_s‘𝐵) <s 𝐴))
40	36, 39	bitrd 282	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -_s‘𝐴) <s 𝐵 ↔ ( -_s‘𝐵) <s 𝐴))
41	40	anbi1d 643	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((( -_s‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵) ↔ (( -_s‘𝐵) <s 𝐴 ∧ 𝐴 <s 𝐵)))
42	33, 41	bitrd 282	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abs_s‘𝐴) <s 𝐵 ↔ (( -_s‘𝐵) <s 𝐴 ∧ 𝐴 <s 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6538 Nocsur 28002 <s clts 28003 ≤s cles 28106 -_scnegs 28410 abs_scabss 28628

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-1o 8479 df-2o 8480 df-nadd 8678 df-no 28005 df-lts 28006 df-bday 28007 df-les 28107 df-slts 28149 df-cuts 28151 df-0s 28198 df-made 28218 df-old 28219 df-left 28221 df-right 28222 df-norec 28329 df-norec2 28340 df-adds 28351 df-negs 28412 df-abss 28629

This theorem is used by: remulscllem2 28892

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