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Theorem abslts 28668

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 28455	. . . . . . 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 28659	. . . . . . . 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 28667	. . . . . . . 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 28666	. . . . . . . . 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 5116	. . . . . . 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 28155	. . . . 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 28659	. . . . . . 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 28667	. . . . . . 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 28155	. . . . 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 28665	. . . . 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 5109	. . . . . . 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 5109	. . . . . . 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 28466	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -_s‘𝐴) <s 𝐵 ↔ ( -_s‘𝐵) <s ( -_s‘( -_s‘𝐴))))
37		negnegs 28463	. . . . . 6 ⊢ (𝐴 ∈ No → ( -_s‘( -_s‘𝐴)) = 𝐴)
38	37	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ( -_s‘( -_s‘𝐴)) = 𝐴)
39	38	breq2d 5118	. . . 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 1570 ∈ wcel 2147 class class class wbr 5106 ‘cfv 6561 Nocsur 28030 <s clts 28031 ≤s cles 28134 -_scnegs 28438 abs_scabss 28656

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-1o 8507 df-2o 8508 df-nadd 8706 df-no 28033 df-lts 28034 df-bday 28035 df-les 28135 df-slts 28177 df-cuts 28179 df-0s 28226 df-made 28246 df-old 28247 df-left 28249 df-right 28250 df-norec 28357 df-norec2 28368 df-adds 28379 df-negs 28440 df-abss 28657

This theorem is used by: remulscllem2 28920

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