Theorem sltmul2 28212

Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)

Assertion

Ref	Expression
sltmul2	⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))

Proof of Theorem sltmul2

Step	Hyp	Ref	Expression
1		simpl1l 1223	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐴 ∈ No )
2		simpl3 1192	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐶 ∈ No )
3		simpl2 1191	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐵 ∈ No )
4	2, 3	subscld 28108	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐶 -s 𝐵) ∈ No )
5		simpl1r 1224	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 0s <s 𝐴)
6		simp2 1136	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No )
7		simp3 1137	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐶 ∈ No )
8	6, 7	posdifsd 28142	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ↔ 0s <s (𝐶 -s 𝐵)))
9	8	biimpa 476	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 0s <s (𝐶 -s 𝐵))
10	1, 4, 5, 9	mulsgt0d 28186	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 0s <s (𝐴 ·s (𝐶 -s 𝐵)))
11	1, 2, 3	subsdid 28199	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s (𝐶 -s 𝐵)) = ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵)))
12	10, 11	breqtrd 5174	. . 3 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 0s <s ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵)))
13	1, 3	mulscld 28176	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) ∈ No )
14	1, 2	mulscld 28176	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐶) ∈ No )
15	13, 14	posdifsd 28142	. . 3 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ 0s <s ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵))))
16	12, 15	mpbird 257	. 2 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶))
17		simp1l 1196	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐴 ∈ No )
18	17, 7	mulscld 28176	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐶) ∈ No )
19		sltirr 27806	. . . . . . 7 ⊢ ((𝐴 ·s 𝐶) ∈ No → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶))
20	18, 19	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶))
21		oveq2 7439	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝐴 ·s 𝐵) = (𝐴 ·s 𝐶))
22	21	breq1d 5158	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶)))
23	22	notbid 318	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶)))
24	20, 23	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 = 𝐶 → ¬ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))
25	24	con2d 134	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) → ¬ 𝐵 = 𝐶))
26	25	imp 406	. . 3 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 𝐵 = 𝐶)
27	17, 6	mulscld 28176	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
28		sltasym 27808	. . . . . . 7 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐶) ∈ No ) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵)))
29	27, 18, 28	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵)))
30	29	imp 406	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵))
31		simpl1l 1223	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐴 ∈ No )
32	31	adantr 480	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 𝐴 ∈ No )
33		simpll2 1212	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 𝐵 ∈ No )
34		simpll3 1213	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 𝐶 ∈ No )
35	33, 34	subscld 28108	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐵 -s 𝐶) ∈ No )
36		simpl1r 1224	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 0s <s 𝐴)
37	36	adantr 480	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 0s <s 𝐴)
38		simpr 484	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 0s <s (𝐵 -s 𝐶))
39	32, 35, 37, 38	mulsgt0d 28186	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 0s <s (𝐴 ·s (𝐵 -s 𝐶)))
40	32, 33, 34	subsdid 28199	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s (𝐵 -s 𝐶)) = ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶)))
41	40	breq2d 5160	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → ( 0s <s (𝐴 ·s (𝐵 -s 𝐶)) ↔ 0s <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶))))
42	18	ad2antrr 726	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s 𝐶) ∈ No )
43	27	ad2antrr 726	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s 𝐵) ∈ No )
44	42, 43	posdifsd 28142	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → ((𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵) ↔ 0s <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶))))
45	41, 44	bitr4d 282	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → ( 0s <s (𝐴 ·s (𝐵 -s 𝐶)) ↔ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵)))
46	39, 45	mpbid 232	. . . . 5 ⊢ (((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵))
47	30, 46	mtand 816	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 0s <s (𝐵 -s 𝐶))
48		simpl3 1192	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐶 ∈ No )
49		simpl2 1191	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 ∈ No )
50	48, 49	posdifsd 28142	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → (𝐶 <s 𝐵 ↔ 0s <s (𝐵 -s 𝐶)))
51	47, 50	mtbird 325	. . 3 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 𝐶 <s 𝐵)
52		sltlin 27809	. . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 <s 𝐵))
53	49, 48, 52	syl2anc 584	. . 3 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → (𝐵 <s 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 <s 𝐵))
54	26, 51, 53	ecase23d 1472	. 2 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 <s 𝐶)
55	16, 54	impbida 801	1 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   0_s c0s 27882   -_s csubs 28067   ·_s cmuls 28147

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  df-muls 28148

This theorem is referenced by:  sltmul2d  28213