Theorem sltmul2 28215

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 1224	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐴 ∈ No )
2		simpl3 1193	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐶 ∈ No )
3		simpl2 1192	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐵 ∈ No )
4	2, 3	subscld 28111	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐶 -s 𝐵) ∈ No )
5		simpl1r 1225	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 0s <s 𝐴)
6		simp2 1137	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No )
7		simp3 1138	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐶 ∈ No )
8	6, 7	posdifsd 28145	. . . . . 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 28189	. . . 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 28202	. . . 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 5192	. . 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 28179	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) ∈ No )
14	1, 2	mulscld 28179	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐶) ∈ No )
15	13, 14	posdifsd 28145	. . 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 1197	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐴 ∈ No )
18	17, 7	mulscld 28179	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐶) ∈ No )
19		sltirr 27809	. . . . . . 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 7456	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝐴 ·s 𝐵) = (𝐴 ·s 𝐶))
22	21	breq1d 5176	. . . . . . 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 28179	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
28		sltasym 27811	. . . . . . 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 583	. . . . . 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 1224	. . . . . . . 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 1213	. . . . . . . 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 1214	. . . . . . . 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 28111	. . . . . . 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 1225	. . . . . . . 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 28189	. . . . . 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 28202	. . . . . . . 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 5178	. . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 28145	. . . . . . 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 815	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 0s <s (𝐵 -s 𝐶))
48		simpl3 1193	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐶 ∈ No )
49		simpl2 1192	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 ∈ No )
50	48, 49	posdifsd 28145	. . . 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 27812	. . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 <s 𝐵))
53	49, 48, 52	syl2anc 583	. . 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 1473	. 2 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 <s 𝐶)
55	16, 54	impbida 800	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 1086   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   0_s c0s 27885   -_s csubs 28070   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151

This theorem is referenced by:  sltmul2d  28216