Theorem ltmuls2 28177

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

Assertion

Ref	Expression
ltmuls2	⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))

Proof of Theorem ltmuls2

Step	Hyp	Ref	Expression
1		simpl1l 1226	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐴 ∈ No )
2		simpl3 1195	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐶 ∈ No )
3		simpl2 1194	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 𝐵 ∈ No )
4	2, 3	subscld 28069	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐶 -s 𝐵) ∈ No )
5		simpl1r 1227	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → 0s <s 𝐴)
6		simp2 1138	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐵 ∈ No )
7		simp3 1139	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐶 ∈ No )
8	6, 7	posdifsd 28104	. . . . . 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 28151	. . . 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 28164	. . . 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 5112	. . 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 28141	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) ∈ No )
14	1, 2	mulscld 28141	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐶) ∈ No )
15	13, 14	posdifsd 28104	. . 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 1199	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → 𝐴 ∈ No )
18	17, 7	mulscld 28141	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐶) ∈ No )
19		ltsirr 27724	. . . . . . 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 7368	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝐴 ·s 𝐵) = (𝐴 ·s 𝐶))
22	21	breq1d 5096	. . . . . . 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 28141	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
28		ltsasym 27726	. . . . . . 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 585	. . . . . 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 1226	. . . . . . . 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 1215	. . . . . . . 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 1216	. . . . . . . 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 28069	. . . . . . 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 1227	. . . . . . . 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 28151	. . . . . 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 28164	. . . . . . . 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 5098	. . . . . . 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 727	. . . . . . . 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 727	. . . . . . . 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 28104	. . . . . . 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 1195	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐶 ∈ No )
49		simpl2 1194	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 ∈ No )
50	48, 49	posdifsd 28104	. . . 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		ltslin 27727	. . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 <s 𝐵))
53	49, 48, 52	syl2anc 585	. . 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 1476	. 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 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  (class class class)co 7360   No csur 27617   <s clts 27618   0_s c0s 27811   -_s csubs 28026   ·_s cmuls 28112

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-muls 28113

This theorem is referenced by:  ltmuls2d  28178