Theorem ltmuls2 28179

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 28071	. . . . 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 28106	. . . . . 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 28153	. . . 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 28166	. . . 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 5126	. . 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 28143	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) ∈ No )
14	1, 2	mulscld 28143	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐶) ∈ No )
15	13, 14	posdifsd 28106	. . 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 28143	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐶) ∈ No )
19		ltsirr 27726	. . . . . . 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 7376	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝐴 ·s 𝐵) = (𝐴 ·s 𝐶))
22	21	breq1d 5110	. . . . . . 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 28143	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
28		ltsasym 27728	. . . . . . 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 28071	. . . . . . 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 28153	. . . . . 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 28166	. . . . . . . 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 5112	. . . . . . 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 28106	. . . . . . 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 28106	. . . 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 27729	. . . 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 5100  (class class class)co 7368   No csur 27619   <s clts 27620   0_s c0s 27813   -_s csubs 28028   ·_s cmuls 28114

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115

This theorem is referenced by:  ltmuls2d  28180