Metamath Proof Explorer

Theorem mulsproplem4

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of A and a member of the old set of B is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025)

Ref Expression
Hypotheses mulsproplem.1 ⊢ φ → ∀ a ∈ No ∀ b ∈ No ∀ c ∈ No ∀ d ∈ No ∀ e ∈ No ∀ f ∈ No bday ⁡ a + bday ⁡ b ∪ bday ⁡ c + bday ⁡ e ∪ bday ⁡ d + bday ⁡ f ∪ bday ⁡ c + bday ⁡ f ∪ bday ⁡ d + bday ⁡ e ∈ bday ⁡ A + bday ⁡ B ∪ bday ⁡ C + bday ⁡ E ∪ bday ⁡ D + bday ⁡ F ∪ bday ⁡ C + bday ⁡ F ∪ bday ⁡ D + bday ⁡ E → a ⋅ s b ∈ No ∧ c < s d ∧ e < s f → c ⋅ s f - s c ⋅ s e < s d ⋅ s f - s d ⋅ s e
mulsproplem4.1 ⊢ φ → X ∈ Old ⁡ bday ⁡ A
mulsproplem4.2 ⊢ φ → Y ∈ Old ⁡ bday ⁡ B
Assertion mulsproplem4 ⊢ φ → X ⋅ s Y ∈ No

Proof

Step Hyp Ref Expression
1 mulsproplem.1 ⊢ φ → ∀ a ∈ No ∀ b ∈ No ∀ c ∈ No ∀ d ∈ No ∀ e ∈ No ∀ f ∈ No bday ⁡ a + bday ⁡ b ∪ bday ⁡ c + bday ⁡ e ∪ bday ⁡ d + bday ⁡ f ∪ bday ⁡ c + bday ⁡ f ∪ bday ⁡ d + bday ⁡ e ∈ bday ⁡ A + bday ⁡ B ∪ bday ⁡ C + bday ⁡ E ∪ bday ⁡ D + bday ⁡ F ∪ bday ⁡ C + bday ⁡ F ∪ bday ⁡ D + bday ⁡ E → a ⋅ s b ∈ No ∧ c < s d ∧ e < s f → c ⋅ s f - s c ⋅ s e < s d ⋅ s f - s d ⋅ s e
2 mulsproplem4.1 ⊢ φ → X ∈ Old ⁡ bday ⁡ A
3 mulsproplem4.2 ⊢ φ → Y ∈ Old ⁡ bday ⁡ B
4 2 oldnod ⊢ φ → X ∈ No
5 3 oldnod ⊢ φ → Y ∈ No
6 0no ⊢ 0 s ∈ No
7 6 a1i ⊢ φ → 0 s ∈ No
8 bday0 ⊢ bday ⁡ 0 s = ∅
9 8 8 oveq12i ⊢ bday ⁡ 0 s + bday ⁡ 0 s = ∅ + ∅
10 0elon ⊢ ∅ ∈ On
11 naddrid ⊢ ∅ ∈ On → ∅ + ∅ = ∅
12 10 11 ax-mp ⊢ ∅ + ∅ = ∅
13 9 12 eqtri ⊢ bday ⁡ 0 s + bday ⁡ 0 s = ∅
14 13 13 uneq12i ⊢ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s = ∅ ∪ ∅
15 un0 ⊢ ∅ ∪ ∅ = ∅
16 14 15 eqtri ⊢ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s = ∅
17 16 16 uneq12i ⊢ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s = ∅ ∪ ∅
18 17 15 eqtri ⊢ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s = ∅
19 18 uneq2i ⊢ bday ⁡ X + bday ⁡ Y ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s = bday ⁡ X + bday ⁡ Y ∪ ∅
20 un0 ⊢ bday ⁡ X + bday ⁡ Y ∪ ∅ = bday ⁡ X + bday ⁡ Y
21 19 20 eqtri ⊢ bday ⁡ X + bday ⁡ Y ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s = bday ⁡ X + bday ⁡ Y
22 oldbdayim ⊢ X ∈ Old ⁡ bday ⁡ A → bday ⁡ X ∈ bday ⁡ A
23 2 22 syl ⊢ φ → bday ⁡ X ∈ bday ⁡ A
24 oldbdayim ⊢ Y ∈ Old ⁡ bday ⁡ B → bday ⁡ Y ∈ bday ⁡ B
25 3 24 syl ⊢ φ → bday ⁡ Y ∈ bday ⁡ B
26 bdayon ⊢ bday ⁡ A ∈ On
27 bdayon ⊢ bday ⁡ B ∈ On
28 naddel12 ⊢ bday ⁡ A ∈ On ∧ bday ⁡ B ∈ On → bday ⁡ X ∈ bday ⁡ A ∧ bday ⁡ Y ∈ bday ⁡ B → bday ⁡ X + bday ⁡ Y ∈ bday ⁡ A + bday ⁡ B
29 26 27 28 mp2an ⊢ bday ⁡ X ∈ bday ⁡ A ∧ bday ⁡ Y ∈ bday ⁡ B → bday ⁡ X + bday ⁡ Y ∈ bday ⁡ A + bday ⁡ B
30 23 25 29 syl2anc ⊢ φ → bday ⁡ X + bday ⁡ Y ∈ bday ⁡ A + bday ⁡ B
31 elun1 ⊢ bday ⁡ X + bday ⁡ Y ∈ bday ⁡ A + bday ⁡ B → bday ⁡ X + bday ⁡ Y ∈ bday ⁡ A + bday ⁡ B ∪ bday ⁡ C + bday ⁡ E ∪ bday ⁡ D + bday ⁡ F ∪ bday ⁡ C + bday ⁡ F ∪ bday ⁡ D + bday ⁡ E
32 30 31 syl ⊢ φ → bday ⁡ X + bday ⁡ Y ∈ bday ⁡ A + bday ⁡ B ∪ bday ⁡ C + bday ⁡ E ∪ bday ⁡ D + bday ⁡ F ∪ bday ⁡ C + bday ⁡ F ∪ bday ⁡ D + bday ⁡ E
33 21 32 eqeltrid ⊢ φ → bday ⁡ X + bday ⁡ Y ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∪ bday ⁡ 0 s + bday ⁡ 0 s ∈ bday ⁡ A + bday ⁡ B ∪ bday ⁡ C + bday ⁡ E ∪ bday ⁡ D + bday ⁡ F ∪ bday ⁡ C + bday ⁡ F ∪ bday ⁡ D + bday ⁡ E
34 1 4 5 7 7 7 7 33 mulsproplem1 ⊢ φ → X ⋅ s Y ∈ No ∧ 0 s < s 0 s ∧ 0 s < s 0 s → 0 s ⋅ s 0 s - s 0 s ⋅ s 0 s < s 0 s ⋅ s 0 s - s 0 s ⋅ s 0 s
35 34 simpld ⊢ φ → X ⋅ s Y ∈ No