Theorem nosupinfsep 33862

Description: Given two sets of surreals, a surreal 𝑊 separates them iff its restriction to the maximum of dom 𝑆 and dom 𝑇 separates them. Corollary 4.4 of [Lipparini] p. 7. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypotheses

Ref	Expression
nosupinfsep.1	⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
nosupinfsep.2	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

Assertion

Ref	Expression
nosupinfsep	⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))

Distinct variable groups:   𝐴,𝑎,𝑔,𝑢,𝑣,𝑥,𝑦   𝐵,𝑏,𝑔   𝑢,𝐵   𝑣,𝑏,𝐵,𝑥,𝑦   𝑢,𝑔,𝑣,𝑥,𝑦   𝑆,𝑎   𝑆,𝑏,𝑔,𝑥   𝑇,𝑎   𝑇,𝑏,𝑔,𝑥   𝑣,𝑢,𝑥,𝑦   𝑊,𝑎   𝑊,𝑏,𝑔,𝑥   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑏)   𝐵(𝑎)   𝑆(𝑦,𝑣,𝑢)   𝑇(𝑦,𝑣,𝑢)   𝑊(𝑦,𝑣,𝑢)

Proof of Theorem nosupinfsep

Step	Hyp	Ref	Expression
1		ssun1 4102	. . . . . . . 8 ⊢ dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇)
2		resabs1 5910	. . . . . . . 8 ⊢ (dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇) → ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) = (𝑊 ↾ dom 𝑆))
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) = (𝑊 ↾ dom 𝑆)
4	3	breq1i 5077	. . . . . 6 ⊢ (((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ (𝑊 ↾ dom 𝑆) <s 𝑆)
5	4	notbii 319	. . . . 5 ⊢ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
6		ssun2 4103	. . . . . . . 8 ⊢ dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇)
7		resabs1 5910	. . . . . . . 8 ⊢ (dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇) → ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) = (𝑊 ↾ dom 𝑇))
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) = (𝑊 ↾ dom 𝑇)
9	8	breq2i 5078	. . . . . 6 ⊢ (𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ 𝑇 <s (𝑊 ↾ dom 𝑇))
10	9	notbii 319	. . . . 5 ⊢ (¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))
11	5, 10	anbi12i 626	. . . 4 ⊢ ((¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)) ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
12	11	bicomi 223	. . 3 ⊢ ((¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)) ↔ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)))
13	12	a1i 11	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)) ↔ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇))))
14		simp1l 1195	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ⊆ No )
15		simp1r 1196	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ∈ V)
16		simp3 1136	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝑊 ∈ No )
17		nosupinfsep.1	. . . . 5 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
18	17	nosupbnd2 33846	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
19	14, 15, 16, 18	syl3anc 1369	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
20		simp2l 1197	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐵 ⊆ No )
21		simp2r 1198	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐵 ∈ V)
22		nosupinfsep.2	. . . . 5 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
23	22	noinfbnd2 33861	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
24	20, 21, 16, 23	syl3anc 1369	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
25	19, 24	anbi12d 630	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
26	17	nosupno 33833	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
27		nodmon 33780	. . . . . . . 8 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom 𝑆 ∈ On)
29	28	3ad2ant1 1131	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑆 ∈ On)
30	22	noinfno 33848	. . . . . . . 8 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
31		nodmon 33780	. . . . . . . 8 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → dom 𝑇 ∈ On)
33	32	3ad2ant2 1132	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑇 ∈ On)
34		onun2 6355	. . . . . 6 ⊢ ((dom 𝑆 ∈ On ∧ dom 𝑇 ∈ On) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
35	29, 33, 34	syl2anc 583	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
36		noreson 33790	. . . . 5 ⊢ ((𝑊 ∈ No ∧ (dom 𝑆 ∪ dom 𝑇) ∈ On) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
37	16, 35, 36	syl2anc 583	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
38	17	nosupbnd2 33846	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ ¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆))
39	14, 15, 37, 38	syl3anc 1369	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ ¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆))
40	20, 21, 37	3jca 1126	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ))
41	22	noinfbnd2 33861	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ) → (∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)))
42	40, 41	syl 17	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇)))
43	39, 42	anbi12d 630	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏) ↔ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇))))
44	13, 25, 43	3bitr4d 310	1 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ifcif 4456  {csn 4558  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580   ↾ cres 5582  Oncon0 6251  suc csuc 6253  ℩cio 6374  ‘cfv 6418  ℩crio 7211  1_oc1o 8260  2_oc2o 8261   No csur 33770   <s cslt 33771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775

This theorem is referenced by:  noetalem1  33871