Theorem nosupinfsep 27798

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   𝑊,𝑎,𝑥   𝑆,𝑎   𝑇,𝑏   𝑊,𝑏,𝑔   𝑆,𝑏,𝑔,𝑥   𝑢,𝐵,𝑦   𝑇,𝑎,𝑔,𝑥   𝐵,𝑏,𝑔,𝑣,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 4132	. . . . . . . 8 ⊢ dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇)
2		resabs1 5994	. . . . . . . 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 5109	. . . . . 6 ⊢ (((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ (𝑊 ↾ dom 𝑆) <s 𝑆)
5	4	notbii 322	. . . . 5 ⊢ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
6		ssun2 4133	. . . . . . . 8 ⊢ dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇)
7		resabs1 5994	. . . . . . . 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 5110	. . . . . 6 ⊢ (𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ 𝑇 <s (𝑊 ↾ dom 𝑇))
10	9	notbii 322	. . . . 5 ⊢ (¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))
11	5, 10	anbi12i 637	. . . 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 226	. . 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 1212	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ⊆ No )
15		simp1r 1213	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ∈ V)
16		simp3 1152	. . . 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 27782	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
19	14, 15, 16, 18	syl3anc 1392	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
20		simp2l 1214	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐵 ⊆ No )
21		simp2r 1215	. . . 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 27797	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
24	20, 21, 16, 23	syl3anc 1392	. . 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 641	. 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 27769	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
27		nodmon 27716	. . . . . . . 8 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom 𝑆 ∈ On)
29	28	3ad2ant1 1147	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑆 ∈ On)
30	22	noinfno 27784	. . . . . . . 8 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
31		nodmon 27716	. . . . . . . 8 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → dom 𝑇 ∈ On)
33	32	3ad2ant2 1148	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑇 ∈ On)
34		onun2 6458	. . . . . 6 ⊢ ((dom 𝑆 ∈ On ∧ dom 𝑇 ∈ On) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
35	29, 33, 34	syl2anc 593	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
36		noreson 27726	. . . . 5 ⊢ ((𝑊 ∈ No ∧ (dom 𝑆 ∪ dom 𝑇) ∈ On) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
37	16, 35, 36	syl2anc 593	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
38	17	nosupbnd2 27782	. . . 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 1392	. . 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 1142	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ))
41	22	noinfbnd2 27797	. . . 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 641	. 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 313	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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ∪ cun 3904   ⊆ wss 3906  ifcif 4482  {csn 4584  ⟨cop 4590   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5649   ↾ cres 5651  Oncon0 6348  suc csuc 6350  ℩cio 6477  ‘cfv 6523  ℩crio 7354  1_oc1o 8432  2_oc2o 8433   No csur 27706   <s clts 27707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-1o 8439  df-2o 8440  df-no 27709  df-lts 27710  df-bday 27711

This theorem is referenced by:  noetalem1  27807