Theorem nosupinfsep 27777

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 4178	. . . . . . . 8 ⊢ dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇)
2		resabs1 6024	. . . . . . . 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 5150	. . . . . 6 ⊢ (((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ (𝑊 ↾ dom 𝑆) <s 𝑆)
5	4	notbii 320	. . . . 5 ⊢ (¬ ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑆) <s 𝑆 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
6		ssun2 4179	. . . . . . . 8 ⊢ dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇)
7		resabs1 6024	. . . . . . . 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 5151	. . . . . 6 ⊢ (𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ 𝑇 <s (𝑊 ↾ dom 𝑇))
10	9	notbii 320	. . . . 5 ⊢ (¬ 𝑇 <s ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↾ dom 𝑇) ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))
11	5, 10	anbi12i 628	. . . 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 224	. . 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 1198	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ⊆ No )
15		simp1r 1199	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ∈ V)
16		simp3 1139	. . . 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 27761	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
19	14, 15, 16, 18	syl3anc 1373	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
20		simp2l 1200	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐵 ⊆ No )
21		simp2r 1201	. . . 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 27776	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
24	20, 21, 16, 23	syl3anc 1373	. . 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 632	. 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 27748	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
27		nodmon 27695	. . . . . . . 8 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom 𝑆 ∈ On)
29	28	3ad2ant1 1134	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑆 ∈ On)
30	22	noinfno 27763	. . . . . . . 8 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
31		nodmon 27695	. . . . . . . 8 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → dom 𝑇 ∈ On)
33	32	3ad2ant2 1135	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑇 ∈ On)
34		onun2 6492	. . . . . 6 ⊢ ((dom 𝑆 ∈ On ∧ dom 𝑇 ∈ On) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
35	29, 33, 34	syl2anc 584	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (dom 𝑆 ∪ dom 𝑇) ∈ On)
36		noreson 27705	. . . . 5 ⊢ ((𝑊 ∈ No ∧ (dom 𝑆 ∪ dom 𝑇) ∈ On) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
37	16, 35, 36	syl2anc 584	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No )
38	17	nosupbnd2 27761	. . . 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 1373	. . 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 1129	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ))
41	22	noinfbnd2 27776	. . . 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 632	. 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 311	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  ifcif 4525  {csn 4626  ⟨cop 4632   class class class wbr 5143   ↦ cmpt 5225  dom cdm 5685   ↾ cres 5687  Oncon0 6384  suc csuc 6386  ℩cio 6512  ‘cfv 6561  ℩crio 7387  1_oc1o 8499  2_oc2o 8500   No csur 27684   <s cslt 27685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689

This theorem is referenced by:  noetalem1  27786