Theorem nosupinfsep 33935

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 4106	. . . . . . . 8 ⊢ dom 𝑆 ⊆ (dom 𝑆 ∪ dom 𝑇)
2		resabs1 5921	. . . . . . . 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 5081	. . . . . 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 4107	. . . . . . . 8 ⊢ dom 𝑇 ⊆ (dom 𝑆 ∪ dom 𝑇)
7		resabs1 5921	. . . . . . . 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 5082	. . . . . 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 627	. . . 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 1196	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ⊆ No )
15		simp1r 1197	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐴 ∈ V)
16		simp3 1137	. . . 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 33919	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
19	14, 15, 16, 18	syl3anc 1370	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ↔ ¬ (𝑊 ↾ dom 𝑆) <s 𝑆))
20		simp2l 1198	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → 𝐵 ⊆ No )
21		simp2r 1199	. . . 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 33934	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ 𝑊 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑊 <s 𝑏 ↔ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇)))
24	20, 21, 16, 23	syl3anc 1370	. . 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 631	. 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 33906	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
27		nodmon 33853	. . . . . . . 8 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom 𝑆 ∈ On)
29	28	3ad2ant1 1132	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑆 ∈ On)
30	22	noinfno 33921	. . . . . . . 8 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
31		nodmon 33853	. . . . . . . 8 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → dom 𝑇 ∈ On)
33	32	3ad2ant2 1133	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → dom 𝑇 ∈ On)
34		onun2 6370	. . . . . 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 33863	. . . . 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 33919	. . . 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 1370	. . 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 1127	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → (𝐵 ⊆ No ∧ 𝐵 ∈ V ∧ (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∈ No ))
41	22	noinfbnd2 33934	. . . 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 631	. 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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ifcif 4459  {csn 4561  ⟨cop 4567   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589   ↾ cres 5591  Oncon0 6266  suc csuc 6268  ℩cio 6389  ‘cfv 6433  ℩crio 7231  1_oc1o 8290  2_oc2o 8291   No csur 33843   <s cslt 33844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848

This theorem is referenced by:  noetalem1  33944