Theorem cutsun12 27860

Description: Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)

Assertion

Ref	Expression
cutsun12	⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))

Proof of Theorem cutsun12

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1148	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s 𝐵)
2		cutcuts 27851	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
3	1, 2	syl 17	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4	3	simp2d 1155	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
5		simp2 1149	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s {(𝐴 |s 𝐵)})
6		sltsun1 27858	. . . . 5 ⊢ ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝐶 <<s {(𝐴 |s 𝐵)}) → (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)})
7	4, 5, 6	syl2anc 593	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)})
8	3	simp3d 1156	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐵)
9		simp3 1150	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐷)
10		sltsun2 27859	. . . . 5 ⊢ (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))
11	8, 9, 10	syl2anc 593	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))
12		ovex 7425	. . . . . 6 ⊢ (𝐴 |s 𝐵) ∈ V
13	12	snnz 4734	. . . . 5 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
14		sltstr 27857	. . . . 5 ⊢ (((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷) ∧ {(𝐴 |s 𝐵)} ≠ ∅) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
15	13, 14	mp3an3 1470	. . . 4 ⊢ (((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
16	7, 11, 15	syl2anc 593	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
17		cutsval 27850	. . 3 ⊢ ((𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
18	16, 17	syl 17	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
19		vex 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20	19	elima 6051	. . . . . . . . 9 ⊢ (𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∃𝑧 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}𝑧 bday 𝑥)
21		sneq 4591	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
22	21	breq2d 5111	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {𝑧}))
23	21	breq1d 5109	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ({𝑦} <<s (𝐵 ∪ 𝐷) ↔ {𝑧} <<s (𝐵 ∪ 𝐷)))
24	22, 23	anbi12d 641	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷)) ↔ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))))
25	24	rexrab 3658	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}𝑧 bday 𝑥 ↔ ∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥))
26	20, 25	bitri 277	. . . . . . . 8 ⊢ (𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥))
27		simplr 778	. . . . . . . . . . . . 13 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝑧 ∈ No )
28		bdayfn 27818	. . . . . . . . . . . . . 14 ⊢ bday Fn No
29		fnbrfvb 6913	. . . . . . . . . . . . . 14 ⊢ (( bday Fn No ∧ 𝑧 ∈ No ) → (( bday ‘𝑧) = 𝑥 ↔ 𝑧 bday 𝑥))
30	28, 29	mpan 700	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ No → (( bday ‘𝑧) = 𝑥 ↔ 𝑧 bday 𝑥))
31	27, 30	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (( bday ‘𝑧) = 𝑥 ↔ 𝑧 bday 𝑥))
32		simpll1 1225	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝐴 <<s 𝐵)
33		cutbday 27854	. . . . . . . . . . . . . . 15 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
34	32, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
35		simprl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝐴 ∪ 𝐶) <<s {𝑧})
36		ssun1 4130	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐶)
37		ssslts1 27843	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ 𝐴 ⊆ (𝐴 ∪ 𝐶)) → 𝐴 <<s {𝑧})
38	36, 37	mpan2 701	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∪ 𝐶) <<s {𝑧} → 𝐴 <<s {𝑧})
39	35, 38	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝐴 <<s {𝑧})
40		simprr 782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → {𝑧} <<s (𝐵 ∪ 𝐷))
41		ssun1 4130	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
42		ssslts2 27844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑧} <<s (𝐵 ∪ 𝐷) ∧ 𝐵 ⊆ (𝐵 ∪ 𝐷)) → {𝑧} <<s 𝐵)
43	41, 42	mpan2 701	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑧} <<s (𝐵 ∪ 𝐷) → {𝑧} <<s 𝐵)
44	40, 43	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → {𝑧} <<s 𝐵)
45	39, 44	jca 519	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵))
46	21	breq2d 5111	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
47	21	breq1d 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ({𝑦} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
48	46, 47	anbi12d 641	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
49	48	elrab 3650	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑧 ∈ No ∧ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
50	27, 45, 49	sylanbrc 592	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
51		ssrab2 4033	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52		fnfvima 7213	. . . . . . . . . . . . . . . . 17 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No ∧ 𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
53	28, 51, 52	mp3an12 1471	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} → ( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
54	50, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
55		intss1 4920	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑧))
56	54, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑧))
57	34, 56	eqsstrd 3970	. . . . . . . . . . . . 13 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧))
58		sseq2 3962	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧) ↔ ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
59	58	biimpd 231	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
60	59	com12 32	. . . . . . . . . . . . 13 ⊢ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧) → (( bday ‘𝑧) = 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
61	57, 60	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (( bday ‘𝑧) = 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
62	31, 61	sylbird 262	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝑧 bday 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
63	62	ex 416	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) → (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) → (𝑧 bday 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)))
64	63	impd 414	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) → ((((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
65	64	rexlimdva 3162	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
66	26, 65	biimtrid 244	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
67	66	ralrimiv 3152	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∀𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)
68		ssint 4921	. . . . . 6 ⊢ (( bday ‘(𝐴 |s 𝐵)) ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∀𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)
69	67, 68	sylibr 236	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
70	3	simp1d 1154	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
71	7, 11	jca 519	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)))
72		sneq 4591	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
73	72	breq2d 5111	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)}))
74	72	breq1d 5109	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s (𝐵 ∪ 𝐷) ↔ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)))
75	73, 74	anbi12d 641	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 |s 𝐵) → (((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷)) ↔ ((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))))
76	75	elrab 3650	. . . . . . . 8 ⊢ ((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ↔ ((𝐴 |s 𝐵) ∈ No ∧ ((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))))
77	70, 71, 76	sylanbrc 592	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})
78		ssrab2 4033	. . . . . . . 8 ⊢ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ⊆ No
79		fnfvima 7213	. . . . . . . 8 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ⊆ No ∧ (𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
80	28, 78, 79	mp3an12 1471	. . . . . . 7 ⊢ ((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
81	77, 80	syl 17	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
82		intss1 4920	. . . . . 6 ⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) → ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
84	69, 83	eqssd 3953	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
85		conway 27849	. . . . . 6 ⊢ ((𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷) → ∃!𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
86	16, 85	syl 17	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∃!𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
87		fveqeq2 6872	. . . . . 6 ⊢ (𝑥 = (𝐴 |s 𝐵) → (( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
88	87	riota2 7374	. . . . 5 ⊢ (((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ∧ ∃!𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) → (( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) = (𝐴 |s 𝐵)))
89	77, 86, 88	syl2anc 593	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) = (𝐴 |s 𝐵)))
90	84, 89	mpbid 234	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) = (𝐴 |s 𝐵))
91	90	eqcomd 2767	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
92	18, 91	eqtr4d 2799	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  ∃!wreu 3364  {crab 3413   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4581  ∩ cint 4904   class class class wbr 5099   “ cima 5648   Fn wfn 6512  ‘cfv 6517  ℩crio 7348  (class class class)co 7392   No csur 27681   bday cbday 27683   <<s cslts 27827   |s ccuts 27829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1o 8432  df-2o 8433  df-no 27684  df-lts 27685  df-bday 27686  df-slts 27828  df-cuts 27830

This theorem is referenced by: (None)