Theorem cutsun12 28181

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 1154	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s 𝐵)
2		cutcuts 28172	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
3	1, 2	syl 18	. . . . . 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 1161	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
5		simp2 1155	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s {(𝐴 |s 𝐵)})
6		sltsun1 28179	. . . . 5 ⊢ ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝐶 <<s {(𝐴 |s 𝐵)}) → (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)})
7	4, 5, 6	syl2anc 596	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)})
8	3	simp3d 1162	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐵)
9		simp3 1156	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐷)
10		sltsun2 28180	. . . . 5 ⊢ (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))
11	8, 9, 10	syl2anc 596	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))
12		ovex 7453	. . . . . 6 ⊢ (𝐴 |s 𝐵) ∈ V
13	12	snnz 4733	. . . . 5 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
14		sltstr 28178	. . . . 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 1479	. . . 4 ⊢ (((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
16	7, 11, 15	syl2anc 596	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
17		cutsval 28171	. . 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 18	. 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 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20	19	elima 6060	. . . . . . . . 9 ⊢ (𝑥 ∈ (bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∃𝑧 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}𝑧bday𝑥)
21		sneq 4590	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
22	21	breq2d 5114	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {𝑧}))
23	21	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ({𝑦} <<s (𝐵 ∪ 𝐷) ↔ {𝑧} <<s (𝐵 ∪ 𝐷)))
24	22, 23	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷)) ↔ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))))
25	24	rexrab 3653	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}𝑧bday𝑥 ↔ ∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧bday𝑥))
26	20, 25	bitri 278	. . . . . . . 8 ⊢ (𝑥 ∈ (bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧bday𝑥))
27		simplr 781	. . . . . . . . . . . . 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 28139	. . . . . . . . . . . . . 14 ⊢ bday Fn No
29		fnbrfvb 6935	. . . . . . . . . . . . . 14 ⊢ ((bday Fn No ∧ 𝑧 ∈ No) → ((bday‘𝑧) = 𝑥 ↔ 𝑧bday𝑥))
30	28, 29	mpan 703	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ No → ((bday‘𝑧) = 𝑥 ↔ 𝑧bday𝑥))
31	27, 30	syl 18	. . . . . . . . . . . 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 ↔ 𝑧bday𝑥))
32		simpll1 1231	. . . . . . . . . . . . . . 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 28175	. . . . . . . . . . . . . . 15 ⊢ (𝐴 <<s 𝐵 → (bday‘(𝐴 |s 𝐵)) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
34	32, 33	syl 18	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐶)
37		ssslts1 28164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ 𝐴 ⊆ (𝐴 ∪ 𝐶)) → 𝐴 <<s {𝑧})
38	36, 37	mpan2 704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∪ 𝐶) <<s {𝑧} → 𝐴 <<s {𝑧})
39	35, 38	syl 18	. . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
42		ssslts2 28165	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑧} <<s (𝐵 ∪ 𝐷) ∧ 𝐵 ⊆ (𝐵 ∪ 𝐷)) → {𝑧} <<s 𝐵)
43	41, 42	mpan2 704	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑧} <<s (𝐵 ∪ 𝐷) → {𝑧} <<s 𝐵)
44	40, 43	syl 18	. . . . . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
47	21	breq1d 5112	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ({𝑦} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
48	46, 47	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
49	48	elrab 3644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑧 ∈ No ∧ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
50	27, 45, 49	sylanbrc 595	. . . . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52		fnfvima 7239	. . . . . . . . . . . . . . . . 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 1480	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 4922	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝑧) = 𝑥 → ((bday‘(𝐴 |s 𝐵)) ⊆ (bday‘𝑧) ↔ (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥))
59	58	biimpd 232	. . . . . . . . . . . . . 14 ⊢ ((bday‘𝑧) = 𝑥 → ((bday‘(𝐴 |s 𝐵)) ⊆ (bday‘𝑧) → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥))
60	59	com12 33	. . . . . . . . . . . . 13 ⊢ ((bday‘(𝐴 |s 𝐵)) ⊆ (bday‘𝑧) → ((bday‘𝑧) = 𝑥 → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥))
61	57, 60	syl 18	. . . . . . . . . . . 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 263	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝑧bday𝑥 → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥))
63	62	ex 418	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No) → (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) → (𝑧bday𝑥 → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥)))
64	63	impd 416	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No) → ((((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧bday𝑥) → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥))
65	64	rexlimdva 3163	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧bday𝑥) → (bday‘(𝐴 |s 𝐵)) ⊆ 𝑥))
66	26, 65	biimtrid 245	. . . . . . 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 3153	. . . . . 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 4923	. . . . . 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 237	. . . . 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 1160	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ No)
71	7, 11	jca 521	. . . . . . . 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 4590	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
73	72	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)}))
74	72	breq1d 5112	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s (𝐵 ∪ 𝐷) ↔ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)))
75	73, 74	anbi12d 644	. . . . . . . . 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 3644	. . . . . . . 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 595	. . . . . . 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 4005	. . . . . . . 8 ⊢ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ⊆ No
79		fnfvima 7239	. . . . . . . 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 1480	. . . . . . 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 18	. . . . . 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 4922	. . . . . 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 18	. . . . 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 3925	. . . 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 28170	. . . . . 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 18	. . . . 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 6894	. . . . . 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 7402	. . . . 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 596	. . . 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 235	. . 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 2766	. 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 2798	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  {crab 3412   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4584  ∩ cint 4906   class class class wbr 5102   “ cima 5653   Fn wfn 6533  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151

This theorem is used by: (None)