addsproplem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > addsproplem2		Structured version Visualization version GIF version

Theorem addsproplem2 28050

Description: Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)

**Hypotheses**
Ref	Expression
addsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
addsproplem2.2	⊢ (𝜑 → 𝑋 ∈ No )
addsproplem2.3	⊢ (𝜑 → 𝑌 ∈ No )

**Assertion**
Ref	Expression
addsproplem2	⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))

Distinct variable groups: 𝑋,𝑞,𝑡 𝑋,𝑝,𝑤 𝑌,𝑝,𝑤 𝑌,𝑞,𝑡 𝑥,𝑍,𝑦,𝑧 𝜑,𝑝,𝑟,𝑤 𝑋,𝑙,𝑚,𝑟,𝑠,𝑥,𝑦,𝑧 𝑌,𝑙,𝑚,𝑟,𝑠,𝑥,𝑦,𝑧 𝜑,𝑙,𝑞,𝑚,𝑠 𝜑,𝑡,𝑟,𝑠 𝑝,𝑙,𝑞,𝑟 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧) 𝑍(𝑤,𝑡,𝑚,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem addsproplem2 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6874	. . . . 5 ⊢ ( L ‘𝑋) ∈ V
2	1	abrexex 7937	. . . 4 ⊢ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∈ V)
4		fvex 6874	. . . . 5 ⊢ ( L ‘𝑌) ∈ V
5	4	abrexex 7937	. . . 4 ⊢ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ∈ V
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ∈ V)
7	3, 6	unexd 7731	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∈ V)
8		fvex 6874	. . . . 5 ⊢ ( R ‘𝑋) ∈ V
9	8	abrexex 7937	. . . 4 ⊢ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∈ V)
11		fvex 6874	. . . . 5 ⊢ ( R ‘𝑌) ∈ V
12	11	abrexex 7937	. . . 4 ⊢ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ∈ V
13	12	a1i 11	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ∈ V)
14	10, 13	unexd 7731	. 2 ⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ∈ V)
15		addsproplem.1	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
16	15	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
17		leftno 27957	. . . . . . . . 9 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ No )
18	17	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑙 ∈ No )
19		addsproplem2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
20	19	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑌 ∈ No )
21		0no 27889	. . . . . . . . 9 ⊢ 0_s ∈ No
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 0_s ∈ No )
23		bday0 27891	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0_s ) = ∅
24	23	oveq2i 7401	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = (( bday ‘𝑙) +no ∅)
25		bdayon 27832	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑙) ∈ On
26		naddrid 8647	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑙) ∈ On → (( bday ‘𝑙) +no ∅) = ( bday ‘𝑙))
27	25, 26	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ∅) = ( bday ‘𝑙)
28	24, 27	eqtri 2784	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = ( bday ‘𝑙)
29	28	uneq2i 4116	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙))
30		bdayon 27832	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑌) ∈ On
31		naddword1 8655	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)))
32	25, 30, 31	mp2an 702	. . . . . . . . . . 11 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌))
33		ssequn2 4139	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
34	32, 33	mpbi 232	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
35	29, 34	eqtri 2784	. . . . . . . . 9 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑌))
36		leftold 27955	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ ( O ‘( bday ‘𝑋)))
37		bdayon 27832	. . . . . . . . . . . . . 14 ⊢ ( bday ‘𝑋) ∈ On
38		oldbday 27981	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑙 ∈ No ) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
39	37, 17, 38	sylancr 596	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
40	36, 39	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → ( bday ‘𝑙) ∈ ( bday ‘𝑋))
41		naddel1 8651	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
42	25, 37, 30, 41	mp3an 1481	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
43	40, 42	sylib 220	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
44	43	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
45		elun1 4132	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
46	44, 45	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
47	35, 46	eqeltrid 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
48	16, 18, 20, 22, 47	addsproplem1 28049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑙) <s ( 0_s +_s 𝑙))))
49	48	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑙 +_s 𝑌) ∈ No )
50		eleq1a 2856	. . . . . 6 ⊢ ((𝑙 +_s 𝑌) ∈ No → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
51	49, 50	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
52	51	rexlimdva 3162	. . . 4 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
53	52	abssdv 4018	. . 3 ⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ⊆ No )
54	15	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
55		addsproplem2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ No )
56	55	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑋 ∈ No )
57		leftno 27957	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ No )
58	57	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑚 ∈ No )
59	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 0_s ∈ No )
60	23	oveq2i 7401	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = (( bday ‘𝑋) +no ∅)
61		naddrid 8647	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑋) ∈ On → (( bday ‘𝑋) +no ∅) = ( bday ‘𝑋))
62	37, 61	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ∅) = ( bday ‘𝑋)
63	60, 62	eqtri 2784	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = ( bday ‘𝑋)
64	63	uneq2i 4116	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋))
65		bdayon 27832	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑚) ∈ On
66		naddword1 8655	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)))
67	37, 65, 66	mp2an 702	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚))
68		ssequn2 4139	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
69	67, 68	mpbi 232	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
70	64, 69	eqtri 2784	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
71		leftold 27955	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
72		oldbday 27981	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
73	30, 57, 72	sylancr 596	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
74	71, 73	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ( L ‘𝑌) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
75		naddel2 8652	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
76	65, 30, 37, 75	mp3an 1481	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
77	74, 76	sylib 220	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ( L ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
78	77	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
79		elun1 4132	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
80	78, 79	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
81	70, 80	eqeltrid 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
82	54, 56, 58, 59, 81	addsproplem1 28049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
83	82	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑋 +_s 𝑚) ∈ No )
84		eleq1a 2856	. . . . . 6 ⊢ ((𝑋 +_s 𝑚) ∈ No → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
85	83, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
86	85	rexlimdva 3162	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
87	86	abssdv 4018	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ⊆ No )
88	53, 87	unssd 4142	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ⊆ No )
89	15	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
90		rightno 27958	. . . . . . . . 9 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ No )
91	90	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑟 ∈ No )
92	19	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑌 ∈ No )
93	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 0_s ∈ No )
94	23	oveq2i 7401	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = (( bday ‘𝑟) +no ∅)
95		bdayon 27832	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑟) ∈ On
96		naddrid 8647	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑟) ∈ On → (( bday ‘𝑟) +no ∅) = ( bday ‘𝑟))
97	95, 96	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ∅) = ( bday ‘𝑟)
98	94, 97	eqtri 2784	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = ( bday ‘𝑟)
99	98	uneq2i 4116	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟))
100		naddword1 8655	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)))
101	95, 30, 100	mp2an 702	. . . . . . . . . . 11 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌))
102		ssequn2 4139	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
103	101, 102	mpbi 232	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
104	99, 103	eqtri 2784	. . . . . . . . 9 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑌))
105		rightold 27956	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ ( O ‘( bday ‘𝑋)))
106		oldbday 27981	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑟 ∈ No ) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
107	37, 90, 106	sylancr 596	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
108	105, 107	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ( R ‘𝑋) → ( bday ‘𝑟) ∈ ( bday ‘𝑋))
109		naddel1 8651	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
110	95, 37, 30, 109	mp3an 1481	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
111	108, 110	sylib 220	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
112	111	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
113		elun1 4132	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
114	112, 113	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
115	104, 114	eqeltrid 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
116	89, 91, 92, 93, 115	addsproplem1 28049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
117	116	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑟 +_s 𝑌) ∈ No )
118		eleq1a 2856	. . . . . 6 ⊢ ((𝑟 +_s 𝑌) ∈ No → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
119	117, 118	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
120	119	rexlimdva 3162	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
121	120	abssdv 4018	. . 3 ⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ⊆ No )
122	15	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
123	55	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑋 ∈ No )
124		rightno 27958	. . . . . . . . 9 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ No )
125	124	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑠 ∈ No )
126	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 0_s ∈ No )
127	63	uneq2i 4116	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋))
128		bdayon 27832	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑠) ∈ On
129		naddword1 8655	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)))
130	37, 128, 129	mp2an 702	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠))
131		ssequn2 4139	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
132	130, 131	mpbi 232	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
133	127, 132	eqtri 2784	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑠))
134		rightold 27956	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ ( O ‘( bday ‘𝑌)))
135		oldbday 27981	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑠 ∈ No ) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
136	30, 124, 135	sylancr 596	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
137	134, 136	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ( R ‘𝑌) → ( bday ‘𝑠) ∈ ( bday ‘𝑌))
138		naddel2 8652	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
139	128, 30, 37, 138	mp3an 1481	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
140	137, 139	sylib 220	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ( R ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
141	140	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
142		elun1 4132	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
143	141, 142	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
144	133, 143	eqeltrid 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
145	122, 123, 125, 126, 144	addsproplem1 28049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((𝑋 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑋) <s ( 0_s +_s 𝑋))))
146	145	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑋 +_s 𝑠) ∈ No )
147		eleq1a 2856	. . . . . 6 ⊢ ((𝑋 +_s 𝑠) ∈ No → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
148	146, 147	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
149	148	rexlimdva 3162	. . . 4 ⊢ (𝜑 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
150	149	abssdv 4018	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ⊆ No )
151	121, 150	unssd 4142	. 2 ⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ⊆ No )
152		elun 4104	. . . . . . 7 ⊢ (𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ↔ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∨ 𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}))
153		vex 3457	. . . . . . . . 9 ⊢ 𝑎 ∈ V
154		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑙 +_s 𝑌) ↔ 𝑎 = (𝑙 +_s 𝑌)))
155	154	rexbidv 3185	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌)))
156	153, 155	elab 3637	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌))
157		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑞 = 𝑎 → (𝑞 = (𝑋 +_s 𝑚) ↔ 𝑎 = (𝑋 +_s 𝑚)))
158	157	rexbidv 3185	. . . . . . . . 9 ⊢ (𝑞 = 𝑎 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
159	153, 158	elab 3637	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚))
160	156, 159	orbi12i 925	. . . . . . 7 ⊢ ((𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∨ 𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
161	152, 160	bitri 277	. . . . . 6 ⊢ (𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
162		elun 4104	. . . . . . 7 ⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ↔ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))
163		vex 3457	. . . . . . . . 9 ⊢ 𝑏 ∈ V
164		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +_s 𝑌) ↔ 𝑏 = (𝑟 +_s 𝑌)))
165	164	rexbidv 3185	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
166	163, 165	elab 3637	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌))
167		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑋 +_s 𝑠) ↔ 𝑏 = (𝑋 +_s 𝑠)))
168	167	rexbidv 3185	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
169	163, 168	elab 3637	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))
170	166, 169	orbi12i 925	. . . . . . 7 ⊢ ((𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ↔ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
171	162, 170	bitri 277	. . . . . 6 ⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ↔ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
172	161, 171	anbi12i 637	. . . . 5 ⊢ ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) ↔ ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)) ∧ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))))
173		anddi 1023	. . . . 5 ⊢ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)) ∧ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ↔ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))))
174	172, 173	bitri 277	. . . 4 ⊢ ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) ↔ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))))
175		reeanv 3233	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
176		lltr 27942	. . . . . . . . . . . 12 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
177	176	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ( L ‘𝑋) <<s ( R ‘𝑋))
178		simprl 780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ ( L ‘𝑋))
179		simprr 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ ( R ‘𝑋))
180	177, 178, 179	sltssepcd 27852	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 <s 𝑟)
181	15	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
182	19	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
183	17	ad2antrl 738	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ No )
184	90	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
185		naddcom 8646	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
186	30, 25, 185	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
187	43	ad2antrl 738	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
188	186, 187	eqeltrid 2865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
189		naddcom 8646	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
190	30, 95, 189	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
191	111	ad2antll 739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
192	190, 191	eqeltrid 2865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
193		naddcl 8640	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On)
194	30, 25, 193	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On
195		naddcl 8640	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On)
196	30, 95, 195	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On
197		naddcl 8640	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On)
198	37, 30, 197	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On
199		onunel 6447	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On ∧ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
200	194, 196, 198, 199	mp3an 1481	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
201	188, 192, 200	sylanbrc 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
202		elun1 4132	. . . . . . . . . . . . 13 ⊢ (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
203	201, 202	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
204	181, 182, 183, 184, 203	addsproplem1 28049	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑌 +_s 𝑙) ∈ No ∧ (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))))
205	204	simprd 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
206	180, 205	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))
207		breq12 5102	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
208	206, 207	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
209	208	rexlimdvva 3218	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
210	175, 209	biimtrrid 245	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
211		reeanv 3233	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
212	49	adantrr 727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) ∈ No )
213	15	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
214	17	ad2antrl 738	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 ∈ No )
215	124	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
216	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 0_s ∈ No )
217	28	uneq2i 4116	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙))
218		naddword1 8655	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)))
219	25, 128, 218	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠))
220		ssequn2 4139	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
221	219, 220	mpbi 232	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
222	217, 221	eqtri 2784	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑠))
223		naddel1 8651	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠))))
224	25, 37, 128, 223	mp3an 1481	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
225	40, 224	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
226	225	ad2antrl 738	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
227	140	ad2antll 739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
228		ontr1 6387	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On → (((( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
229	198, 228	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
230	226, 227, 229	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
231		elun1 4132	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
232	230, 231	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
233	222, 232	eqeltrid 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
234	213, 214, 215, 216, 233	addsproplem1 28049	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑙) <s ( 0_s +_s 𝑙))))
235	234	simpld 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) ∈ No )
236	146	adantrl 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) ∈ No )
237		rightgt 27934	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑌 <s 𝑠)
238	237	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 <s 𝑠)
239	19	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 ∈ No )
240	43	ad2antrl 738	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
241		naddcl 8640	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On)
242	25, 30, 241	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On
243		naddcl 8640	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On)
244	25, 128, 243	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On
245		onunel 6447	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On ∧ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
246	242, 244, 198, 245	mp3an 1481	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
247	240, 230, 246	sylanbrc 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
248		elun1 4132	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
249	247, 248	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
250	213, 214, 239, 215, 249	addsproplem1 28049	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))))
251	250	simprd 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙)))
252	238, 251	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))
253	214, 239	addscomd 28047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) = (𝑌 +_s 𝑙))
254	214, 215	addscomd 28047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) = (𝑠 +_s 𝑙))
255	252, 253, 254	3brtr4d 5129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑙 +_s 𝑠))
256		leftlt 27933	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 <s 𝑋)
257	256	ad2antrl 738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 <s 𝑋)
258	55	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
259		naddcom 8646	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
260	128, 25, 259	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
261	260, 230	eqeltrid 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
262		naddcom 8646	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
263	128, 37, 262	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
264	263, 227	eqeltrid 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
265		naddcl 8640	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On)
266	128, 25, 265	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On
267		naddcl 8640	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On)
268	128, 37, 267	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On
269		onunel 6447	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On ∧ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
270	266, 268, 198, 269	mp3an 1481	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
271	261, 264, 270	sylanbrc 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
272		elun1 4132	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
273	271, 272	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
274	213, 215, 214, 258, 273	addsproplem1 28049	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑠 +_s 𝑙) ∈ No ∧ (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))))
275	274	simprd 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠)))
276	257, 275	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))
277	212, 235, 236, 255, 276	ltstrd 27814	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠))
278		breq12 5102	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠)))
279	277, 278	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
280	279	rexlimdvva 3218	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
281	211, 280	biimtrrid 245	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
282	210, 281	jaod 870	. . . . 5 ⊢ (𝜑 → (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) → 𝑎 <s 𝑏))
283		reeanv 3233	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) ↔ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
284	15	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
285	55	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 ∈ No )
286	57	ad2antrl 738	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ No )
287	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 0_s ∈ No )
288	77	ad2antrl 738	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
289	288, 79	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
290	70, 289	eqeltrid 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
291	284, 285, 286, 287, 290	addsproplem1 28049	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
292	291	simpld 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) ∈ No )
293	90	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
294	98	uneq2i 4116	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟))
295		naddword1 8655	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)))
296	95, 65, 295	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚))
297		ssequn2 4139	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
298	296, 297	mpbi 232	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
299	294, 298	eqtri 2784	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑚))
300		naddel1 8651	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚))))
301	95, 37, 65, 300	mp3an 1481	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
302	108, 301	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
303	302	ad2antll 739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
304		ontr1 6387	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On → (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
305	198, 304	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
306	303, 288, 305	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
307		elun1 4132	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
308	306, 307	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
309	299, 308	eqeltrid 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
310	284, 293, 286, 287, 309	addsproplem1 28049	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑟) <s ( 0_s +_s 𝑟))))
311	310	simpld 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) ∈ No )
312	19	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
313	111	ad2antll 739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
314	313, 113	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
315	104, 314	eqeltrid 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
316	284, 293, 312, 287, 315	addsproplem1 28049	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
317	316	simpld 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) ∈ No )
318		rightval 27930	. . . . . . . . . . . . . . . 16 ⊢ ( R ‘𝑋) = {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟}
319	318	eleq2i 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ ( R ‘𝑋) ↔ 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
320	319	biimpi 218	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
321	320	ad2antll 739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
322		rabid 3434	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟} ↔ (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
323	321, 322	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
324	323	simprd 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 <s 𝑟)
325		naddcom 8646	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
326	65, 37, 325	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
327	326, 288	eqeltrid 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
328		naddcom 8646	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
329	65, 95, 328	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
330	329, 306	eqeltrid 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
331		naddcl 8640	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On)
332	65, 37, 331	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On
333		naddcl 8640	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On)
334	65, 95, 333	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On
335		onunel 6447	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On ∧ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
336	332, 334, 198, 335	mp3an 1481	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
337	327, 330, 336	sylanbrc 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
338		elun1 4132	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
339	337, 338	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
340	284, 286, 285, 293, 339	addsproplem1 28049	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑚 +_s 𝑋) ∈ No ∧ (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))))
341	340	simprd 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚)))
342	324, 341	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))
343		leftval 27929	. . . . . . . . . . . . . . . . 17 ⊢ ( L ‘𝑌) = {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌}
344	343	eleq2i 2853	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ( L ‘𝑌) ↔ 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
345	344	biimpi 218	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
346	345	ad2antrl 738	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
347		rabid 3434	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌} ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
348	346, 347	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
349	348	simprd 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 <s 𝑌)
350		naddcl 8640	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On)
351	95, 65, 350	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On
352		naddcl 8640	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On)
353	95, 30, 352	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On
354		onunel 6447	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On ∧ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
355	351, 353, 198, 354	mp3an 1481	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
356	306, 313, 355	sylanbrc 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
357		elun1 4132	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
358	356, 357	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
359	284, 293, 286, 312, 358	addsproplem1 28049	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑚) ∈ No ∧ (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))))
360	359	simprd 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟)))
361	349, 360	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))
362	293, 286	addscomd 28047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) = (𝑚 +_s 𝑟))
363	293, 312	addscomd 28047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) = (𝑌 +_s 𝑟))
364	361, 362, 363	3brtr4d 5129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) <s (𝑟 +_s 𝑌))
365	292, 311, 317, 342, 364	ltstrd 27814	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌))
366		breq12 5102	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌)))
367	365, 366	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
368	367	rexlimdvva 3218	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
369	283, 368	biimtrrid 245	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
370		reeanv 3233	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) ↔ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
371		lltr 27942	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑌) <<s ( R ‘𝑌)
372	371	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ( L ‘𝑌) <<s ( R ‘𝑌))
373		simprl 780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ ( L ‘𝑌))
374		simprr 782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ ( R ‘𝑌))
375	372, 373, 374	sltssepcd 27852	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 <s 𝑠)
376	15	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
377	55	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
378	57	ad2antrl 738	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ No )
379	124	ad2antll 739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
380	77	ad2antrl 738	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
381	140	ad2antll 739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
382		naddcl 8640	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On)
383	37, 65, 382	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On
384		naddcl 8640	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On)
385	37, 128, 384	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On
386		onunel 6447	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
387	383, 385, 198, 386	mp3an 1481	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
388	380, 381, 387	sylanbrc 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
389		elun1 4132	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
390	388, 389	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
391	376, 377, 378, 379, 390	addsproplem1 28049	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))))
392	391	simprd 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋)))
393	375, 392	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))
394	377, 378	addscomd 28047	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) = (𝑚 +_s 𝑋))
395	377, 379	addscomd 28047	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) = (𝑠 +_s 𝑋))
396	393, 394, 395	3brtr4d 5129	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠))
397		breq12 5102	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠)))
398	396, 397	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
399	398	rexlimdvva 3218	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
400	370, 399	biimtrrid 245	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
401	369, 400	jaod 870	. . . . 5 ⊢ (𝜑 → (((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) → 𝑎 <s 𝑏))
402	282, 401	jaod 870	. . . 4 ⊢ (𝜑 → ((((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))) → 𝑎 <s 𝑏))
403	174, 402	biimtrid 244	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) → 𝑎 <s 𝑏))
404	403	3impib 1128	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) → 𝑎 <s 𝑏)
405	7, 14, 88, 151, 404	sltsd 27848	1 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 {cab 2739 ∀wral 3075 ∃wrex 3085 {crab 3413 Vcvv 3453 ∪ cun 3900 ⊆ wss 3902 ∅c0 4283 class class class wbr 5097 Oncon0 6340 ‘cfv 6515 (class class class)co 7390 +no cnadd 8628 No csur 27691 <s clts 27692 bday cbday 27693 <<s cslts 27837 0_s c0s 27885 O cold 27903 L cleft 27905 R cright 27906 +_s cadds 28039

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 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712

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 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-1o 8430 df-2o 8431 df-nadd 8629 df-no 27694 df-lts 27695 df-bday 27696 df-slts 27838 df-cuts 27840 df-0s 27887 df-made 27907 df-old 27908 df-left 27910 df-right 27911 df-norec2 28029 df-adds 28040

This theorem is referenced by: addsproplem3 28051

W3C validator