Theorem addbday 28386

Description: The birthday of the sum of two surreals is less than or equal to the natural ordinal sum of their individual birthdays. Theorem 6.1 of [Gonshor] p. 95. (Contributed by Scott Fenton, 12-Aug-2025.)

Assertion

Ref	Expression
addbday	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Proof of Theorem addbday

Dummy variables 𝑥 𝑦 𝑥_𝑂 𝑦_𝑂 𝑥_𝐿 𝑦_𝐿 𝑥_𝑅 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 7435	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(𝑥𝑂 +s 𝑦)))
2		fveq2 6877	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘𝑥) = ( bday ‘𝑥𝑂))
3	2	oveq1d 7427	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
4	1, 3	sseq12d 3964	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘(𝑥 +s 𝑦)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦))))
5		oveq2 7420	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
6	5	fveq2d 6881	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ( bday ‘(𝑥𝑂 +s 𝑦)) = ( bday ‘(𝑥𝑂 +s 𝑦𝑂)))
7		fveq2 6877	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → ( bday ‘𝑦) = ( bday ‘𝑦𝑂))
8	7	oveq2d 7428	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)))
9	6, 8	sseq12d 3964	. 2 ⊢ (𝑦 = 𝑦𝑂 → (( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂))))
10		fvoveq1 7435	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘(𝑥 +s 𝑦𝑂)) = ( bday ‘(𝑥𝑂 +s 𝑦𝑂)))
11	2	oveq1d 7427	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)))
12	10, 11	sseq12d 3964	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂))))
13		fvoveq1 7435	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(𝐴 +s 𝑦)))
14		fveq2 6877	. . . 4 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
15	14	oveq1d 7427	. . 3 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝑦)))
16	13, 15	sseq12d 3964	. 2 ⊢ (𝑥 = 𝐴 → (( bday ‘(𝑥 +s 𝑦)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝐴 +s 𝑦)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦))))
17		oveq2 7420	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
18	17	fveq2d 6881	. . 3 ⊢ (𝑦 = 𝐵 → ( bday ‘(𝐴 +s 𝑦)) = ( bday ‘(𝐴 +s 𝐵)))
19		fveq2 6877	. . . 4 ⊢ (𝑦 = 𝐵 → ( bday ‘𝑦) = ( bday ‘𝐵))
20	19	oveq2d 7428	. . 3 ⊢ (𝑦 = 𝐵 → (( bday ‘𝐴) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
21	18, 20	sseq12d 3964	. 2 ⊢ (𝑦 = 𝐵 → (( bday ‘(𝐴 +s 𝑦)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵))))
22		addsval2 28331	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑥 +s 𝑦) = (({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) |s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})))
23	22	fveq2d 6881	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) |s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))))
24	23	adantr 486	. . . 4 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) |s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))))
25		simpl 488	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → 𝑥 ∈ No )
26		simpr 490	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → 𝑦 ∈ No )
27	25, 26	addcuts2 28347	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) <<s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))
28		imaundi 6139	. . . . . . 7 ⊢ ( bday “ (({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∪ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) = (( bday “ ({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ∪ ( bday “ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})))
29		imaundi 6139	. . . . . . . 8 ⊢ ( bday “ ({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) = (( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))
30		imaundi 6139	. . . . . . . 8 ⊢ ( bday “ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) = (( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))
31	29, 30	uneq12i 4113	. . . . . . 7 ⊢ (( bday “ ({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ∪ ( bday “ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) = ((( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ∪ (( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})))
32	28, 31	eqtri 2784	. . . . . 6 ⊢ ( bday “ (({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∪ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) = ((( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ∪ (( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})))
33		simplr 781	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → 𝑦 ∈ No )
34		simpr2 1214	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
35		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → 𝑦 ∈ No )
36		leftssno 28241	. . . . . . . . . . . . . . . . . . 19 ⊢ ( L ‘𝑥) ⊆ No
37		rightssno 28242	. . . . . . . . . . . . . . . . . . 19 ⊢ ( R ‘𝑥) ⊆ No
38	36, 37	unssi 4137	. . . . . . . . . . . . . . . . . 18 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) ⊆ No
39	38	sseli 3927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) → 𝑥𝑂 ∈ No )
40	39	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → 𝑥𝑂 ∈ No )
41	35, 40	addscomd 28335	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑦 +s 𝑥𝑂) = (𝑥𝑂 +s 𝑦))
42	41	fveq2d 6881	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → ( bday ‘(𝑦 +s 𝑥𝑂)) = ( bday ‘(𝑥𝑂 +s 𝑦)))
43		bdayon 28120	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑦) ∈ On
44		bdayon 28120	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑥𝑂) ∈ On
45		naddcom 8676	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑦) ∈ On ∧ ( bday ‘𝑥𝑂) ∈ On) → (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
46	43, 44, 45	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦))
47	46	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
48	42, 47	sseq12d 3964	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (( bday ‘(𝑦 +s 𝑥𝑂)) ⊆ (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦))))
49	48	ralbidva 3184	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑦 +s 𝑥𝑂)) ⊆ (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦))))
50	49	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑦 +s 𝑥𝑂)) ⊆ (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦))))
51	34, 50	mpbird 260	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑦 +s 𝑥𝑂)) ⊆ (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)))
52		ssun1 4124	. . . . . . . . . 10 ⊢ ( L ‘𝑥) ⊆ (( L ‘𝑥) ∪ ( R ‘𝑥))
53	33, 51, 52	addbdaylem 28385	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)}) ⊆ (( bday ‘𝑦) +no ( bday ‘𝑥)))
54	36	sseli 3927	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ No )
55	54	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ No )
56		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑦 ∈ No )
57	55, 56	addscomd 28335	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s 𝑦) = (𝑦 +s 𝑥𝐿))
58	57	eqeq2d 2772	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑧 = (𝑥𝐿 +s 𝑦) ↔ 𝑧 = (𝑦 +s 𝑥𝐿)))
59	58	rexbidva 3185	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)))
60	59	abbidv 2827	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} = {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)})
61	60	imaeq2d 6054	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) = ( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)}))
62	61	adantr 486	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) = ( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)}))
63		bdayon 28120	. . . . . . . . . . 11 ⊢ ( bday ‘𝑥) ∈ On
64		naddcom 8676	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑦) +no ( bday ‘𝑥)))
65	63, 43, 64	mp2an 705	. . . . . . . . . 10 ⊢ (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑦) +no ( bday ‘𝑥))
66	65	a1i 11	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑦) +no ( bday ‘𝑥)))
67	53, 62, 66	3sstr4d 3986	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
68		simpll 779	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → 𝑥 ∈ No )
69		simpr3 1215	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))
70		ssun1 4124	. . . . . . . . 9 ⊢ ( L ‘𝑦) ⊆ (( L ‘𝑦) ∪ ( R ‘𝑦))
71	68, 69, 70	addbdaylem 28385	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
72	67, 71	unssd 4138	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → (( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
73		ssun2 4125	. . . . . . . . . 10 ⊢ ( R ‘𝑥) ⊆ (( L ‘𝑥) ∪ ( R ‘𝑥))
74	33, 51, 73	addbdaylem 28385	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)}) ⊆ (( bday ‘𝑦) +no ( bday ‘𝑥)))
75	37	sseli 3927	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ No )
76	75	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ No )
77		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑦 ∈ No )
78	76, 77	addscomd 28335	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s 𝑦) = (𝑦 +s 𝑥𝑅))
79	78	eqeq2d 2772	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑧 = (𝑥𝑅 +s 𝑦) ↔ 𝑧 = (𝑦 +s 𝑥𝑅)))
80	79	rexbidva 3185	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)))
81	80	abbidv 2827	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} = {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)})
82	81	imaeq2d 6054	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) = ( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)}))
83	82	adantr 486	. . . . . . . . 9 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) = ( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)}))
84	74, 83, 66	3sstr4d 3986	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
85		ssun2 4125	. . . . . . . . 9 ⊢ ( R ‘𝑦) ⊆ (( L ‘𝑦) ∪ ( R ‘𝑦))
86	68, 69, 85	addbdaylem 28385	. . . . . . . 8 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
87	84, 86	unssd 4138	. . . . . . 7 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → (( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
88	72, 87	unssd 4138	. . . . . 6 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ((( bday “ {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)})) ∪ (( bday “ {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)}) ∪ ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
89	32, 88	eqsstrid 3969	. . . . 5 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday “ (({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∪ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
90		naddcl 8670	. . . . . . 7 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘𝑥) +no ( bday ‘𝑦)) ∈ On)
91	63, 43, 90	mp2an 705	. . . . . 6 ⊢ (( bday ‘𝑥) +no ( bday ‘𝑦)) ∈ On
92		cutbdaybnd 28163	. . . . . 6 ⊢ ((({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) <<s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∧ (( bday ‘𝑥) +no ( bday ‘𝑦)) ∈ On ∧ ( bday “ (({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∪ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦))) → ( bday ‘(({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) |s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
93	91, 92	mp3an2 1478	. . . . 5 ⊢ ((({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) <<s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∧ ( bday “ (({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) ∪ ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦))) → ( bday ‘(({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) |s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
94	27, 89, 93	syl2an2r 698	. . . 4 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday ‘(({𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}) |s ({𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑦𝐿 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑦𝐿)}))) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
95	24, 94	eqsstrd 3965	. . 3 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)))) → ( bday ‘(𝑥 +s 𝑦)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)))
96	95	ex 418	. 2 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂))) → ( bday ‘(𝑥 +s 𝑦)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦))))
97	4, 9, 12, 16, 21, 96	no2inds 28323	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∪ cun 3897   ⊆ wss 3899   class class class wbr 5103   “ cima 5654  Oncon0 6355  ‘cfv 6531  (class class class)co 7412   +no cnadd 8658   No csur 27979   bday cbday 27981   <<s cslts 28125   |s ccuts 28127   L cleft 28193   R cright 28194   +_s cadds 28327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-0s 28175  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec2 28317  df-adds 28328

This theorem is used by:  addonbday  28647  bdaypw2bnd  28833  z12bdaylem2  28839  z12bdaylem  28852