Theorem addsbday 28059

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
addsbday	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Proof of Theorem addsbday

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 7468	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(𝑥𝑂 +s 𝑦)))
2		fveq2 6919	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘𝑥) = ( bday ‘𝑥𝑂))
3	2	oveq1d 7460	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
4	1, 3	sseq12d 4036	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘(𝑥 +s 𝑦)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦))))
5		oveq2 7453	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
6	5	fveq2d 6923	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ( bday ‘(𝑥𝑂 +s 𝑦)) = ( bday ‘(𝑥𝑂 +s 𝑦𝑂)))
7		fveq2 6919	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → ( bday ‘𝑦) = ( bday ‘𝑦𝑂))
8	7	oveq2d 7461	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)))
9	6, 8	sseq12d 4036	. 2 ⊢ (𝑦 = 𝑦𝑂 → (( bday ‘(𝑥𝑂 +s 𝑦)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂))))
10		fvoveq1 7468	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘(𝑥 +s 𝑦𝑂)) = ( bday ‘(𝑥𝑂 +s 𝑦𝑂)))
11	2	oveq1d 7460	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)))
12	10, 11	sseq12d 4036	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘(𝑥 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ↔ ( bday ‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂))))
13		fvoveq1 7468	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(𝐴 +s 𝑦)))
14		fveq2 6919	. . . 4 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
15	14	oveq1d 7460	. . 3 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝑦)))
16	13, 15	sseq12d 4036	. 2 ⊢ (𝑥 = 𝐴 → (( bday ‘(𝑥 +s 𝑦)) ⊆ (( bday ‘𝑥) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝐴 +s 𝑦)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦))))
17		oveq2 7453	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
18	17	fveq2d 6923	. . 3 ⊢ (𝑦 = 𝐵 → ( bday ‘(𝐴 +s 𝑦)) = ( bday ‘(𝐴 +s 𝐵)))
19		fveq2 6919	. . . 4 ⊢ (𝑦 = 𝐵 → ( bday ‘𝑦) = ( bday ‘𝐵))
20	19	oveq2d 7461	. . 3 ⊢ (𝑦 = 𝐵 → (( bday ‘𝐴) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
21	18, 20	sseq12d 4036	. 2 ⊢ (𝑦 = 𝐵 → (( bday ‘(𝐴 +s 𝑦)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦)) ↔ ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵))))
22		addsval2 28005	. . . . . 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 6923	. . . . 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 480	. . . 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 482	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → 𝑥 ∈ No )
26		simpr 484	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → 𝑦 ∈ No )
27	25, 26	addscut2 28021	. . . . 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 6180	. . . . . . 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 6180	. . . . . . . 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 6180	. . . . . . . 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 4183	. . . . . . 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 2762	. . . . . 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 768	. . . . . . . . . 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 1195	. . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → 𝑦 ∈ No )
36		leftssno 27928	. . . . . . . . . . . . . . . . . . 19 ⊢ ( L ‘𝑥) ⊆ No
37		rightssno 27929	. . . . . . . . . . . . . . . . . . 19 ⊢ ( R ‘𝑥) ⊆ No
38	36, 37	unssi 4208	. . . . . . . . . . . . . . . . . 18 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) ⊆ No
39	38	sseli 3998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) → 𝑥𝑂 ∈ No )
40	39	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → 𝑥𝑂 ∈ No )
41	35, 40	addscomd 28009	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑦 +s 𝑥𝑂) = (𝑥𝑂 +s 𝑦))
42	41	fveq2d 6923	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → ( bday ‘(𝑦 +s 𝑥𝑂)) = ( bday ‘(𝑥𝑂 +s 𝑦)))
43		bdayelon 27830	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑦) ∈ On
44		bdayelon 27830	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑥𝑂) ∈ On
45		naddcom 8734	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑦) ∈ On ∧ ( bday ‘𝑥𝑂) ∈ On) → (( bday ‘𝑦) +no ( bday ‘𝑥𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
46	43, 44, 45	mp2an 691	. . . . . . . . . . . . . . 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 4036	. . . . . . . . . . . . 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 3178	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 257	. . . . . . . . . 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 4195	. . . . . . . . . 10 ⊢ ( L ‘𝑥) ⊆ (( L ‘𝑥) ∪ ( R ‘𝑥))
53	33, 51, 52	addsbdaylem 28058	. . . . . . . . 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 3998	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ No )
55	54	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑥𝐿 ∈ No )
56		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → 𝑦 ∈ No )
57	55, 56	addscomd 28009	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑥𝐿 +s 𝑦) = (𝑦 +s 𝑥𝐿))
58	57	eqeq2d 2745	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝐿 ∈ ( L ‘𝑥)) → (𝑧 = (𝑥𝐿 +s 𝑦) ↔ 𝑧 = (𝑦 +s 𝑥𝐿)))
59	58	rexbidva 3179	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)))
60	59	abbidv 2805	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} = {𝑧 ∣ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)})
61	60	imaeq2d 6088	. . . . . . . . . 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 480	. . . . . . . . 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		bdayelon 27830	. . . . . . . . . . 11 ⊢ ( bday ‘𝑥) ∈ On
64		naddcom 8734	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑦) +no ( bday ‘𝑥)))
65	63, 43, 64	mp2an 691	. . . . . . . . . 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 4050	. . . . . . . 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 766	. . . . . . . . 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 1196	. . . . . . . . 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 4195	. . . . . . . . 9 ⊢ ( L ‘𝑦) ⊆ (( L ‘𝑦) ∪ ( R ‘𝑦))
71	68, 69, 70	addsbdaylem 28058	. . . . . . . 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 4209	. . . . . . 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 4196	. . . . . . . . . 10 ⊢ ( R ‘𝑥) ⊆ (( L ‘𝑥) ∪ ( R ‘𝑥))
74	33, 51, 73	addsbdaylem 28058	. . . . . . . . 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 3998	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅 ∈ ( R ‘𝑥) → 𝑥𝑅 ∈ No )
76	75	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑥𝑅 ∈ No )
77		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → 𝑦 ∈ No )
78	76, 77	addscomd 28009	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑥𝑅 +s 𝑦) = (𝑦 +s 𝑥𝑅))
79	78	eqeq2d 2745	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ 𝑥𝑅 ∈ ( R ‘𝑥)) → (𝑧 = (𝑥𝑅 +s 𝑦) ↔ 𝑧 = (𝑦 +s 𝑥𝑅)))
80	79	rexbidva 3179	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦) ↔ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)))
81	80	abbidv 2805	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} = {𝑧 ∣ ∃𝑥𝑅 ∈ ( R ‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)})
82	81	imaeq2d 6088	. . . . . . . . . 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 480	. . . . . . . . 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 4050	. . . . . . . 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 4196	. . . . . . . . 9 ⊢ ( R ‘𝑦) ⊆ (( L ‘𝑦) ∪ ( R ‘𝑦))
86	68, 69, 85	addsbdaylem 28058	. . . . . . . 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 4209	. . . . . . 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 4209	. . . . . 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 4051	. . . . 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 8729	. . . . . . 7 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘𝑥) +no ( bday ‘𝑦)) ∈ On)
91	63, 43, 90	mp2an 691	. . . . . 6 ⊢ (( bday ‘𝑥) +no ( bday ‘𝑦)) ∈ On
92		scutbdaybnd 27869	. . . . . 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 1449	. . . . 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 684	. . . 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 4041	. . 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 412	. 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 27997	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2103  {cab 2711  ∀wral 3063  ∃wrex 3072   ∪ cun 3968   ⊆ wss 3970   class class class wbr 5169   “ cima 5702  Oncon0 6394  ‘cfv 6572  (class class class)co 7445   +no cnadd 8717   No csur 27693   bday cbday 27695   <<s csslt 27834   |s cscut 27836   L cleft 27893   R cright 27894   +_s cadds 28001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-1o 8518  df-2o 8519  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sslt 27835  df-scut 27837  df-0s 27878  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec2 27991  df-adds 28002

This theorem is referenced by: (None)