Theorem addbday 28409

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 7443	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (bday‘(𝑥 +s 𝑦)) = (bday‘(𝑥𝑂 +s 𝑦)))
2		fveq2 6885	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (bday‘𝑥) = (bday‘𝑥𝑂))
3	2	oveq1d 7435	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝑥𝑂) +no (bday‘𝑦)))
4	1, 3	sseq12d 3941	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘(𝑥 +s 𝑦)) ⊆ ((bday‘𝑥) +no (bday‘𝑦)) ↔ (bday‘(𝑥𝑂 +s 𝑦)) ⊆ ((bday‘𝑥𝑂) +no (bday‘𝑦))))
5		oveq2 7428	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
6	5	fveq2d 6889	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (bday‘(𝑥𝑂 +s 𝑦)) = (bday‘(𝑥𝑂 +s 𝑦𝑂)))
7		fveq2 6885	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (bday‘𝑦) = (bday‘𝑦𝑂))
8	7	oveq2d 7436	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((bday‘𝑥𝑂) +no (bday‘𝑦)) = ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂)))
9	6, 8	sseq12d 3941	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((bday‘(𝑥𝑂 +s 𝑦)) ⊆ ((bday‘𝑥𝑂) +no (bday‘𝑦)) ↔ (bday‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂))))
10		fvoveq1 7443	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (bday‘(𝑥 +s 𝑦𝑂)) = (bday‘(𝑥𝑂 +s 𝑦𝑂)))
11	2	oveq1d 7435	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘𝑥) +no (bday‘𝑦𝑂)) = ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂)))
12	10, 11	sseq12d 3941	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘(𝑥 +s 𝑦𝑂)) ⊆ ((bday‘𝑥) +no (bday‘𝑦𝑂)) ↔ (bday‘(𝑥𝑂 +s 𝑦𝑂)) ⊆ ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂))))
13		fvoveq1 7443	. . 3 ⊢ (𝑥 = 𝐴 → (bday‘(𝑥 +s 𝑦)) = (bday‘(𝐴 +s 𝑦)))
14		fveq2 6885	. . . 4 ⊢ (𝑥 = 𝐴 → (bday‘𝑥) = (bday‘𝐴))
15	14	oveq1d 7435	. . 3 ⊢ (𝑥 = 𝐴 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝑦)))
16	13, 15	sseq12d 3941	. 2 ⊢ (𝑥 = 𝐴 → ((bday‘(𝑥 +s 𝑦)) ⊆ ((bday‘𝑥) +no (bday‘𝑦)) ↔ (bday‘(𝐴 +s 𝑦)) ⊆ ((bday‘𝐴) +no (bday‘𝑦))))
17		oveq2 7428	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
18	17	fveq2d 6889	. . 3 ⊢ (𝑦 = 𝐵 → (bday‘(𝐴 +s 𝑦)) = (bday‘(𝐴 +s 𝐵)))
19		fveq2 6885	. . . 4 ⊢ (𝑦 = 𝐵 → (bday‘𝑦) = (bday‘𝐵))
20	19	oveq2d 7436	. . 3 ⊢ (𝑦 = 𝐵 → ((bday‘𝐴) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝐵)))
21	18, 20	sseq12d 3941	. 2 ⊢ (𝑦 = 𝐵 → ((bday‘(𝐴 +s 𝑦)) ⊆ ((bday‘𝐴) +no (bday‘𝑦)) ↔ (bday‘(𝐴 +s 𝐵)) ⊆ ((bday‘𝐴) +no (bday‘𝐵))))
22		addsval2 28354	. . . . . 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 6889	. . . . 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 28370	. . . . 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 6140	. . . . . . 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 6140	. . . . . . . 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 6140	. . . . . . . 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 4165	. . . . . . 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 2783	. . . . . 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 28264	. . . . . . . . . . . . . . . . . . 19 ⊢ (L‘𝑥) ⊆ No
37		rightssno 28265	. . . . . . . . . . . . . . . . . . 19 ⊢ (R‘𝑥) ⊆ No
38	36, 37	unssi 4189	. . . . . . . . . . . . . . . . . 18 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) ⊆ No
39	38	sseli 3904	. . . . . . . . . . . . . . . . 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 28358	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))) → (𝑦 +s 𝑥𝑂) = (𝑥𝑂 +s 𝑦))
42	41	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))) → (bday‘(𝑦 +s 𝑥𝑂)) = (bday‘(𝑥𝑂 +s 𝑦)))
43		bdayon 28143	. . . . . . . . . . . . . . . 16 ⊢ (bday‘𝑦) ∈ On
44		bdayon 28143	. . . . . . . . . . . . . . . 16 ⊢ (bday‘𝑥𝑂) ∈ On
45		naddcom 8695	. . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . 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 3183	. . . . . . . . . . . 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 4176	. . . . . . . . . 10 ⊢ (L‘𝑥) ⊆ ((L‘𝑥) ∪ (R‘𝑥))
53	33, 51, 52	addbdaylem 28408	. . . . . . . . 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 3904	. . . . . . . . . . . . . . . 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 28358	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑥𝐿 +s 𝑦) = (𝑦 +s 𝑥𝐿))
58	57	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑥𝐿 ∈ (L‘𝑥)) → (𝑧 = (𝑥𝐿 +s 𝑦) ↔ 𝑧 = (𝑦 +s 𝑥𝐿)))
59	58	rexbidva 3184	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∃𝑥𝐿 ∈ (L‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦) ↔ ∃𝑥𝐿 ∈ (L‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)))
60	59	abbidv 2826	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → {𝑧 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑧 = (𝑥𝐿 +s 𝑦)} = {𝑧 ∣ ∃𝑥𝐿 ∈ (L‘𝑥)𝑧 = (𝑦 +s 𝑥𝐿)})
61	60	imaeq2d 6051	. . . . . . . . . 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 28143	. . . . . . . . . . 11 ⊢ (bday‘𝑥) ∈ On
64		naddcom 8695	. . . . . . . . . . 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 3963	. . . . . . . 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 4176	. . . . . . . . 9 ⊢ (L‘𝑦) ⊆ ((L‘𝑦) ∪ (R‘𝑦))
71	68, 69, 70	addbdaylem 28408	. . . . . . . 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 4190	. . . . . . 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 4177	. . . . . . . . . 10 ⊢ (R‘𝑥) ⊆ ((L‘𝑥) ∪ (R‘𝑥))
74	33, 51, 73	addbdaylem 28408	. . . . . . . . 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 3904	. . . . . . . . . . . . . . . 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 28358	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑥𝑅 +s 𝑦) = (𝑦 +s 𝑥𝑅))
79	78	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑥𝑅 ∈ (R‘𝑥)) → (𝑧 = (𝑥𝑅 +s 𝑦) ↔ 𝑧 = (𝑦 +s 𝑥𝑅)))
80	79	rexbidva 3184	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∃𝑥𝑅 ∈ (R‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦) ↔ ∃𝑥𝑅 ∈ (R‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)))
81	80	abbidv 2826	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → {𝑧 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑧 = (𝑥𝑅 +s 𝑦)} = {𝑧 ∣ ∃𝑥𝑅 ∈ (R‘𝑥)𝑧 = (𝑦 +s 𝑥𝑅)})
82	81	imaeq2d 6051	. . . . . . . . . 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 3963	. . . . . . . 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 4177	. . . . . . . . 9 ⊢ (R‘𝑦) ⊆ ((L‘𝑦) ∪ (R‘𝑦))
86	68, 69, 85	addbdaylem 28408	. . . . . . . 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 4190	. . . . . . 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 4190	. . . . . 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 3946	. . . . 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 8689	. . . . . . 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 28186	. . . . . 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 3942	. . 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 28346	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 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149   class class class wbr 5102   “ cima 5653  Oncon0 6362  ‘cfv 6538  (class class class)co 7420   +no cnadd 8677  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150  Lcleft 28216  Rcright 28217   +_s cadds 28350

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-adds 28351

This theorem is used by:  addonbday  28670  bdaypw2bnd  28856  z12bdaylem2  28862  z12bdaylem  28875