Theorem readdscl 28729

Description: The surreal reals are closed under addition. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
readdscl	⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 +s 𝐵) ∈ ℝs)

Proof of Theorem readdscl

Dummy variables 𝑛 𝑚 𝑝 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addscl 28211	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	adantr 486	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → (𝐴 +s 𝐵) ∈ No )
3		nnaddscl 28576	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 +s 𝑚) ∈ ℕs)
4	3	adantr 486	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))) → (𝑛 +s 𝑚) ∈ ℕs)
5	4	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → (𝑛 +s 𝑚) ∈ ℕs)
6		simprll 791	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝑛 ∈ ℕs)
7	6	nnnod 28556	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝑛 ∈ No )
8		simprlr 792	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝑚 ∈ ℕs)
9	8	nnnod 28556	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝑚 ∈ No )
10		negsdi 28280	. . . . . . . . . 10 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No ) → ( -us ‘(𝑛 +s 𝑚)) = (( -us ‘𝑛) +s ( -us ‘𝑚)))
11	7, 9, 10	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ( -us ‘(𝑛 +s 𝑚)) = (( -us ‘𝑛) +s ( -us ‘𝑚)))
12	7	negscld 28267	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ( -us ‘𝑛) ∈ No )
13	9	negscld 28267	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ( -us ‘𝑚) ∈ No )
14		simpll 779	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝐴 ∈ No )
15		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝐵 ∈ No )
16		simprll 791	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))) → ( -us ‘𝑛) <s 𝐴)
17	16	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ( -us ‘𝑛) <s 𝐴)
18		simprrl 793	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))) → ( -us ‘𝑚) <s 𝐵)
19	18	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ( -us ‘𝑚) <s 𝐵)
20	12, 13, 14, 15, 17, 19	lt2addsd 28243	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → (( -us ‘𝑛) +s ( -us ‘𝑚)) <s (𝐴 +s 𝐵))
21	11, 20	eqbrtrd 5138	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵))
22		simprlr 792	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))) → 𝐴 <s 𝑛)
23	22	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝐴 <s 𝑛)
24		simprrr 794	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))) → 𝐵 <s 𝑚)
25	24	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → 𝐵 <s 𝑚)
26	14, 15, 7, 9, 23, 25	lt2addsd 28243	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → (𝐴 +s 𝐵) <s (𝑛 +s 𝑚))
27		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑛 +s 𝑚) → ( -us ‘𝑝) = ( -us ‘(𝑛 +s 𝑚)))
28	27	breq1d 5124	. . . . . . . . . 10 ⊢ (𝑝 = (𝑛 +s 𝑚) → (( -us ‘𝑝) <s (𝐴 +s 𝐵) ↔ ( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵)))
29		breq2 5118	. . . . . . . . . 10 ⊢ (𝑝 = (𝑛 +s 𝑚) → ((𝐴 +s 𝐵) <s 𝑝 ↔ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚)))
30	28, 29	anbi12d 644	. . . . . . . . 9 ⊢ (𝑝 = (𝑛 +s 𝑚) → ((( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝) ↔ (( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚))))
31	30	rspcev 3584	. . . . . . . 8 ⊢ (((𝑛 +s 𝑚) ∈ ℕs ∧ (( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚))) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝))
32	5, 21, 26, 31	syl12anc 850	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) ∧ ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝))
33	32	expr 462	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝)))
34	33	rexlimdvva 3225	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝)))
35		simpl 488	. . . . . . 7 ⊢ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛))
36		simpl 488	. . . . . . 7 ⊢ ((∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})) → ∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚))
37	35, 36	anim12i 625	. . . . . 6 ⊢ (((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
38		reeanv 3240	. . . . . 6 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)) ↔ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
39	37, 38	sylibr 237	. . . . 5 ⊢ (((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚)))
40	34, 39	impel 515	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝))
41		simpr 490	. . . . . 6 ⊢ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) → 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
42		simpr 490	. . . . . 6 ⊢ ((∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})) → 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))
43	41, 42	anim12i 625	. . . . 5 ⊢ (((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))
44		simpll 779	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → 𝐴 ∈ No )
45		recut 28724	. . . . . . . 8 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
46	44, 45	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
47		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → 𝐵 ∈ No )
48		recut 28724	. . . . . . . 8 ⊢ (𝐵 ∈ No → {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} <<s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})
49	47, 48	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} <<s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})
50		simprl 783	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
51		simprr 785	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))
52	46, 49, 50, 51	addsunif 28232	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (𝐴 +s 𝐵) = (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)})))
53		ovex 7456	. . . . . . . . . . . . . . 15 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
54		oveq1 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
55	54	eqeq2d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵)))
56	53, 55	ceqsexv 3506	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
57		simpll 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
58		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → 𝐵 ∈ No )
59		1no 28040	. . . . . . . . . . . . . . . . . . 19 ⊢ 1s ∈ No
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
61		nnno 28554	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
62		nnne0s 28567	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
63	60, 61, 62	divscld 28454	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
64	63	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
65	57, 58, 64	addsubsd 28312	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
66	65	eqeq2d 2777	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵)))
67	56, 66	bitr4id 293	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
68	67	rexbidva 3190	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
69		r19.41v 3198	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
70	69	exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
71		rexcom4 3295	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
72		eqeq1 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
73	72	rexbidv 3192	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
74	73	rexab 3661	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
75	70, 71, 74	3bitr4ri 307	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
76		oveq2 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
77	76	oveq2d 7439	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)))
78	77	eqeq2d 2777	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
79	78	cbvrexvw 3247	. . . . . . . . . . . 12 ⊢ (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)))
80	68, 75, 79	3bitr4g 317	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))))
81	80	abbidv 2832	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))})
82		ovex 7456	. . . . . . . . . . . . . . 15 ⊢ (𝐵 -s ( 1s /su 𝑚)) ∈ V
83		oveq2 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
84	83	eqeq2d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚)))))
85	82, 84	ceqsexv 3506	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
86		simpll 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → 𝐴 ∈ No )
87		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → 𝐵 ∈ No )
88	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 1s ∈ No )
89		nnno 28554	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
90		nnne0s 28567	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
91	88, 89, 90	divscld 28454	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
92	91	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
93	86, 87, 92	addsubsassd 28311	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
94	93	eqeq2d 2777	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚)))))
95	85, 94	bitr4id 293	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → (∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
96	95	rexbidva 3190	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑚 ∈ ℕs ∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
97		r19.41v 3198	. . . . . . . . . . . . . 14 ⊢ (∃𝑚 ∈ ℕs (𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ (∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
98	97	exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑚 ∈ ℕs (𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
99		rexcom4 3295	. . . . . . . . . . . . 13 ⊢ (∃𝑚 ∈ ℕs ∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡∃𝑚 ∈ ℕs (𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
100		eqeq1 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
101	100	rexbidv 3192	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
102	101	rexab 3661	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
103	98, 99, 102	3bitr4ri 307	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑚 ∈ ℕs ∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
104		oveq2 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑚 → ( 1s /su 𝑝) = ( 1s /su 𝑚))
105	104	oveq2d 7439	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑚 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)))
106	105	eqeq2d 2777	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
107	106	cbvrexvw 3247	. . . . . . . . . . . 12 ⊢ (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)))
108	96, 103, 107	3bitr4g 317	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))))
109	108	abbidv 2832	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))})
110	81, 109	uneq12d 4126	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))}))
111		unidm 4114	. . . . . . . . 9 ⊢ ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))}
112	110, 111	eqtrdi 2817	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))})
113		ovex 7456	. . . . . . . . . . . . . . 15 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
114		oveq1 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵))
115	114	eqeq2d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵)))
116	113, 115	ceqsexv 3506	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵))
117	57, 64, 58	adds32d 28237	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
118	117	eqeq2d 2777	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
119	116, 118	bitrid 286	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → (∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
120	119	rexbidva 3190	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
121		r19.41v 3198	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ (∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
122	121	exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
123		rexcom4 3295	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ ∃𝑡∃𝑛 ∈ ℕs (𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
124		eqeq1 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
125	124	rexbidv 3192	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
126	125	rexab 3661	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑡(∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
127	122, 123, 126	3bitr4ri 307	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑛 ∈ ℕs ∃𝑡(𝑡 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)))
128	76	oveq2d 7439	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
129	128	eqeq2d 2777	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
130	129	cbvrexvw 3247	. . . . . . . . . . . 12 ⊢ (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
131	120, 127, 130	3bitr4g 317	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))))
132	131	abbidv 2832	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))})
133		ovex 7456	. . . . . . . . . . . . . . 15 ⊢ (𝐵 +s ( 1s /su 𝑚)) ∈ V
134		oveq2 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
135	134	eqeq2d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚)))))
136	133, 135	ceqsexv 3506	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
137	86, 87, 92	addsassd 28236	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
138	137	eqeq2d 2777	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚)))))
139	136, 138	bitr4id 293	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → (∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
140	139	rexbidva 3190	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑚 ∈ ℕs ∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
141		r19.41v 3198	. . . . . . . . . . . . . 14 ⊢ (∃𝑚 ∈ ℕs (𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ (∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
142	141	exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑡∃𝑚 ∈ ℕs (𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
143		rexcom4 3295	. . . . . . . . . . . . 13 ⊢ (∃𝑚 ∈ ℕs ∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ ∃𝑡∃𝑚 ∈ ℕs (𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
144		eqeq1 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
145	144	rexbidv 3192	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
146	145	rexab 3661	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑡(∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
147	142, 143, 146	3bitr4ri 307	. . . . . . . . . . . 12 ⊢ (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑚 ∈ ℕs ∃𝑡(𝑡 = (𝐵 +s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)))
148	104	oveq2d 7439	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑚 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)))
149	148	eqeq2d 2777	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
150	149	cbvrexvw 3247	. . . . . . . . . . . 12 ⊢ (∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ ∃𝑚 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)))
151	140, 147, 150	3bitr4g 317	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡) ↔ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))))
152	151	abbidv 2832	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)} = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))})
153	132, 152	uneq12d 4126	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
154		unidm 4114	. . . . . . . . 9 ⊢ ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))} ∪ {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}
155	153, 154	eqtrdi 2817	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) = {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))})
156	112, 155	oveq12d 7441	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)})) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
157	156	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)}) |s ({𝑧 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}𝑧 = (𝑡 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑡 ∈ {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}𝑧 = (𝐴 +s 𝑡)})) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
158	52, 157	eqtrd 2801	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))) → (𝐴 +s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
159	43, 158	sylan2 605	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → (𝐴 +s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))
160	2, 40, 159	jca32 525	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})) ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ((𝐴 +s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝) ∧ (𝐴 +s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))))
161	160	an4s 673	. 2 ⊢ (((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) ∧ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))) → ((𝐴 +s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝) ∧ (𝐴 +s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))))
162		elreno 28721	. . 3 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))
163		elreno 28721	. . 3 ⊢ (𝐵 ∈ ℝs ↔ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))}))))
164	162, 163	anbi12i 640	. 2 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) ↔ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) ∧ (𝐵 ∈ No ∧ (∃𝑚 ∈ ℕs (( -us ‘𝑚) <s 𝐵 ∧ 𝐵 <s 𝑚) ∧ 𝐵 = ({𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚))} |s {𝑦 ∣ ∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚))})))))
165		elreno 28721	. 2 ⊢ ((𝐴 +s 𝐵) ∈ ℝs ↔ ((𝐴 +s 𝐵) ∈ No ∧ (∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 +s 𝐵) ∧ (𝐴 +s 𝐵) <s 𝑝) ∧ (𝐴 +s 𝐵) = ({𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝))} |s {𝑧 ∣ ∃𝑝 ∈ ℕs 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝))}))))
166	161, 164, 165	3imtr4i 295	1 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 +s 𝐵) ∈ ℝs)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2744  ∃wrex 3092   ∪ cun 3906   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423   No csur 27841   <s clts 27842   <<s cslts 27987   |s ccuts 27989   1_s c1s 28036   +_s cadds 28189   -u_s cnegs 28249   -_s csubs 28250   /_su cdivs 28417  ℕ_scnns 28543  ℝ_screno 28719

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620  ax-dc 10448

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-1s 28038  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-muls 28337  df-divs 28418  df-n0s 28544  df-nns 28545  df-reno 28720

This theorem is used by: (None)