Theorem readdscl 28650

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 28132	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	adantr 485	. . . 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 28497	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 +s 𝑚) ∈ ℕs)
4	3	adantr 485	. . . . . . . . 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 486	. . . . . . . 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 790	. . . . . . . . . . 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 28477	. . . . . . . . . 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 791	. . . . . . . . . . 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 28477	. . . . . . . . . 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 28201	. . . . . . . . . 10 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No ) → ( -us ‘(𝑛 +s 𝑚)) = (( -us ‘𝑛) +s ( -us ‘𝑚)))
11	7, 9, 10	syl2anc 595	. . . . . . . . 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 28188	. . . . . . . . . 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 28188	. . . . . . . . . 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 778	. . . . . . . . . 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 780	. . . . . . . . . 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 790	. . . . . . . . . . 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 486	. . . . . . . . . 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 792	. . . . . . . . . . 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 486	. . . . . . . . . 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 28164	. . . . . . . . 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 5127	. . . . . . . 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 791	. . . . . . . . . 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 486	. . . . . . . . 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 793	. . . . . . . . . 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 486	. . . . . . . . 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 28164	. . . . . . . 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 6871	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑛 +s 𝑚) → ( -us ‘𝑝) = ( -us ‘(𝑛 +s 𝑚)))
28	27	breq1d 5115	. . . . . . . . . 10 ⊢ (𝑝 = (𝑛 +s 𝑚) → (( -us ‘𝑝) <s (𝐴 +s 𝐵) ↔ ( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵)))
29		breq2 5109	. . . . . . . . . 10 ⊢ (𝑝 = (𝑛 +s 𝑚) → ((𝐴 +s 𝐵) <s 𝑝 ↔ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚)))
30	28, 29	anbi12d 643	. . . . . . . . 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 849	. . . . . . 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 461	. . . . . 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 3222	. . . . 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 487	. . . . . . 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 487	. . . . . . 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 624	. . . . . 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 3237	. . . . . 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 514	. . . 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 489	. . . . . 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 489	. . . . . 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 624	. . . . 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 778	. . . . . . . 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 28645	. . . . . . . 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 780	. . . . . . . 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 28645	. . . . . . . 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 782	. . . . . . 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 784	. . . . . . 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 28153	. . . . . 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 7433	. . . . . . . . . . . . . . 15 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
54		oveq1 7407	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
55	54	eqeq2d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵)))
56	53, 55	ceqsexv 3505	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
57		simpll 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
58		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → 𝐵 ∈ No )
59		1no 27961	. . . . . . . . . . . . . . . . . . 19 ⊢ 1s ∈ No
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
61		nnno 28475	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
62		nnne0s 28488	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
63	60, 61, 62	divscld 28375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
64	63	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
65	57, 58, 64	addsubsd 28233	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
66	65	eqeq2d 2776	. . . . . . . . . . . . . 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 3187	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . . 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 1871	. . . . . . . . . . . . 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 3292	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
73	72	rexbidv 3189	. . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
77	76	oveq2d 7416	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)))
78	77	eqeq2d 2776	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
79	78	cbvrexvw 3244	. . . . . . . . . . . 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 2831	. . . . . . . . . 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 7433	. . . . . . . . . . . . . . 15 ⊢ (𝐵 -s ( 1s /su 𝑚)) ∈ V
83		oveq2 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
84	83	eqeq2d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚)))))
85	82, 84	ceqsexv 3505	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
86		simpll 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → 𝐴 ∈ No )
87		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → 𝐵 ∈ No )
88	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 1s ∈ No )
89		nnno 28475	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
90		nnne0s 28488	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
91	88, 89, 90	divscld 28375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
92	91	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
93	86, 87, 92	addsubsassd 28232	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
94	93	eqeq2d 2776	. . . . . . . . . . . . . 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 3187	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . . 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 1871	. . . . . . . . . . . . 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 3292	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
101	100	rexbidv 3189	. . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑚 → ( 1s /su 𝑝) = ( 1s /su 𝑚))
105	104	oveq2d 7416	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑚 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)))
106	105	eqeq2d 2776	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
107	106	cbvrexvw 3244	. . . . . . . . . . . 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 2831	. . . . . . . . . 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 4125	. . . . . . . . 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 4113	. . . . . . . . 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 2816	. . . . . . . 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 7433	. . . . . . . . . . . . . . 15 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
114		oveq1 7407	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵))
115	114	eqeq2d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵)))
116	113, 115	ceqsexv 3505	. . . . . . . . . . . . . 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 28158	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
118	117	eqeq2d 2776	. . . . . . . . . . . . . 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 3187	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . . 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 1871	. . . . . . . . . . . . 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 3292	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
125	124	rexbidv 3189	. . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
129	128	eqeq2d 2776	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
130	129	cbvrexvw 3244	. . . . . . . . . . . 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 2831	. . . . . . . . . 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 7433	. . . . . . . . . . . . . . 15 ⊢ (𝐵 +s ( 1s /su 𝑚)) ∈ V
134		oveq2 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
135	134	eqeq2d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚)))))
136	133, 135	ceqsexv 3505	. . . . . . . . . . . . . 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 28157	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
138	137	eqeq2d 2776	. . . . . . . . . . . . . 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 3187	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . . 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 1871	. . . . . . . . . . . . 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 3292	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
145	144	rexbidv 3189	. . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑚 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)))
149	148	eqeq2d 2776	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
150	149	cbvrexvw 3244	. . . . . . . . . . . 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 2831	. . . . . . . . . 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 4125	. . . . . . . . 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 4113	. . . . . . . . 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 2816	. . . . . . . 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 7418	. . . . . . 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 485	. . . . . 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 2800	. . . . 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 604	. . . 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 524	. . 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 672	. 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 28642	. . 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 28642	. . 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 639	. 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 28642	. 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∃wrex 3089   ∪ cun 3905   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400   No csur 27762   <s clts 27763   <<s cslts 27908   |s ccuts 27910   1_s c1s 27957   +_s cadds 28110   -u_s cnegs 28170   -_s csubs 28171   /_su cdivs 28338  ℕ_scnns 28464  ℝ_screno 28640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-dc 10418

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-muls 28258  df-divs 28339  df-n0s 28465  df-nns 28466  df-reno 28641

This theorem is referenced by: (None)