Theorem readdscl 28350

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 27888	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	adantr 480	. . . 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 28238	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 +s 𝑚) ∈ ℕs)
4	3	adantr 480	. . . . . . . . 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 481	. . . . . . . 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 778	. . . . . . . . . . 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	nnsnod 28219	. . . . . . . . . 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 779	. . . . . . . . . . 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	nnsnod 28219	. . . . . . . . . 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 27956	. . . . . . . . . 10 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No ) → ( -us ‘(𝑛 +s 𝑚)) = (( -us ‘𝑛) +s ( -us ‘𝑚)))
11	7, 9, 10	syl2anc 584	. . . . . . . . 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 27943	. . . . . . . . . 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 27943	. . . . . . . . . 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 766	. . . . . . . . . 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 768	. . . . . . . . . 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 778	. . . . . . . . . . 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 481	. . . . . . . . . 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 780	. . . . . . . . . . 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 481	. . . . . . . . . 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	slt2addd 27920	. . . . . . . . 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 5129	. . . . . . . 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 779	. . . . . . . . . 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 481	. . . . . . . . 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 781	. . . . . . . . . 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 481	. . . . . . . . 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	slt2addd 27920	. . . . . . . 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 6858	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑛 +s 𝑚) → ( -us ‘𝑝) = ( -us ‘(𝑛 +s 𝑚)))
28	27	breq1d 5117	. . . . . . . . . 10 ⊢ (𝑝 = (𝑛 +s 𝑚) → (( -us ‘𝑝) <s (𝐴 +s 𝐵) ↔ ( -us ‘(𝑛 +s 𝑚)) <s (𝐴 +s 𝐵)))
29		breq2 5111	. . . . . . . . . 10 ⊢ (𝑝 = (𝑛 +s 𝑚) → ((𝐴 +s 𝐵) <s 𝑝 ↔ (𝐴 +s 𝐵) <s (𝑛 +s 𝑚)))
30	28, 29	anbi12d 632	. . . . . . . . 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 3588	. . . . . . . 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 836	. . . . . . 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 456	. . . . . 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 3194	. . . . 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 482	. . . . . . 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 482	. . . . . . 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 613	. . . . . 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 3209	. . . . . 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 234	. . . . 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 505	. . . 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 484	. . . . . 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 484	. . . . . 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 613	. . . . 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 766	. . . . . . . 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 28347	. . . . . . . 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 17	. . . . . . 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 768	. . . . . . . 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 28347	. . . . . . . 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 17	. . . . . . 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 770	. . . . . . 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 772	. . . . . . 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 27909	. . . . . 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 7420	. . . . . . . . . . . . . . 15 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
54		oveq1 7394	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
55	54	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵)))
56	53, 55	ceqsexv 3498	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝑡 +s 𝐵)) ↔ 𝑧 = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
57		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
58		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → 𝐵 ∈ No )
59		1sno 27739	. . . . . . . . . . . . . . . . . . 19 ⊢ 1s ∈ No
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
61		nnsno 28217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
62		nnne0s 28229	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
63	60, 61, 62	divscld 28126	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
64	63	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
65	57, 58, 64	addsubsd 27986	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)) = ((𝐴 -s ( 1s /su 𝑛)) +s 𝐵))
66	65	eqeq2d 2740	. . . . . . . . . . . . . 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 290	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 3167	. . . . . . . . . . . . . 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 1848	. . . . . . . . . . . . 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 3264	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
73	72	rexbidv 3157	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 -s ( 1s /su 𝑛))))
74	73	rexab 3666	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
77	76	oveq2d 7403	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛)))
78	77	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑛))))
79	78	cbvrexvw 3216	. . . . . . . . . . . 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 314	. . . . . . . . . . 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 2795	. . . . . . . . . 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 7420	. . . . . . . . . . . . . . 15 ⊢ (𝐵 -s ( 1s /su 𝑚)) ∈ V
83		oveq2 7395	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
84	83	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐵 -s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚)))))
85	82, 84	ceqsexv 3498	. . . . . . . . . . . . . 14 ⊢ (∃𝑡(𝑡 = (𝐵 -s ( 1s /su 𝑚)) ∧ 𝑧 = (𝐴 +s 𝑡)) ↔ 𝑧 = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
86		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → 𝐴 ∈ No )
87		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → 𝐵 ∈ No )
88	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 1s ∈ No )
89		nnsno 28217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
90		nnne0s 28229	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
91	88, 89, 90	divscld 28126	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
92	91	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
93	86, 87, 92	addsubsassd 27985	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 -s ( 1s /su 𝑚))))
94	93	eqeq2d 2740	. . . . . . . . . . . . . 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 290	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 3167	. . . . . . . . . . . . . 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 1848	. . . . . . . . . . . . 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 3264	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
101	100	rexbidv 3157	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 -s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑡 = (𝐵 -s ( 1s /su 𝑚))))
102	101	rexab 3666	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑚 → ( 1s /su 𝑝) = ( 1s /su 𝑚))
105	104	oveq2d 7403	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑚 → ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚)))
106	105	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) -s ( 1s /su 𝑚))))
107	106	cbvrexvw 3216	. . . . . . . . . . . 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 314	. . . . . . . . . . 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 2795	. . . . . . . . . 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 4132	. . . . . . . . 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 4120	. . . . . . . . 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 2780	. . . . . . . 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 7420	. . . . . . . . . . . . . . 15 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
114		oveq1 7394	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑡 +s 𝐵) = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵))
115	114	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = (𝑡 +s 𝐵) ↔ 𝑧 = ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵)))
116	113, 115	ceqsexv 3498	. . . . . . . . . . . . . 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 27914	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) +s 𝐵) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
118	117	eqeq2d 2740	. . . . . . . . . . . . . 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 283	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 3167	. . . . . . . . . . . . . 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 1848	. . . . . . . . . . . . 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 3264	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
125	124	rexbidv 3157	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑡 = (𝐴 +s ( 1s /su 𝑛))))
126	125	rexab 3666	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 7403	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛)))
129	128	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑛 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑛))))
130	129	cbvrexvw 3216	. . . . . . . . . . . 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 314	. . . . . . . . . . 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 2795	. . . . . . . . . 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 7420	. . . . . . . . . . . . . . 15 ⊢ (𝐵 +s ( 1s /su 𝑚)) ∈ V
134		oveq2 7395	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝐴 +s 𝑡) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
135	134	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝐵 +s ( 1s /su 𝑚)) → (𝑧 = (𝐴 +s 𝑡) ↔ 𝑧 = (𝐴 +s (𝐵 +s ( 1s /su 𝑚)))))
136	133, 135	ceqsexv 3498	. . . . . . . . . . . . . 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 27913	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑚 ∈ ℕs) → ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)) = (𝐴 +s (𝐵 +s ( 1s /su 𝑚))))
138	137	eqeq2d 2740	. . . . . . . . . . . . . 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 290	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 3167	. . . . . . . . . . . . . 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 1848	. . . . . . . . . . . . 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 3264	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
145	144	rexbidv 3157	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (∃𝑚 ∈ ℕs 𝑦 = (𝐵 +s ( 1s /su 𝑚)) ↔ ∃𝑚 ∈ ℕs 𝑡 = (𝐵 +s ( 1s /su 𝑚))))
146	145	rexab 3666	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 7403	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑚 → ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚)))
149	148	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑚 → (𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑝)) ↔ 𝑧 = ((𝐴 +s 𝐵) +s ( 1s /su 𝑚))))
150	149	cbvrexvw 3216	. . . . . . . . . . . 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 314	. . . . . . . . . . 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 2795	. . . . . . . . . 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 4132	. . . . . . . . 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 4120	. . . . . . . . 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 2780	. . . . . . . 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 7405	. . . . . . 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 480	. . . . . 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 2764	. . . . 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 593	. . . 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 515	. . 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 660	. 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 28346	. . 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 28346	. . 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 628	. 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 28346	. 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 292	1 ⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 +s 𝐵) ∈ ℝs)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ∪ cun 3912   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387   No csur 27551   <s cslt 27552   <<s csslt 27692   |s cscut 27694   1_s c1s 27735   +_s cadds 27866   -u_s cnegs 27925   -_s csubs 27926   /_su cdivs 28090  ℕ_scnns 28207  ℝ_screno 28344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-dc 10399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-norec2 27856  df-adds 27867  df-negs 27927  df-subs 27928  df-muls 28010  df-divs 28091  df-n0s 28208  df-nns 28209  df-reno 28345

This theorem is referenced by: (None)