Theorem renegscl 28742

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

Assertion

Ref	Expression
renegscl	⊢ (𝐴 ∈ ℝs → ( -us ‘𝐴) ∈ ℝs)

Proof of Theorem renegscl

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

Step	Hyp	Ref	Expression
1		negscl 28280	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
2	1	adantr 486	. . 3 ⊢ ((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))}))) → ( -us ‘𝐴) ∈ No )
3		nnno 28568	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
4	3	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝑛 ∈ No )
5	4	negscld 28281	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘𝑛) ∈ No )
6		simpl 488	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
7	5, 6	ltnegsd 28291	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝑛) <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘( -us ‘𝑛))))
8		negnegs 28288	. . . . . . . . . . 11 ⊢ (𝑛 ∈ No → ( -us ‘( -us ‘𝑛)) = 𝑛)
9	4, 8	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( -us ‘𝑛)) = 𝑛)
10	9	breq2d 5123	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) <s ( -us ‘( -us ‘𝑛)) ↔ ( -us ‘𝐴) <s 𝑛))
11	7, 10	bitrd 282	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝑛) <s 𝐴 ↔ ( -us ‘𝐴) <s 𝑛))
12	6, 4	ltnegsd 28291	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -us ‘𝑛) <s ( -us ‘𝐴)))
13	11, 12	anbi12d 644	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -us ‘𝐴) <s 𝑛 ∧ ( -us ‘𝑛) <s ( -us ‘𝐴))))
14	13	biancomd 469	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛)))
15	14	rexbidva 3189	. . . . 5 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛)))
16	15	biimpa 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛))
17	16	adantrr 730	. . 3 ⊢ ((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))}))) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛))
18		recut 28738	. . . . . 6 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
19	18	adantr 486	. . . . 5 ⊢ ((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))}))) → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
20		simprr 785	. . . . 5 ⊢ ((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))}))) → 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
21	19, 20	negsunif 28299	. . . 4 ⊢ ((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))}))) → ( -us ‘𝐴) = (( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})))
22		negsfn 28267	. . . . . . . . 9 ⊢ -us Fn No
23		sltsss2 28010	. . . . . . . . . 10 ⊢ ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
24	18, 23	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
25		fvelimab 6957	. . . . . . . . 9 ⊢ (( -us Fn No ∧ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No ) → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦))
26	22, 24, 25	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦))
27		eqeq1 2769	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
28	27	rexbidv 3191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexab 3660	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
30		rexcom4 3294	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
31		ovex 7452	. . . . . . . . . . . . 13 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
32		fveqeq2 6894	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us ‘𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
33	31, 32	ceqsexv 3505	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
34	33	rexbii 3114	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
35		r19.41v 3197	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
36	35	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
37	30, 34, 36	3bitr3ri 305	. . . . . . . . . 10 ⊢ (∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
38	29, 37	bitri 278	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
39		1no 28054	. . . . . . . . . . . . . . . . 17 ⊢ 1s ∈ No
40	39	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
41		nnne0s 28581	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
42	40, 3, 41	divscld 28468	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
43	42	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44		negsdi 28294	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
45	43, 44	syldan 603	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
46	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘𝐴) ∈ No )
47	46, 43	subsvald 28305	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) -s ( 1s /su 𝑛)) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
48	45, 47	eqtr4d 2803	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) -s ( 1s /su 𝑛)))
49	48	eqeq1d 2767	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us ‘𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50		eqcom 2772	. . . . . . . . . . 11 ⊢ ((( -us ‘𝐴) -s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛)))
51	49, 50	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))))
52	51	rexbidva 3189	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))))
53	38, 52	bitrid 286	. . . . . . . 8 ⊢ (𝐴 ∈ No → (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))))
54	26, 53	bitrd 282	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))))
55	54	eqabdv 2898	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))})
56		sltsss1 28009	. . . . . . . . . 10 ⊢ ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
57	18, 56	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
58		fvelimab 6957	. . . . . . . . 9 ⊢ (( -us Fn No ∧ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No ) → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦))
59	22, 57, 58	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦))
60		eqeq1 2769	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
61	60	rexbidv 3191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
62	61	rexab 3660	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
63		rexcom4 3294	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
64		ovex 7452	. . . . . . . . . . . . 13 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
65		fveqeq2 6894	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us ‘𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
66	64, 65	ceqsexv 3505	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
67	66	rexbii 3114	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
68		r19.41v 3197	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
69	68	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦))
70	63, 67, 69	3bitr3ri 305	. . . . . . . . . 10 ⊢ (∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
71	62, 70	bitri 278	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
72	6, 43	subsvald 28305	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
73	72	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
74	43	negscld 28281	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75		negsdi 28294	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ ( -us ‘( 1s /su 𝑛)) ∈ No ) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
76	74, 75	syldan 603	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
77		negnegs 28288	. . . . . . . . . . . . . . 15 ⊢ (( 1s /su 𝑛) ∈ No → ( -us ‘( -us ‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
78	43, 77	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( -us ‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
79	78	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
80	73, 76, 79	3eqtrd 2804	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
81	80	eqeq1d 2767	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us ‘𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82		eqcom 2772	. . . . . . . . . . 11 ⊢ ((( -us ‘𝐴) +s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
83	81, 82	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))))
84	83	rexbidva 3189	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))))
85	71, 84	bitrid 286	. . . . . . . 8 ⊢ (𝐴 ∈ No → (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))))
86	59, 85	bitrd 282	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))))
87	86	eqabdv 2898	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))})
88	55, 87	oveq12d 7437	. . . . 5 ⊢ (𝐴 ∈ No → (( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))}))
89	88	adantr 486	. . . 4 ⊢ ((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))}))) → (( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))}))
90	21, 89	eqtrd 2800	. . 3 ⊢ ((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))}))) → ( -us ‘𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))}))
91	2, 17, 90	jca32 525	. 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))}))) → (( -us ‘𝐴) ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛) ∧ ( -us ‘𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))}))))
92		elreno 28735	. 2 ⊢ (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))}))))
93		elreno 28735	. 2 ⊢ (( -us ‘𝐴) ∈ ℝs ↔ (( -us ‘𝐴) ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛) ∧ ( -us ‘𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))}))))
94	91, 92, 93	3imtr4i 295	1 ⊢ (𝐴 ∈ ℝs → ( -us ‘𝐴) ∈ ℝs)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∃wrex 3091   ⊆ wss 3906   class class class wbr 5111   “ cima 5666   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   No csur 27855   <s clts 27856   <<s cslts 28001   |s ccuts 28003   1_s c1s 28050   +_s cadds 28203   -u_s cnegs 28263   -_s csubs 28264   /_su cdivs 28431  ℕ_scnns 28557  ℝ_screno 28733

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 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-dc 10445

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-nadd 8658  df-no 27858  df-lts 27859  df-bday 27860  df-les 27960  df-slts 28002  df-cuts 28004  df-0s 28051  df-1s 28052  df-made 28071  df-old 28072  df-left 28074  df-right 28075  df-norec 28182  df-norec2 28193  df-adds 28204  df-negs 28265  df-subs 28266  df-muls 28351  df-divs 28432  df-n0s 28558  df-nns 28559  df-reno 28734

This theorem is used by: (None)