Theorem renegscl 28893

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 → ( -s‘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 28431	. . . 4 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
2	1	adantr 486	. . 3 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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))}))) → ( -s‘𝐴) ∈ No)
3		nnno 28719	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No)
4	3	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝑛 ∈ No)
5	4	negscld 28432	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘𝑛) ∈ No)
6		simpl 488	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No)
7	5, 6	ltnegsd 28442	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝑛) <s 𝐴 ↔ ( -s‘𝐴) <s ( -s‘( -s‘𝑛))))
8		negnegs 28439	. . . . . . . . . . 11 ⊢ (𝑛 ∈ No → ( -s‘( -s‘𝑛)) = 𝑛)
9	4, 8	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘( -s‘𝑛)) = 𝑛)
10	9	breq2d 5114	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝐴) <s ( -s‘( -s‘𝑛)) ↔ ( -s‘𝐴) <s 𝑛))
11	7, 10	bitrd 282	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝑛) <s 𝐴 ↔ ( -s‘𝐴) <s 𝑛))
12	6, 4	ltnegsd 28442	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -s‘𝑛) <s ( -s‘𝐴)))
13	11, 12	anbi12d 644	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -s‘𝐴) <s 𝑛 ∧ ( -s‘𝑛) <s ( -s‘𝐴))))
14	13	biancomd 469	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -s‘𝑛) <s ( -s‘𝐴) ∧ ( -s‘𝐴) <s 𝑛)))
15	14	rexbidva 3184	. . . . 5 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ ∃𝑛 ∈ ℕs (( -s‘𝑛) <s ( -s‘𝐴) ∧ ( -s‘𝐴) <s 𝑛)))
16	15	biimpa 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)) → ∃𝑛 ∈ ℕs (( -s‘𝑛) <s ( -s‘𝐴) ∧ ( -s‘𝐴) <s 𝑛))
17	16	adantrr 730	. . 3 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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 (( -s‘𝑛) <s ( -s‘𝐴) ∧ ( -s‘𝐴) <s 𝑛))
18		recut 28889	. . . . . 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 (( -s‘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 (( -s‘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 28450	. . . 4 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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))}))) → ( -s‘𝐴) = (( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))})))
22		negsfn 28418	. . . . . . . . 9 ⊢ -s Fn No
23		sltsss2 28161	. . . . . . . . . 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 6962	. . . . . . . . 9 ⊢ (( -s Fn No ∧ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No) → (𝑦 ∈ ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦))
26	22, 24, 25	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦))
27		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
28	27	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexab 3652	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦 ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
30		rexcom4 3289	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
31		ovex 7457	. . . . . . . . . . . . 13 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
32		fveqeq2 6899	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -s‘𝑧) = 𝑦 ↔ ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
33	31, 32	ceqsexv 3498	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
34	33	rexbii 3109	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
35		r19.41v 3192	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
36	35	exbii 1880	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
37	30, 34, 36	3bitr3ri 305	. . . . . . . . . 10 ⊢ (∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
38	29, 37	bitri 278	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
39		1no 28205	. . . . . . . . . . . . . . . . 17 ⊢ 1s ∈ No
40	39	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 1s ∈ No)
41		nnne0s 28732	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
42	40, 3, 41	divscld 28619	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No)
43	42	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No)
44		negsdi 28445	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No) → ( -s‘(𝐴 +s ( 1s /su 𝑛))) = (( -s‘𝐴) +s ( -s‘( 1s /su 𝑛))))
45	43, 44	syldan 603	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 +s ( 1s /su 𝑛))) = (( -s‘𝐴) +s ( -s‘( 1s /su 𝑛))))
46	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘𝐴) ∈ No)
47	46, 43	subsvald 28456	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝐴) −s ( 1s /su 𝑛)) = (( -s‘𝐴) +s ( -s‘( 1s /su 𝑛))))
48	45, 47	eqtr4d 2798	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 +s ( 1s /su 𝑛))) = (( -s‘𝐴) −s ( 1s /su 𝑛)))
49	48	eqeq1d 2762	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -s‘𝐴) −s ( 1s /su 𝑛)) = 𝑦))
50		eqcom 2767	. . . . . . . . . . 11 ⊢ ((( -s‘𝐴) −s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛)))
51	49, 50	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))))
52	51	rexbidva 3184	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))))
53	38, 52	bitrid 286	. . . . . . . 8 ⊢ (𝐴 ∈ No → (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))))
54	26, 53	bitrd 282	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))))
55	54	eqabdv 2893	. . . . . 6 ⊢ (𝐴 ∈ No → ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))})
56		sltsss1 28160	. . . . . . . . . 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 6962	. . . . . . . . 9 ⊢ (( -s Fn No ∧ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ⊆ No) → (𝑦 ∈ ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦))
59	22, 57, 58	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦))
60		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 −s ( 1s /su 𝑛))))
61	60	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 −s ( 1s /su 𝑛))))
62	61	rexab 3652	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦 ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
63		rexcom4 3289	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
64		ovex 7457	. . . . . . . . . . . . 13 ⊢ (𝐴 −s ( 1s /su 𝑛)) ∈ V
65		fveqeq2 6899	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 −s ( 1s /su 𝑛)) → (( -s‘𝑧) = 𝑦 ↔ ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦))
66	64, 65	ceqsexv 3498	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦)
67	66	rexbii 3109	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕs ∃𝑧(𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦)
68		r19.41v 3192	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕs (𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
69	68	exbii 1880	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑛 ∈ ℕs (𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦))
70	63, 67, 69	3bitr3ri 305	. . . . . . . . . 10 ⊢ (∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦)
71	62, 70	bitri 278	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦)
72	6, 43	subsvald 28456	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 −s ( 1s /su 𝑛)) = (𝐴 +s ( -s‘( 1s /su 𝑛))))
73	72	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 −s ( 1s /su 𝑛))) = ( -s‘(𝐴 +s ( -s‘( 1s /su 𝑛)))))
74	43	negscld 28432	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘( 1s /su 𝑛)) ∈ No)
75		negsdi 28445	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ ( -s‘( 1s /su 𝑛)) ∈ No) → ( -s‘(𝐴 +s ( -s‘( 1s /su 𝑛)))) = (( -s‘𝐴) +s ( -s‘( -s‘( 1s /su 𝑛)))))
76	74, 75	syldan 603	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 +s ( -s‘( 1s /su 𝑛)))) = (( -s‘𝐴) +s ( -s‘( -s‘( 1s /su 𝑛)))))
77		negnegs 28439	. . . . . . . . . . . . . . 15 ⊢ (( 1s /su 𝑛) ∈ No → ( -s‘( -s‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
78	43, 77	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘( -s‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
79	78	oveq2d 7440	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝐴) +s ( -s‘( -s‘( 1s /su 𝑛)))) = (( -s‘𝐴) +s ( 1s /su 𝑛)))
80	73, 76, 79	3eqtrd 2799	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 −s ( 1s /su 𝑛))) = (( -s‘𝐴) +s ( 1s /su 𝑛)))
81	80	eqeq1d 2762	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦 ↔ (( -s‘𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82		eqcom 2767	. . . . . . . . . . 11 ⊢ ((( -s‘𝐴) +s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛)))
83	81, 82	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))))
84	83	rexbidva 3184	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))))
85	71, 84	bitrid 286	. . . . . . . 8 ⊢ (𝐴 ∈ No → (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ( -s‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))))
86	59, 85	bitrd 282	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝑦 ∈ ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))))
87	86	eqabdv 2893	. . . . . 6 ⊢ (𝐴 ∈ No → ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))})
88	55, 87	oveq12d 7442	. . . . 5 ⊢ (𝐴 ∈ No → (( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))})) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))}))
89	88	adantr 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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))}))) → (( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -s “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))})) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))}))
90	21, 89	eqtrd 2795	. . 3 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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))}))) → ( -s‘𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))}))
91	2, 17, 90	jca32 525	. 2 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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))}))) → (( -s‘𝐴) ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s ( -s‘𝐴) ∧ ( -s‘𝐴) <s 𝑛) ∧ ( -s‘𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))}))))
92		elreno 28886	. 2 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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 28886	. 2 ⊢ (( -s‘𝐴) ∈ ℝs ↔ (( -s‘𝐴) ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s ( -s‘𝐴) ∧ ( -s‘𝐴) <s 𝑛) ∧ ( -s‘𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) −s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -s‘𝐴) +s ( 1s /su 𝑛))}))))
94	91, 92, 93	3imtr4i 295	1 ⊢ (𝐴 ∈ ℝs → ( -s‘𝐴) ∈ ℝs)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ⊆ wss 3896   class class class wbr 5102   “ cima 5654   Fn wfn 6538  ‘cfv 6543  (class class class)co 7424  Nocsur 28006   <s clts 28007   <<s cslts 28152   |s ccuts 28154   1_s c1s 28201   +_s cadds 28354   -_scnegs 28414   −_s csubs 28415   /_su cdivs 28582  ℕ_scnns 28708  ℝ_screno 28884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-dc 10531

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-nadd 8682  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-1s 28203  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-muls 28502  df-divs 28583  df-n0s 28709  df-nns 28710  df-reno 28885

This theorem is used by: (None)