Theorem renegscl 28941

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 28479	. . . 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 28767	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No)
4	3	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝑛 ∈ No)
5	4	negscld 28480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘𝑛) ∈ No)
6		simpl 488	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No)
7	5, 6	ltnegsd 28490	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝑛) <s 𝐴 ↔ ( -s‘𝐴) <s ( -s‘( -s‘𝑛))))
8		negnegs 28487	. . . . . . . . . . 11 ⊢ (𝑛 ∈ No → ( -s‘( -s‘𝑛)) = 𝑛)
9	4, 8	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘( -s‘𝑛)) = 𝑛)
10	9	breq2d 5120	. . . . . . . . 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 28490	. . . . . . . 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 3188	. . . . 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 28937	. . . . . 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 28498	. . . 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 28466	. . . . . . . . 9 ⊢ -s Fn No
23		sltsss2 28209	. . . . . . . . . 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 6988	. . . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
28	27	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexab 3656	. . . . . . . . . 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 3293	. . . . . . . . . . 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 7484	. . . . . . . . . . . . 13 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
32		fveqeq2 6925	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -s‘𝑧) = 𝑦 ↔ ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
33	31, 32	ceqsexv 3502	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
34	33	rexbii 3113	. . . . . . . . . . 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 3196	. . . . . . . . . . . 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 1881	. . . . . . . . . . 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 28253	. . . . . . . . . . . . . . . . 17 ⊢ 1s ∈ No
40	39	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 1s ∈ No)
41		nnne0s 28780	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
42	40, 3, 41	divscld 28667	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No)
43	42	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No)
44		negsdi 28493	. . . . . . . . . . . . . 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 28504	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝐴) −s ( 1s /su 𝑛)) = (( -s‘𝐴) +s ( -s‘( 1s /su 𝑛))))
48	45, 47	eqtr4d 2801	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 +s ( 1s /su 𝑛))) = (( -s‘𝐴) −s ( 1s /su 𝑛)))
49	48	eqeq1d 2765	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -s‘𝐴) −s ( 1s /su 𝑛)) = 𝑦))
50		eqcom 2770	. . . . . . . . . . 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 3188	. . . . . . . . 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 2897	. . . . . 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 28208	. . . . . . . . . 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 6988	. . . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 −s ( 1s /su 𝑛))))
61	60	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 −s ( 1s /su 𝑛))))
62	61	rexab 3656	. . . . . . . . . 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 3293	. . . . . . . . . . 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 7484	. . . . . . . . . . . . 13 ⊢ (𝐴 −s ( 1s /su 𝑛)) ∈ V
65		fveqeq2 6925	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 −s ( 1s /su 𝑛)) → (( -s‘𝑧) = 𝑦 ↔ ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦))
66	64, 65	ceqsexv 3502	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 −s ( 1s /su 𝑛)) ∧ ( -s‘𝑧) = 𝑦) ↔ ( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦)
67	66	rexbii 3113	. . . . . . . . . . 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 3196	. . . . . . . . . . . 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 1881	. . . . . . . . . . 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 28504	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 −s ( 1s /su 𝑛)) = (𝐴 +s ( -s‘( 1s /su 𝑛))))
73	72	fveq2d 6920	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 −s ( 1s /su 𝑛))) = ( -s‘(𝐴 +s ( -s‘( 1s /su 𝑛)))))
74	43	negscld 28480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘( 1s /su 𝑛)) ∈ No)
75		negsdi 28493	. . . . . . . . . . . . . 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 28487	. . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘𝐴) +s ( -s‘( -s‘( 1s /su 𝑛)))) = (( -s‘𝐴) +s ( 1s /su 𝑛)))
80	73, 76, 79	3eqtrd 2802	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -s‘(𝐴 −s ( 1s /su 𝑛))) = (( -s‘𝐴) +s ( 1s /su 𝑛)))
81	80	eqeq1d 2765	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -s‘(𝐴 −s ( 1s /su 𝑛))) = 𝑦 ↔ (( -s‘𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82		eqcom 2770	. . . . . . . . . . 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 3188	. . . . . . . . 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 2897	. . . . . 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 7469	. . . . 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 2798	. . 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 28934	. 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 28934	. 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 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108   “ cima 5661   Fn wfn 6561  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   <s clts 28055   <<s cslts 28200   |s ccuts 28202   1_s c1s 28249   +_s cadds 28402   -_scnegs 28462   −_s csubs 28463   /_su cdivs 28630  ℕ_scnns 28756  ℝ_screno 28932

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-dc 10563

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550  df-divs 28631  df-n0s 28757  df-nns 28758  df-reno 28933

This theorem is used by: (None)