Theorem renegscl 28508

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 28046	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
2	1	adantr 481	. . 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 28334	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
4	3	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝑛 ∈ No )
5	4	negscld 28047	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘𝑛) ∈ No )
6		simpl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
7	5, 6	ltnegsd 28057	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝑛) <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘( -us ‘𝑛))))
8		negnegs 28054	. . . . . . . . . . 11 ⊢ (𝑛 ∈ No → ( -us ‘( -us ‘𝑛)) = 𝑛)
9	4, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( -us ‘𝑛)) = 𝑛)
10	9	breq2d 5084	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) <s ( -us ‘( -us ‘𝑛)) ↔ ( -us ‘𝐴) <s 𝑛))
11	7, 10	bitrd 280	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝑛) <s 𝐴 ↔ ( -us ‘𝐴) <s 𝑛))
12	6, 4	ltnegsd 28057	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -us ‘𝑛) <s ( -us ‘𝐴)))
13	11, 12	anbi12d 638	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -us ‘𝐴) <s 𝑛 ∧ ( -us ‘𝑛) <s ( -us ‘𝐴))))
14	13	biancomd 464	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛)))
15	14	rexbidva 3161	. . . . 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 477	. . . 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 723	. . 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 28504	. . . . . 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 481	. . . . 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 778	. . . . 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 28065	. . . 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 28033	. . . . . . . . 9 ⊢ -us Fn No
23		sltsss2 27776	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
25		fvelimab 6899	. . . . . . . . 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 593	. . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
28	27	rexbidv 3163	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexab 3636	. . . . . . . . . 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 3266	. . . . . . . . . . 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 7389	. . . . . . . . . . . . 13 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
32		fveqeq2 6836	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us ‘𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
33	31, 32	ceqsexv 3479	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
34	33	rexbii 3086	. . . . . . . . . . 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 3169	. . . . . . . . . . . 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 1855	. . . . . . . . . . 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 303	. . . . . . . . . 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 276	. . . . . . . . 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 27820	. . . . . . . . . . . . . . . . 17 ⊢ 1s ∈ No
40	39	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
41		nnne0s 28347	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
42	40, 3, 41	divscld 28234	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
43	42	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44		negsdi 28060	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
45	43, 44	syldan 597	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
46	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘𝐴) ∈ No )
47	46, 43	subsvald 28071	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) -s ( 1s /su 𝑛)) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
48	45, 47	eqtr4d 2777	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) -s ( 1s /su 𝑛)))
49	48	eqeq1d 2741	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us ‘𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50		eqcom 2746	. . . . . . . . . . 11 ⊢ ((( -us ‘𝐴) -s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛)))
51	49, 50	bitrdi 288	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))))
52	51	rexbidva 3161	. . . . . . . . 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 284	. . . . . . . 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 280	. . . . . . 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 2872	. . . . . 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 27775	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
58		fvelimab 6899	. . . . . . . . 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 593	. . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
61	60	rexbidv 3163	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
62	61	rexab 3636	. . . . . . . . . 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 3266	. . . . . . . . . . 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 7389	. . . . . . . . . . . . 13 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
65		fveqeq2 6836	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us ‘𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
66	64, 65	ceqsexv 3479	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
67	66	rexbii 3086	. . . . . . . . . . 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 3169	. . . . . . . . . . . 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 1855	. . . . . . . . . . 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 303	. . . . . . . . . 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 276	. . . . . . . . 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 28071	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
73	72	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
74	43	negscld 28047	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75		negsdi 28060	. . . . . . . . . . . . . 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 597	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
77		negnegs 28054	. . . . . . . . . . . . . . 15 ⊢ (( 1s /su 𝑛) ∈ No → ( -us ‘( -us ‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
78	43, 77	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( -us ‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
79	78	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
80	73, 76, 79	3eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
81	80	eqeq1d 2741	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us ‘𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82		eqcom 2746	. . . . . . . . . . 11 ⊢ ((( -us ‘𝐴) +s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
83	81, 82	bitrdi 288	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))))
84	83	rexbidva 3161	. . . . . . . . 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 284	. . . . . . . 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 280	. . . . . . 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 2872	. . . . . 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 7374	. . . . 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 481	. . . 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 2774	. . 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 520	. 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 28501	. 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 28501	. 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 293	1 ⊢ (𝐴 ∈ ℝs → ( -us ‘𝐴) ∈ ℝs)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∃wrex 3063   ⊆ wss 3883   class class class wbr 5072   “ cima 5621   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356   No csur 27621   <s clts 27622   <<s cslts 27767   |s ccuts 27769   1_s c1s 27816   +_s cadds 27969   -u_s cnegs 28029   -_s csubs 28030   /_su cdivs 28197  ℕ_scnns 28323  ℝ_screno 28499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-dc 10359

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-nadd 8592  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-muls 28117  df-divs 28198  df-n0s 28324  df-nns 28325  df-reno 28500

This theorem is referenced by: (None)