Theorem renegscl 28349

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 27942	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
2	1	adantr 480	. . 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		nnsno 28217	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
4	3	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝑛 ∈ No )
5	4	negscld 27943	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘𝑛) ∈ No )
6		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
7	5, 6	sltnegd 27953	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝑛) <s 𝐴 ↔ ( -us ‘𝐴) <s ( -us ‘( -us ‘𝑛))))
8		negnegs 27950	. . . . . . . . . . 11 ⊢ (𝑛 ∈ No → ( -us ‘( -us ‘𝑛)) = 𝑛)
9	4, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( -us ‘𝑛)) = 𝑛)
10	9	breq2d 5119	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) <s ( -us ‘( -us ‘𝑛)) ↔ ( -us ‘𝐴) <s 𝑛))
11	7, 10	bitrd 279	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝑛) <s 𝐴 ↔ ( -us ‘𝐴) <s 𝑛))
12	6, 4	sltnegd 27953	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -us ‘𝑛) <s ( -us ‘𝐴)))
13	11, 12	anbi12d 632	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -us ‘𝐴) <s 𝑛 ∧ ( -us ‘𝑛) <s ( -us ‘𝐴))))
14	13	biancomd 463	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ↔ (( -us ‘𝑛) <s ( -us ‘𝐴) ∧ ( -us ‘𝐴) <s 𝑛)))
15	14	rexbidva 3155	. . . . 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 476	. . . 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 717	. . 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 28347	. . . . . 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 480	. . . . 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 772	. . . . 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 27961	. . . 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 27929	. . . . . . . . 9 ⊢ -us Fn No
23		ssltss2 27701	. . . . . . . . . 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 6933	. . . . . . . . 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 587	. . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
28	27	rexbidv 3157	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexab 3666	. . . . . . . . . 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 3264	. . . . . . . . . . 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 7420	. . . . . . . . . . . . 13 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
32		fveqeq2 6867	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us ‘𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
33	31, 32	ceqsexv 3498	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
34	33	rexbii 3076	. . . . . . . . . . 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 3167	. . . . . . . . . . . 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 1848	. . . . . . . . . . 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 302	. . . . . . . . . 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 275	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us ‘𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
39		1sno 27739	. . . . . . . . . . . . . . . . 17 ⊢ 1s ∈ No
40	39	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
41		nnne0s 28229	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
42	40, 3, 41	divscld 28126	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
43	42	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44		negsdi 27956	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
45	43, 44	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
46	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘𝐴) ∈ No )
47	46, 43	subsvald 27965	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) -s ( 1s /su 𝑛)) = (( -us ‘𝐴) +s ( -us ‘( 1s /su 𝑛))))
48	45, 47	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us ‘𝐴) -s ( 1s /su 𝑛)))
49	48	eqeq1d 2731	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us ‘𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50		eqcom 2736	. . . . . . . . . . 11 ⊢ ((( -us ‘𝐴) -s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛)))
51	49, 50	bitrdi 287	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))))
52	51	rexbidva 3155	. . . . . . . . 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 283	. . . . . . . 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 279	. . . . . . 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 2861	. . . . . 6 ⊢ (𝐴 ∈ No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us ‘𝐴) -s ( 1s /su 𝑛))})
56		ssltss1 27700	. . . . . . . . . 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 6933	. . . . . . . . 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 587	. . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
61	60	rexbidv 3157	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
62	61	rexab 3666	. . . . . . . . . 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 3264	. . . . . . . . . . 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 7420	. . . . . . . . . . . . 13 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
65		fveqeq2 6867	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us ‘𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
66	64, 65	ceqsexv 3498	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us ‘𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
67	66	rexbii 3076	. . . . . . . . . . 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 3167	. . . . . . . . . . . 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 1848	. . . . . . . . . . 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 302	. . . . . . . . . 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 275	. . . . . . . . 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 27965	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
73	72	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
74	43	negscld 27943	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75		negsdi 27956	. . . . . . . . . . . . . 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 591	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
77		negnegs 27950	. . . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))) = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
80	73, 76, 79	3eqtrd 2768	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
81	80	eqeq1d 2731	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us ‘𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82		eqcom 2736	. . . . . . . . . . 11 ⊢ ((( -us ‘𝐴) +s ( 1s /su 𝑛)) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛)))
83	81, 82	bitrdi 287	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ 𝑦 = (( -us ‘𝐴) +s ( 1s /su 𝑛))))
84	83	rexbidva 3155	. . . . . . . . 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 283	. . . . . . . 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 279	. . . . . . 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 2861	. . . . . 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 7405	. . . . 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 480	. . . 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 2764	. . 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 515	. 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 28346	. 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 28346	. 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 292	1 ⊢ (𝐴 ∈ ℝs → ( -us ‘𝐴) ∈ ℝs)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3914   class class class wbr 5107   “ cima 5641   Fn wfn 6506  ‘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)