supminfxr2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > supminfxr2		Structured version Visualization version GIF version

Theorem supminfxr2 46224

Description: The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypothesis**
Ref	Expression
supminfxr2.1	⊢ (𝜑 → 𝐴 ⊆ ℝ^*)

**Assertion**
Ref	Expression
supminfxr2	⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < ))

Distinct variable group: 𝑥,𝐴 Allowed substitution hint: 𝜑(𝑥)

Proof of Theorem supminfxr2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xnegmnf 13254	. . . . . 6 ⊢ -_𝑒-∞ = +∞
2	1	eqcomi 2775	. . . . 5 ⊢ +∞ = -_𝑒-∞
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -_𝑒-∞)
4		supminfxr2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
5		supxrpnf 13362	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
6	4, 5	sylan 592	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
7		ssrab2 4037	. . . . . . . 8 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*
8	7	a1i 11	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*)
9		xnegeq 13251	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = -_𝑒-∞)
10	1	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒-∞ = +∞)
11	9, 10	eqtrd 2801	. . . . . . . . 9 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = +∞)
12	11	eleq1d 2851	. . . . . . . 8 ⊢ (𝑦 = -∞ → (-_𝑒𝑦 ∈ 𝐴 ↔ +∞ ∈ 𝐴))
13		mnfxr 11284	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ ℝ^*)
15		id 23	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
16	12, 14, 15	elrabd 3655	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
17		infxrmnf 13382	. . . . . . 7 ⊢ (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^* ∧ -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
18	8, 16, 17	syl2anc 596	. . . . . 6 ⊢ (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
19	18	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
20	19	xnegeqd 46192	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -_𝑒-∞)
21	3, 6, 20	3eqtr4d 2811	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
22	4	ssdifssd 4104	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
23	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
24		difssd 4094	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25		id 23	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26		ssnel 45804	. . . . . . . 8 ⊢ (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
27	24, 25, 26	syl2anc 596	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
28	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29		neldifsnd 4766	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
30	23, 28, 29	xrssre 46105	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
31	30	supminfxr 46219	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
32		supxrmnf2 46188	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
33	4, 32	syl 18	. . . . . 6 ⊢ (𝜑 → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
34	33	eqcomd 2772	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
35	34	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
36		rexr 11273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
37	36	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ^*)
38		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
39	38	rexnegd 45902	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 = -𝑦)
40		eldifi 4088	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦 ∈ 𝐴)
41	40	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦 ∈ 𝐴)
42	39, 41	eqeltrd 2866	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 ∈ 𝐴)
43	37, 42	jca 521	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
44		rabid 3440	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ↔ (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
45	43, 44	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
46		renepnf 11275	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47		elsni 4611	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
48	47	necon3ai 2986	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ +∞ → ¬ 𝑦 ∈ {+∞})
49	46, 48	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → ¬ 𝑦 ∈ {+∞})
50	38, 49	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → ¬ 𝑦 ∈ {+∞})
51	45, 50	eldifd 3919	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
52	51	ex 418	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
53	52	rgen 3084	. . . . . . . . . . 11 ⊢ ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
54	53	a1i 11	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
55		nfrab1 3439	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}
56		nfcv 2928	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{+∞}
57	55, 56	nfdif 4087	. . . . . . . . . . 11 ⊢ Ⅎ𝑦({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})
58	57	rabssf 45878	. . . . . . . . . 10 ⊢ ({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ↔ ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
59	54, 58	sylibr 237	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
60		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 ¬ +∞ ∈ 𝐴
61		nfcv 2928	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦ℝ
62		eldifi 4088	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
63	7, 62	sselid 3938	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ^*)
64	63	adantl 487	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ^*)
65	44	simprbi 503	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} → -_𝑒𝑦 ∈ 𝐴)
66	62, 65	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → -_𝑒𝑦 ∈ 𝐴)
67	12	biimpac 484	. . . . . . . . . . . . . . . . 17 ⊢ ((-_𝑒𝑦 ∈ 𝐴 ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
68	67	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69		simpll 779	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
70	68, 69	pm2.65da 829	. . . . . . . . . . . . . . 15 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → ¬ 𝑦 = -∞)
71	70	neqned 2968	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → 𝑦 ≠ -∞)
72	66, 71	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73		eldifsni 4763	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
74	73	adantl 487	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
75	64, 72, 74	xrred 46121	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
76	60, 57, 61, 75	ssdf2 45900	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ)
77	75	rexnegd 45902	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 = -𝑦)
78	66	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ 𝐴)
79	63	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 ∈ ℝ^*)
80		elsni 4611	. . . . . . . . . . . . . . . . . 18 ⊢ (-_𝑒𝑦 ∈ {-∞} → -_𝑒𝑦 = -∞)
81	80	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → -_𝑒𝑦 = -∞)
82		xnegeq 13251	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-_𝑒𝑦 = -_𝑒-∞)
83	1	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-∞ = +∞)
84	82, 83	eqtr2d 2802	. . . . . . . . . . . . . . . . . . 19 ⊢ (-_𝑒𝑦 = -∞ → +∞ = -_𝑒-_𝑒𝑦)
85	84	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → +∞ = -_𝑒-_𝑒𝑦)
86		xnegneg 13258	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → -_𝑒-_𝑒𝑦 = 𝑦)
87	86	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → -_𝑒-_𝑒𝑦 = 𝑦)
88	85, 87	eqtr2d 2802	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → 𝑦 = +∞)
89	79, 81, 88	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
90	73	neneqd 2966	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
91	90	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
92	89, 91	pm2.65da 829	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ -_𝑒𝑦 ∈ {-∞})
93	92	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → ¬ -_𝑒𝑦 ∈ {-∞})
94	78, 93	eldifd 3919	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
95	77, 94	eqeltrrd 2867	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
96	95	ralrimiva 3160	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
97	76, 96	jca 521	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
98	57, 61	ssrabf 45873	. . . . . . . . . 10 ⊢ (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ↔ (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
99	97, 98	sylibr 237	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})})
100	59, 99	eqssd 3957	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
101	100	infeq1d 9448	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ) = inf(({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^*, < ))
102		infxrpnf2 46218	. . . . . . . . 9 ⊢ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^* → inf(({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^, < ) = inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
103	7, 102	ax-mp 5	. . . . . . . 8 ⊢ inf(({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^, < ) = inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < )
104	103	a1i 11	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → inf(({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^, < ) = inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
105	101, 104	eqtr2d 2802	. . . . . 6 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
106	105	xnegeqd 46192	. . . . 5 ⊢ (¬ +∞ ∈ 𝐴 → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
107	106	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
108	31, 35, 107	3eqtr4d 2811	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
109	21, 108	pm2.61dan 825	. 2 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
110		xnegeq 13251	. . . . . . 7 ⊢ (𝑦 = 𝑥 → -_𝑒𝑦 = -_𝑒𝑥)
111	110	eleq1d 2851	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-_𝑒𝑦 ∈ 𝐴 ↔ -_𝑒𝑥 ∈ 𝐴))
112	111	cbvrabv 3429	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} = {𝑥 ∈ ℝ^* ∣ -_𝑒𝑥 ∈ 𝐴}
113	112	infeq1i 9449	. . . 4 ⊢ inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < )
114	113	xnegeqi 46195	. . 3 ⊢ -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < )
115	114	a1i 11	. 2 ⊢ (𝜑 → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < ))
116	109, 115	eqtrd 2801	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 {crab 3419 ∖ cdif 3905 ⊆ wss 3908 {csn 4594 supcsup 9410 infcinf 9411 ℝcr 11117 +∞cpnf 11258 -∞cmnf 11259 ℝ^*cxr 11260 < clt 11261 -cneg 11460 -_𝑒cxne 13152

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 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-po 5574 df-so 5575 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-xneg 13155

This theorem is used by: supminfxrrnmpt 46226 liminfvalxr 46538

W3C validator