supminfxr2 - Mathbox for Glauco Siliprandi

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Theorem supminfxr2 44179

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 13189	. . . . . 6 ⊢ -_𝑒-∞ = +∞
2	1	eqcomi 2742	. . . . 5 ⊢ +∞ = -_𝑒-∞
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -_𝑒-∞)
4		supminfxr2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
5		supxrpnf 13297	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
6	4, 5	sylan 581	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
7		ssrab2 4078	. . . . . . . 8 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*
8	7	a1i 11	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*)
9		xnegeq 13186	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = -_𝑒-∞)
10	1	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒-∞ = +∞)
11	9, 10	eqtrd 2773	. . . . . . . . 9 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = +∞)
12	11	eleq1d 2819	. . . . . . . 8 ⊢ (𝑦 = -∞ → (-_𝑒𝑦 ∈ 𝐴 ↔ +∞ ∈ 𝐴))
13		mnfxr 11271	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ ℝ^*)
15		id 22	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
16	12, 14, 15	elrabd 3686	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
17		infxrmnf 13316	. . . . . . 7 ⊢ (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^* ∧ -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
18	8, 16, 17	syl2anc 585	. . . . . 6 ⊢ (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
19	18	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
20	19	xnegeqd 44147	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -_𝑒-∞)
21	3, 6, 20	3eqtr4d 2783	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
22	4	ssdifssd 4143	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
23	22	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
24		difssd 4133	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25		id 22	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26		ssnel 43727	. . . . . . . 8 ⊢ (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
27	24, 25, 26	syl2anc 585	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
28	27	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29		neldifsnd 4797	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
30	23, 28, 29	xrssre 44058	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
31	30	supminfxr 44174	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
32		supxrmnf2 44143	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
33	4, 32	syl 17	. . . . . 6 ⊢ (𝜑 → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
34	33	eqcomd 2739	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
35	34	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
36		rexr 11260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
37	36	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ^*)
38		simpl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
39	38	rexnegd 43832	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 = -𝑦)
40		eldifi 4127	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦 ∈ 𝐴)
41	40	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦 ∈ 𝐴)
42	39, 41	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 ∈ 𝐴)
43	37, 42	jca 513	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
44		rabid 3453	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ↔ (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
45	43, 44	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
46		renepnf 11262	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47		elsni 4646	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
48	47	necon3ai 2966	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ +∞ → ¬ 𝑦 ∈ {+∞})
49	46, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → ¬ 𝑦 ∈ {+∞})
50	38, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → ¬ 𝑦 ∈ {+∞})
51	45, 50	eldifd 3960	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
52	51	ex 414	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
53	52	rgen 3064	. . . . . . . . . . 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 3452	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}
56		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{+∞}
57	55, 56	nfdif 4126	. . . . . . . . . . 11 ⊢ Ⅎ𝑦({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})
58	57	rabssf 43808	. . . . . . . . . 10 ⊢ ({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ↔ ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
59	54, 58	sylibr 233	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
60		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 ¬ +∞ ∈ 𝐴
61		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦ℝ
62		eldifi 4127	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
63	7, 62	sselid 3981	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ^*)
64	63	adantl 483	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ^*)
65	44	simprbi 498	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} → -_𝑒𝑦 ∈ 𝐴)
66	62, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → -_𝑒𝑦 ∈ 𝐴)
67	12	biimpac 480	. . . . . . . . . . . . . . . . 17 ⊢ ((-_𝑒𝑦 ∈ 𝐴 ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
68	67	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
70	68, 69	pm2.65da 816	. . . . . . . . . . . . . . 15 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → ¬ 𝑦 = -∞)
71	70	neqned 2948	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → 𝑦 ≠ -∞)
72	66, 71	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73		eldifsni 4794	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
74	73	adantl 483	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
75	64, 72, 74	xrred 44075	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
76	60, 57, 61, 75	ssdf2 43830	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ)
77	75	rexnegd 43832	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 = -𝑦)
78	66	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ 𝐴)
79	63	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 ∈ ℝ^*)
80		elsni 4646	. . . . . . . . . . . . . . . . . 18 ⊢ (-_𝑒𝑦 ∈ {-∞} → -_𝑒𝑦 = -∞)
81	80	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → -_𝑒𝑦 = -∞)
82		xnegeq 13186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-_𝑒𝑦 = -_𝑒-∞)
83	1	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-∞ = +∞)
84	82, 83	eqtr2d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (-_𝑒𝑦 = -∞ → +∞ = -_𝑒-_𝑒𝑦)
85	84	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → +∞ = -_𝑒-_𝑒𝑦)
86		xnegneg 13193	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → -_𝑒-_𝑒𝑦 = 𝑦)
87	86	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → -_𝑒-_𝑒𝑦 = 𝑦)
88	85, 87	eqtr2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → 𝑦 = +∞)
89	79, 81, 88	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
90	73	neneqd 2946	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
91	90	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
92	89, 91	pm2.65da 816	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ -_𝑒𝑦 ∈ {-∞})
93	92	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → ¬ -_𝑒𝑦 ∈ {-∞})
94	78, 93	eldifd 3960	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
95	77, 94	eqeltrrd 2835	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
96	95	ralrimiva 3147	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
97	76, 96	jca 513	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
98	57, 61	ssrabf 43803	. . . . . . . . . 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 233	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})})
100	59, 99	eqssd 4000	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
101	100	infeq1d 9472	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ) = inf(({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^*, < ))
102		infxrpnf2 44173	. . . . . . . . 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 2774	. . . . . 6 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
106	105	xnegeqd 44147	. . . . 5 ⊢ (¬ +∞ ∈ 𝐴 → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
107	106	adantl 483	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
108	31, 35, 107	3eqtr4d 2783	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
109	21, 108	pm2.61dan 812	. 2 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
110		xnegeq 13186	. . . . . . 7 ⊢ (𝑦 = 𝑥 → -_𝑒𝑦 = -_𝑒𝑥)
111	110	eleq1d 2819	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-_𝑒𝑦 ∈ 𝐴 ↔ -_𝑒𝑥 ∈ 𝐴))
112	111	cbvrabv 3443	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} = {𝑥 ∈ ℝ^* ∣ -_𝑒𝑥 ∈ 𝐴}
113	112	infeq1i 9473	. . . 4 ⊢ inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < )
114	113	xnegeqi 44150	. . 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 2773	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 ∖ cdif 3946 ⊆ wss 3949 {csn 4629 supcsup 9435 infcinf 9436 ℝcr 11109 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 -cneg 11445 -_𝑒cxne 13089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-xneg 13092

This theorem is referenced by: supminfxrrnmpt 44181 liminfvalxr 44499

W3C validator