supminfxr2 - Mathbox for Glauco Siliprandi

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Theorem supminfxr2 41287

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 12453	. . . . . 6 ⊢ -_𝑒-∞ = +∞
2	1	eqcomi 2804	. . . . 5 ⊢ +∞ = -_𝑒-∞
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -_𝑒-∞)
4		supminfxr2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
5		supxrpnf 12561	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
6	4, 5	sylan 580	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
7		ssrab2 3977	. . . . . . . 8 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*
8	7	a1i 11	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*)
9		xnegeq 12450	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = -_𝑒-∞)
10	1	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒-∞ = +∞)
11	9, 10	eqtrd 2831	. . . . . . . . 9 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = +∞)
12	11	eleq1d 2867	. . . . . . . 8 ⊢ (𝑦 = -∞ → (-_𝑒𝑦 ∈ 𝐴 ↔ +∞ ∈ 𝐴))
13		mnfxr 10545	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ ℝ^*)
15		id 22	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
16	12, 14, 15	elrabd 3620	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
17		infxrmnf 12580	. . . . . . 7 ⊢ (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^* ∧ -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
18	8, 16, 17	syl2anc 584	. . . . . 6 ⊢ (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
19	18	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
20	19	xnegeqd 41253	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -_𝑒-∞)
21	3, 6, 20	3eqtr4d 2841	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
22	4	ssdifssd 4040	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
23	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
24		difssd 4030	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25		id 22	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26		ssnel 40841	. . . . . . . 8 ⊢ (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
27	24, 25, 26	syl2anc 584	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
28	27	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29		neldifsnd 4633	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
30	23, 28, 29	xrssre 41157	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
31	30	supminfxr 41282	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
32		supxrmnf2 41249	. . . . . . 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 2801	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
35	34	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
36		rexr 10533	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
37	36	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ^*)
38		simpl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
39	38	rexnegd 40951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 = -𝑦)
40		eldifi 4024	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦 ∈ 𝐴)
41	40	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦 ∈ 𝐴)
42	39, 41	eqeltrd 2883	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 ∈ 𝐴)
43	37, 42	jca 512	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
44		rabid 3337	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ↔ (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
45	43, 44	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
46		renepnf 10535	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47		elsni 4489	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
48	47	necon3ai 3009	. . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
52	51	ex 413	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
53	52	rgen 3115	. . . . . . . . . . 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 3344	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}
56		nfcv 2949	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{+∞}
57	55, 56	nfdif 4023	. . . . . . . . . . 11 ⊢ Ⅎ𝑦({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})
58	57	rabssf 40925	. . . . . . . . . 10 ⊢ ({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ↔ ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
59	54, 58	sylibr 235	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
60		nfv 1892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 ¬ +∞ ∈ 𝐴
61		nfcv 2949	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦ℝ
62		eldifi 4024	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
63	7, 62	sseldi 3887	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ^*)
64	63	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ^*)
65	44	simprbi 497	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} → -_𝑒𝑦 ∈ 𝐴)
66	62, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → -_𝑒𝑦 ∈ 𝐴)
67	12	biimpac 479	. . . . . . . . . . . . . . . . 17 ⊢ ((-_𝑒𝑦 ∈ 𝐴 ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
68	67	adantll 710	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69		simpll 763	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
70	68, 69	pm2.65da 813	. . . . . . . . . . . . . . 15 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → ¬ 𝑦 = -∞)
71	70	neqned 2991	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → 𝑦 ≠ -∞)
72	66, 71	sylan2 592	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73		eldifsni 4629	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
74	73	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
75	64, 72, 74	xrred 41174	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
76	60, 57, 61, 75	ssdf2 40949	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ)
77	75	rexnegd 40951	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 = -𝑦)
78	66	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ 𝐴)
79	63	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 ∈ ℝ^*)
80		elsni 4489	. . . . . . . . . . . . . . . . . 18 ⊢ (-_𝑒𝑦 ∈ {-∞} → -_𝑒𝑦 = -∞)
81	80	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → -_𝑒𝑦 = -∞)
82		xnegeq 12450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-_𝑒𝑦 = -_𝑒-∞)
83	1	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-∞ = +∞)
84	82, 83	eqtr2d 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (-_𝑒𝑦 = -∞ → +∞ = -_𝑒-_𝑒𝑦)
85	84	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → +∞ = -_𝑒-_𝑒𝑦)
86		xnegneg 12457	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → -_𝑒-_𝑒𝑦 = 𝑦)
87	86	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → -_𝑒-_𝑒𝑦 = 𝑦)
88	85, 87	eqtr2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → 𝑦 = +∞)
89	79, 81, 88	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
90	73	neneqd 2989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
91	90	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
92	89, 91	pm2.65da 813	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ -_𝑒𝑦 ∈ {-∞})
93	92	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → ¬ -_𝑒𝑦 ∈ {-∞})
94	78, 93	eldifd 3870	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
95	77, 94	eqeltrrd 2884	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
96	95	ralrimiva 3149	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
97	76, 96	jca 512	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
98	57, 61	ssrabf 40921	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})})
100	59, 99	eqssd 3906	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
101	100	infeq1d 8787	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ) = inf(({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^*, < ))
102		infxrpnf2 41281	. . . . . . . . 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 2832	. . . . . 6 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
106	105	xnegeqd 41253	. . . . 5 ⊢ (¬ +∞ ∈ 𝐴 → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
107	106	adantl 482	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
108	31, 35, 107	3eqtr4d 2841	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
109	21, 108	pm2.61dan 809	. 2 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
110		xnegeq 12450	. . . . . . 7 ⊢ (𝑦 = 𝑥 → -_𝑒𝑦 = -_𝑒𝑥)
111	110	eleq1d 2867	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-_𝑒𝑦 ∈ 𝐴 ↔ -_𝑒𝑥 ∈ 𝐴))
112	111	cbvrabv 3434	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} = {𝑥 ∈ ℝ^* ∣ -_𝑒𝑥 ∈ 𝐴}
113	112	infeq1i 8788	. . . 4 ⊢ inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < )
114	113	xnegeqi 41256	. . 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 2831	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 {crab 3109 ∖ cdif 3856 ⊆ wss 3859 {csn 4472 supcsup 8750 infcinf 8751 ℝcr 10382 +∞cpnf 10518 -∞cmnf 10519 ℝ^*cxr 10520 < clt 10521 -cneg 10718 -_𝑒cxne 12354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-xneg 12357

This theorem is referenced by: supminfxrrnmpt 41289 liminfvalxr 41606

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