supminfxr2 - Mathbox for Glauco Siliprandi

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Theorem supminfxr2 39699

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 12041	. . . . . 6 ⊢ -_𝑒-∞ = +∞
2	1	eqcomi 2631	. . . . 5 ⊢ +∞ = -_𝑒-∞
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -_𝑒-∞)
4		supminfxr2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
5		supxrpnf 12148	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
6	4, 5	sylan 488	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)
7		ssrab2 3687	. . . . . . . 8 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*
8	7	a1i 11	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^*)
9		xnegeq 12038	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = -_𝑒-∞)
10	1	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -_𝑒-∞ = +∞)
11	9, 10	eqtrd 2656	. . . . . . . . 9 ⊢ (𝑦 = -∞ → -_𝑒𝑦 = +∞)
12	11	eleq1d 2686	. . . . . . . 8 ⊢ (𝑦 = -∞ → (-_𝑒𝑦 ∈ 𝐴 ↔ +∞ ∈ 𝐴))
13		mnfxr 10096	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ ℝ^*)
15		id 22	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
16	12, 14, 15	elrabd 3365	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
17		infxrmnf 12167	. . . . . . 7 ⊢ (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ⊆ ℝ^* ∧ -∞ ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
18	8, 16, 17	syl2anc 693	. . . . . 6 ⊢ (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
19	18	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -∞)
20	19	xnegeqd 39664	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → -_𝑒inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ) = -_𝑒-∞)
21	3, 6, 20	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
22	4	ssdifssd 3748	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
23	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
24		difssd 3738	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25		id 22	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26		ssnel 39204	. . . . . . . 8 ⊢ (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
27	24, 25, 26	syl2anc 693	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
28	27	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29		neldifsnd 4322	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
30	23, 28, 29	xrssre 39564	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
31	30	supminfxr 39694	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
32		supxrmnf2 39660	. . . . . . 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 2628	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
35	34	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
36		rexr 10085	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
37	36	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ^*)
38		simpl 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
39	38	rexnegd 39334	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 = -𝑦)
40		eldifi 3732	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦 ∈ 𝐴)
41	40	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦 ∈ 𝐴)
42	39, 41	eqeltrd 2701	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -_𝑒𝑦 ∈ 𝐴)
43	37, 42	jca 554	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
44		rabid 3116	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ↔ (𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ 𝐴))
45	43, 44	sylibr 224	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
46		renepnf 10087	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47		elsni 4194	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
48	47	necon3ai 2819	. . . . . . . . . . . . . . . 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 3585	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
52	51	ex 450	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
53	52	rgen 2922	. . . . . . . . . . 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 3122	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}
56		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{+∞}
57	55, 56	nfdif 3731	. . . . . . . . . . 11 ⊢ Ⅎ𝑦({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})
58	57	rabssf 39302	. . . . . . . . . 10 ⊢ ({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ↔ ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})))
59	54, 58	sylibr 224	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
60		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 ¬ +∞ ∈ 𝐴
61		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦ℝ
62		eldifi 3732	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴})
63	7, 62	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ^*)
64	63	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ^*)
65	44	simprbi 480	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} → -_𝑒𝑦 ∈ 𝐴)
66	62, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → -_𝑒𝑦 ∈ 𝐴)
67	12	biimpac 503	. . . . . . . . . . . . . . . . 17 ⊢ ((-_𝑒𝑦 ∈ 𝐴 ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
68	67	adantll 750	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
70	68, 69	pm2.65da 600	. . . . . . . . . . . . . . 15 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → ¬ 𝑦 = -∞)
71	70	neqned 2801	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -_𝑒𝑦 ∈ 𝐴) → 𝑦 ≠ -∞)
72	66, 71	sylan2 491	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73		eldifsni 4320	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
74	73	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
75	64, 72, 74	xrred 39581	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
76	60, 57, 61, 75	ssdf2 39331	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ)
77	75	rexnegd 39334	. . . . . . . . . . . . 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 4194	. . . . . . . . . . . . . . . . . 18 ⊢ (-_𝑒𝑦 ∈ {-∞} → -_𝑒𝑦 = -∞)
81	80	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → -_𝑒𝑦 = -∞)
82		xnegeq 12038	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-_𝑒𝑦 = -_𝑒-∞)
83	1	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-_𝑒𝑦 = -∞ → -_𝑒-∞ = +∞)
84	82, 83	eqtr2d 2657	. . . . . . . . . . . . . . . . . . 19 ⊢ (-_𝑒𝑦 = -∞ → +∞ = -_𝑒-_𝑒𝑦)
85	84	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → +∞ = -_𝑒-_𝑒𝑦)
86		xnegneg 12045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → -_𝑒-_𝑒𝑦 = 𝑦)
87	86	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → -_𝑒-_𝑒𝑦 = 𝑦)
88	85, 87	eqtr2d 2657	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑦 = -∞) → 𝑦 = +∞)
89	79, 81, 88	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
90	73	neneqd 2799	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
91	90	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -_𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
92	89, 91	pm2.65da 600	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) → ¬ -_𝑒𝑦 ∈ {-∞})
93	92	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → ¬ -_𝑒𝑦 ∈ {-∞})
94	78, 93	eldifd 3585	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -_𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
95	77, 94	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
96	95	ralrimiva 2966	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
97	76, 96	jca 554	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
98	57, 61	ssrabf 39298	. . . . . . . . . 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 224	. . . . . . . . 9 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})})
100	59, 99	eqssd 3620	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}))
101	100	infeq1d 8383	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ) = inf(({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴} ∖ {+∞}), ℝ^*, < ))
102		infxrpnf2 39693	. . . . . . . . 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 2657	. . . . . 6 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ^, < ))
106	105	xnegeqd 39664	. . . . 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 2666	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
109	21, 108	pm2.61dan 832	. 2 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑦 ∈ ℝ^ ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^*, < ))
110		xnegeq 12038	. . . . . . 7 ⊢ (𝑦 = 𝑥 → -_𝑒𝑦 = -_𝑒𝑥)
111	110	eleq1d 2686	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-_𝑒𝑦 ∈ 𝐴 ↔ -_𝑒𝑥 ∈ 𝐴))
112	111	cbvrabv 3199	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴} = {𝑥 ∈ ℝ^* ∣ -_𝑒𝑥 ∈ 𝐴}
113	112	infeq1i 8384	. . . 4 ⊢ inf({𝑦 ∈ ℝ^* ∣ -_𝑒𝑦 ∈ 𝐴}, ℝ^, < ) = inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < )
114	113	xnegeqi 39667	. . 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 2656	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ^ ∣ -_𝑒𝑥 ∈ 𝐴}, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 supcsup 8346 infcinf 8347 ℝcr 9935 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 -cneg 10267 -_𝑒cxne 11943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-xneg 11946

This theorem is referenced by: supminfxrrnmpt 39701 liminfvalxr 40015

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