Theorem supminfxr2 46572

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 13343	. . . . . 6 ⊢ -e-∞ = +∞
2	1	eqcomi 2769	. . . . 5 ⊢ +∞ = -e-∞
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -e-∞)
4		supminfxr2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
5		supxrpnf 13451	. . . . 5 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
6	4, 5	sylan 592	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
7		ssrab2 4005	. . . . . . . 8 ⊢ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ⊆ ℝ*
8	7	a1i 11	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ⊆ ℝ*)
9		xnegeq 13340	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -e𝑦 = -e-∞)
10	1	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → -e-∞ = +∞)
11	9, 10	eqtrd 2795	. . . . . . . . 9 ⊢ (𝑦 = -∞ → -e𝑦 = +∞)
12	11	eleq1d 2845	. . . . . . . 8 ⊢ (𝑦 = -∞ → ( -e𝑦 ∈ 𝐴 ↔ +∞ ∈ 𝐴))
13		mnfxr 11369	. . . . . . . . 9 ⊢ -∞ ∈ ℝ*
14	13	a1i 11	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ ℝ*)
15		id 23	. . . . . . . 8 ⊢ (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
16	12, 14, 15	elrabd 3646	. . . . . . 7 ⊢ (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴})
17		infxrmnf 13471	. . . . . . 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 46540	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → -einf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴}, ℝ*, < ) = -e-∞)
21	3, 6, 20	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -einf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴}, ℝ*, < ))
22	4	ssdifssd 4083	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ*)
23	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ*)
24		difssd 4073	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25		id 23	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26		ssnel 46153	. . . . . . . 8 ⊢ (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
27	24, 25, 26	syl2anc 596	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
28	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29		neldifsnd 4752	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
30	23, 28, 29	xrssre 46453	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
31	30	supminfxr 46567	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = -einf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
32		supxrmnf2 46536	. . . . . . 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 2766	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
35	34	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
36		rexr 11358	. . . . . . . . . . . . . . . . 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 46251	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -e𝑦 = -𝑦)
40		eldifi 4067	. . . . . . . . . . . . . . . . . 18 ⊢ ( -𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦 ∈ 𝐴)
41	40	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦 ∈ 𝐴)
42	39, 41	eqeltrd 2860	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -e𝑦 ∈ 𝐴)
43	37, 42	jca 521	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ* ∧ -e𝑦 ∈ 𝐴))
44		rabid 3432	. . . . . . . . . . . . . . 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 11360	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47		elsni 4597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
48	47	necon3ai 2980	. . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . 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 3078	. . . . . . . . . . 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 3431	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴}
56		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{+∞}
57	55, 56	nfdif 4066	. . . . . . . . . . 11 ⊢ Ⅎ𝑦({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞})
58	57	rabssf 46227	. . . . . . . . . 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 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 ¬ +∞ ∈ 𝐴
61		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦ℝ
62		eldifi 4067	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴})
63	7, 62	sselid 3906	. . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . . 14 ⊢ ((¬ +∞ ∈ 𝐴 ∧ -e𝑦 ∈ 𝐴) → 𝑦 ≠ -∞)
72	66, 71	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73		eldifsni 4749	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
74	73	adantl 487	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
75	64, 72, 74	xrred 46469	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
76	60, 57, 61, 75	ssdf2 46249	. . . . . . . . . . 11 ⊢ (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}) ⊆ ℝ)
77	75	rexnegd 46251	. . . . . . . . . . . . 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 4597	. . . . . . . . . . . . . . . . . 18 ⊢ ( -e𝑦 ∈ {-∞} → -e𝑦 = -∞)
81	80	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -e𝑦 ∈ {-∞}) → -e𝑦 = -∞)
82		xnegeq 13340	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( -e𝑦 = -∞ → -e -e𝑦 = -e-∞)
83	1	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( -e𝑦 = -∞ → -e-∞ = +∞)
84	82, 83	eqtr2d 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ ( -e𝑦 = -∞ → +∞ = -e -e𝑦)
85	84	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = -∞) → +∞ = -e -e𝑦)
86		xnegneg 13347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ* → -e -e𝑦 = 𝑦)
87	86	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = -∞) → -e -e𝑦 = 𝑦)
88	85, 87	eqtr2d 2796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = -∞) → 𝑦 = +∞)
89	79, 81, 88	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}) ∧ -e𝑦 ∈ {-∞}) → 𝑦 = +∞)
90	73	neneqd 2960	. . . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . 13 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞})) → -e𝑦 ∈ (𝐴 ∖ {-∞}))
95	77, 94	eqeltrrd 2861	. . . . . . . . . . . 12 ⊢ ((¬ +∞ ∈ 𝐴 ∧ 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
96	95	ralrimiva 3154	. . . . . . . . . . 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 46222	. . . . . . . . . 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 3925	. . . . . . . 8 ⊢ (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}))
101	100	infeq1d 9473	. . . . . . 7 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ) = inf(({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} ∖ {+∞}), ℝ*, < ))
102		infxrpnf2 46566	. . . . . . . . 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 2796	. . . . . 6 ⊢ (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴}, ℝ*, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
106	105	xnegeqd 46540	. . . . 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 2805	. . 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 13340	. . . . . . 7 ⊢ (𝑦 = 𝑥 → -e𝑦 = -e𝑥)
111	110	eleq1d 2845	. . . . . 6 ⊢ (𝑦 = 𝑥 → ( -e𝑦 ∈ 𝐴 ↔ -e𝑥 ∈ 𝐴))
112	111	cbvrabv 3422	. . . . 5 ⊢ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴} = {𝑥 ∈ ℝ* ∣ -e𝑥 ∈ 𝐴}
113	112	infeq1i 9474	. . . 4 ⊢ inf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ 𝐴}, ℝ*, < ) = inf({𝑥 ∈ ℝ* ∣ -e𝑥 ∈ 𝐴}, ℝ*, < )
114	113	xnegeqi 46543	. . 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 2795	1 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -einf({𝑥 ∈ ℝ* ∣ -e𝑥 ∈ 𝐴}, ℝ*, < ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∖ cdif 4045  {csn 4584  supcsup 9435  infcinf 9436  ℝcr 11202  +∞cpnf 11343  -∞cmnf 11344  ℝ^*cxr 11345   < clt 11346   -cneg 11545   -_ecxne 13241

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-sup 9437  df-inf 9438  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-xneg 13244

This theorem is used by:  supminfxrrnmpt  46574  liminfvalxr  46886