supminfxr - Mathbox for Glauco Siliprandi

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Theorem supminfxr 41760

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

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

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

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

Proof of Theorem supminfxr Dummy variables 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supeq1 8909	. . . . 5 ⊢ (𝐴 = ∅ → sup(𝐴, ℝ^, < ) = sup(∅, ℝ^, < ))
2		xrsup0 12717	. . . . . 6 ⊢ sup(∅, ℝ^*, < ) = -∞
3	2	a1i 11	. . . . 5 ⊢ (𝐴 = ∅ → sup(∅, ℝ^*, < ) = -∞)
4	1, 3	eqtrd 2856	. . . 4 ⊢ (𝐴 = ∅ → sup(𝐴, ℝ^*, < ) = -∞)
5	4	adantl 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → sup(𝐴, ℝ^*, < ) = -∞)
6		eleq2 2901	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (-𝑥 ∈ 𝐴 ↔ -𝑥 ∈ ∅))
7	6	rabbidv 3480	. . . . . . . . 9 ⊢ (𝐴 = ∅ → {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} = {𝑥 ∈ ℝ ∣ -𝑥 ∈ ∅})
8		noel 4296	. . . . . . . . . . . . 13 ⊢ ¬ -𝑥 ∈ ∅
9	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ¬ -𝑥 ∈ ∅)
10	9	rgen 3148	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ ℝ ¬ -𝑥 ∈ ∅
11		rabeq0 4338	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ ℝ ∣ -𝑥 ∈ ∅} = ∅ ↔ ∀𝑥 ∈ ℝ ¬ -𝑥 ∈ ∅)
12	10, 11	mpbir 233	. . . . . . . . . 10 ⊢ {𝑥 ∈ ℝ ∣ -𝑥 ∈ ∅} = ∅
13	12	a1i 11	. . . . . . . . 9 ⊢ (𝐴 = ∅ → {𝑥 ∈ ℝ ∣ -𝑥 ∈ ∅} = ∅)
14	7, 13	eqtrd 2856	. . . . . . . 8 ⊢ (𝐴 = ∅ → {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} = ∅)
15	14	infeq1d 8941	. . . . . . 7 ⊢ (𝐴 = ∅ → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ) = inf(∅, ℝ^, < ))
16		xrinf0 12732	. . . . . . . 8 ⊢ inf(∅, ℝ^*, < ) = +∞
17	16	a1i 11	. . . . . . 7 ⊢ (𝐴 = ∅ → inf(∅, ℝ^*, < ) = +∞)
18	15, 17	eqtrd 2856	. . . . . 6 ⊢ (𝐴 = ∅ → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = +∞)
19	18	xnegeqd 41731	. . . . 5 ⊢ (𝐴 = ∅ → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -_𝑒+∞)
20		xnegpnf 12603	. . . . . 6 ⊢ -_𝑒+∞ = -∞
21	20	a1i 11	. . . . 5 ⊢ (𝐴 = ∅ → -_𝑒+∞ = -∞)
22	19, 21	eqtrd 2856	. . . 4 ⊢ (𝐴 = ∅ → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -∞)
23	22	adantl 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -∞)
24	5, 23	eqtr4d 2859	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
25		neqne 3024	. . 3 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅)
26		supminfxr.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
27	26	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → 𝐴 ⊆ ℝ)
28		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → 𝐴 ≠ ∅)
29		simpr 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦)
30		negn0 11069	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅)
31		ublbneg 12334	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧)
32		ssrab2 4056	. . . . . . . . . . 11 ⊢ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ
33		infrenegsup 11624	. . . . . . . . . . 11 ⊢ (({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ ∧ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ))
34	32, 33	mp3an1 1444	. . . . . . . . . 10 ⊢ (({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ))
35	30, 31, 34	syl2an 597	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ))
36	35	3impa 1106	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ))
37		elrabi 3675	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}} → 𝑦 ∈ ℝ)
38	37	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}) → 𝑦 ∈ ℝ)
39		ssel2 3962	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
40		negeq 10878	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → -𝑤 = -𝑦)
41	40	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → (-𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ↔ -𝑦 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}))
42	41	elrab3 3681	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}} ↔ -𝑦 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}))
43		renegcl 10949	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
44		negeq 10878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -𝑦 → -𝑥 = --𝑦)
45	44	eleq1d 2897	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = -𝑦 → (-𝑥 ∈ 𝐴 ↔ --𝑦 ∈ 𝐴))
46	45	elrab3 3681	. . . . . . . . . . . . . . 15 ⊢ (-𝑦 ∈ ℝ → (-𝑦 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ↔ --𝑦 ∈ 𝐴))
47	43, 46	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (-𝑦 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ↔ --𝑦 ∈ 𝐴))
48		recn 10627	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
49	48	negnegd 10988	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → --𝑦 = 𝑦)
50	49	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (--𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
51	42, 47, 50	3bitrd 307	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}} ↔ 𝑦 ∈ 𝐴))
52	51	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}} ↔ 𝑦 ∈ 𝐴))
53	38, 39, 52	eqrdav 2820	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → {𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}} = 𝐴)
54	53	supeq1d 8910	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ) = sup(𝐴, ℝ, < ))
55	54	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ) = sup(𝐴, ℝ, < ))
56	55	negeqd 10880	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}}, ℝ, < ) = -sup(𝐴, ℝ, < ))
57	36, 56	eqtrd 2856	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup(𝐴, ℝ, < ))
58		infrecl 11623	. . . . . . . . . . 11 ⊢ (({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ ∧ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
59	32, 58	mp3an1 1444	. . . . . . . . . 10 ⊢ (({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
60	30, 31, 59	syl2an 597	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
61	60	3impa 1106	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
62		suprcl 11601	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ, < ) ∈ ℝ)
63		recn 10627	. . . . . . . . 9 ⊢ (inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℂ)
64		recn 10627	. . . . . . . . 9 ⊢ (sup(𝐴, ℝ, < ) ∈ ℝ → sup(𝐴, ℝ, < ) ∈ ℂ)
65		negcon2 10939	. . . . . . . . 9 ⊢ ((inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℂ ∧ sup(𝐴, ℝ, < ) ∈ ℂ) → (inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup(𝐴, ℝ, < ) ↔ sup(𝐴, ℝ, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < )))
66	63, 64, 65	syl2an 597	. . . . . . . 8 ⊢ ((inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ ∧ sup(𝐴, ℝ, < ) ∈ ℝ) → (inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup(𝐴, ℝ, < ) ↔ sup(𝐴, ℝ, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < )))
67	61, 62, 66	syl2anc 586	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → (inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -sup(𝐴, ℝ, < ) ↔ sup(𝐴, ℝ, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < )))
68	57, 67	mpbid 234	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
69	27, 28, 29, 68	syl3anc 1367	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
70		supxrre 12721	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
71	27, 28, 29, 70	syl3anc 1367	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
72	32	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ)
73	27, 28, 30	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅)
74	29, 31	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧)
75		infxrre 12730	. . . . . . . 8 ⊢ (({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ ∧ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑦 ≤ 𝑧) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
76	72, 73, 74, 75	syl3anc 1367	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
77	76	xnegeqd 41731	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
78	26, 60	sylanl1 678	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
79	78	rexnegd 41432	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
80	77, 79	eqtrd 2856	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ, < ))
81	69, 71, 80	3eqtr4d 2866	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
82		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦)
83		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ ℝ)
84	26	sselda 3967	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
85	84	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
86	83, 85	ltnled 10787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑦 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑦))
87	86	rexbidva 3296	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ 𝐴 ¬ 𝑧 ≤ 𝑦))
88		rexnal 3238	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝐴 ¬ 𝑧 ≤ 𝑦 ↔ ¬ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦)
89	88	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 ¬ 𝑧 ≤ 𝑦 ↔ ¬ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦))
90	87, 89	bitrd 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦))
91	90	ralbidva 3196	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∀𝑦 ∈ ℝ ¬ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦))
92		ralnex 3236	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ℝ ¬ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦 ↔ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦)
93	92	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ¬ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦 ↔ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦))
94	91, 93	bitrd 281	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦))
95	94	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → (∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦))
96	82, 95	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
97		xnegmnf 12604	. . . . . . . . 9 ⊢ -_𝑒-∞ = +∞
98	97	eqcomi 2830	. . . . . . . 8 ⊢ +∞ = -_𝑒-∞
99	98	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → +∞ = -_𝑒-∞)
100		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
101		ressxr 10685	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
102	101	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
103	26, 102	sstrd 3977	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
104		supxrunb2 12714	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ sup(𝐴, ℝ^*, < ) = +∞))
105	103, 104	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ sup(𝐴, ℝ^*, < ) = +∞))
106	105	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → (∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ sup(𝐴, ℝ^*, < ) = +∞))
107	100, 106	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → sup(𝐴, ℝ^*, < ) = +∞)
108		renegcl 10949	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ℝ → -𝑣 ∈ ℝ)
109	108	adantl 484	. . . . . . . . . . . . 13 ⊢ ((∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ∧ 𝑣 ∈ ℝ) → -𝑣 ∈ ℝ)
110		simpl 485	. . . . . . . . . . . . 13 ⊢ ((∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ∧ 𝑣 ∈ ℝ) → ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
111		breq1 5069	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = -𝑣 → (𝑦 < 𝑧 ↔ -𝑣 < 𝑧))
112	111	rexbidv 3297	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -𝑣 → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ 𝐴 -𝑣 < 𝑧))
113	112	rspcva 3621	. . . . . . . . . . . . 13 ⊢ ((-𝑣 ∈ ℝ ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∃𝑧 ∈ 𝐴 -𝑣 < 𝑧)
114	109, 110, 113	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ∧ 𝑣 ∈ ℝ) → ∃𝑧 ∈ 𝐴 -𝑣 < 𝑧)
115	114	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ∧ 𝑣 ∈ ℝ) → ∃𝑧 ∈ 𝐴 -𝑣 < 𝑧)
116		negeq 10878	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = -𝑧 → -𝑥 = --𝑧)
117	116	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = -𝑧 → (-𝑥 ∈ 𝐴 ↔ --𝑧 ∈ 𝐴))
118	84	renegcld 11067	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → -𝑧 ∈ ℝ)
119	118	ad4ant13 749	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → -𝑧 ∈ ℝ)
120	84	recnd 10669	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℂ)
121	120	negnegd 10988	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → --𝑧 = 𝑧)
122		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
123	121, 122	eqeltrd 2913	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → --𝑧 ∈ 𝐴)
124	123	ad4ant13 749	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → --𝑧 ∈ 𝐴)
125	117, 119, 124	elrabd 3682	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → -𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴})
126		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → -𝑣 < 𝑧)
127	108	ad3antlr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → -𝑣 ∈ ℝ)
128	84	ad4ant13 749	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → 𝑧 ∈ ℝ)
129	127, 128	ltnegd 11218	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → (-𝑣 < 𝑧 ↔ -𝑧 < --𝑣))
130	126, 129	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → -𝑧 < --𝑣)
131		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → 𝑣 ∈ ℝ)
132		recn 10627	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ℝ → 𝑣 ∈ ℂ)
133		negneg 10936	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ℂ → --𝑣 = 𝑣)
134	131, 132, 133	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → --𝑣 = 𝑣)
135	130, 134	breqtrd 5092	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → -𝑧 < 𝑣)
136		breq1 5069	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = -𝑧 → (𝑤 < 𝑣 ↔ -𝑧 < 𝑣))
137	136	rspcev 3623	. . . . . . . . . . . . . 14 ⊢ ((-𝑧 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ∧ -𝑧 < 𝑣) → ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣)
138	125, 135, 137	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑣 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ -𝑣 < 𝑧) → ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣)
139	138	rexlimdva2 3287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ℝ) → (∃𝑧 ∈ 𝐴 -𝑣 < 𝑧 → ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣))
140	139	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ∧ 𝑣 ∈ ℝ) → (∃𝑧 ∈ 𝐴 -𝑣 < 𝑧 → ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣))
141	115, 140	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ∧ 𝑣 ∈ ℝ) → ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣)
142	141	ralrimiva 3182	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑣 ∈ ℝ ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣)
143	32, 101	sstri 3976	. . . . . . . . . 10 ⊢ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ^*
144		infxrunb2 41656	. . . . . . . . . 10 ⊢ ({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴} ⊆ ℝ^* → (∀𝑣 ∈ ℝ ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣 ↔ inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -∞))
145	143, 144	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑣 ∈ ℝ ∃𝑤 ∈ {𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}𝑤 < 𝑣 ↔ inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -∞)
146	142, 145	sylib 220	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -∞)
147	146	xnegeqd 41731	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^*, < ) = -_𝑒-∞)
148	99, 107, 147	3eqtr4d 2866	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
149	96, 148	syldan 593	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
150	149	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ ¬ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
151	81, 150	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
152	25, 151	sylan2 594	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))
153	24, 152	pm2.61dan 811	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) = -_𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 supcsup 8904 infcinf 8905 ℂcc 10535 ℝcr 10536 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 -cneg 10871 -_𝑒cxne 12505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-xneg 12508

This theorem is referenced by: supminfxr2 41765

W3C validator