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Theorem xrminmax 12057

Description: Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)

**Assertion**
Ref	Expression
xrminmax	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → inf({𝐴, 𝐵}, ℝ^, < ) = -_esup({-_e𝐴, -_e𝐵}, ℝ^*, < ))

Proof of Theorem xrminmax Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xnegcl 10252	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ^* → -_e𝑧 ∈ ℝ^*)
2		elprg 3729	. . . . . . . . . . . 12 ⊢ (-_e𝑧 ∈ ℝ^* → (-_e𝑧 ∈ {𝐴, 𝐵} ↔ (-_e𝑧 = 𝐴 ∨ -_e𝑧 = 𝐵)))
3	1, 2	syl 14	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ^* → (-_e𝑧 ∈ {𝐴, 𝐵} ↔ (-_e𝑧 = 𝐴 ∨ -_e𝑧 = 𝐵)))
4	3	adantl 277	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (-_e𝑧 ∈ {𝐴, 𝐵} ↔ (-_e𝑧 = 𝐴 ∨ -_e𝑧 = 𝐵)))
5		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → 𝑧 ∈ ℝ^*)
6		simpll 531	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → 𝐴 ∈ ℝ^*)
7	5, 6	xrnegcon1d 12056	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (-_e𝑧 = 𝐴 ↔ -_e𝐴 = 𝑧))
8		eqcom 2240	. . . . . . . . . . . 12 ⊢ (-_e𝐴 = 𝑧 ↔ 𝑧 = -_e𝐴)
9	7, 8	bitrdi 196	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (-_e𝑧 = 𝐴 ↔ 𝑧 = -_e𝐴))
10		simplr 533	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → 𝐵 ∈ ℝ^*)
11	5, 10	xrnegcon1d 12056	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (-_e𝑧 = 𝐵 ↔ -_e𝐵 = 𝑧))
12		eqcom 2240	. . . . . . . . . . . 12 ⊢ (-_e𝐵 = 𝑧 ↔ 𝑧 = -_e𝐵)
13	11, 12	bitrdi 196	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (-_e𝑧 = 𝐵 ↔ 𝑧 = -_e𝐵))
14	9, 13	orbi12d 805	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → ((-_e𝑧 = 𝐴 ∨ -_e𝑧 = 𝐵) ↔ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)))
15	4, 14	bitrd 188	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑧 ∈ ℝ^) → (-_e𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)))
16	15	rabbidva 2809	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}} = {𝑧 ∈ ℝ^* ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)})
17		dfrab2 3508	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)} = ({𝑧 ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)} ∩ ℝ^*)
18		dfpr2 3728	. . . . . . . . . . 11 ⊢ {-_e𝐴, -_e𝐵} = {𝑧 ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)}
19	18	ineq1i 3428	. . . . . . . . . 10 ⊢ ({-_e𝐴, -_e𝐵} ∩ ℝ^) = ({𝑧 ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)} ∩ ℝ^)
20	17, 19	eqtr4i 2262	. . . . . . . . 9 ⊢ {𝑧 ∈ ℝ^* ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)} = ({-_e𝐴, -_e𝐵} ∩ ℝ^*)
21		xnegcl 10252	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → -_e𝐴 ∈ ℝ^*)
22		xnegcl 10252	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → -_e𝐵 ∈ ℝ^*)
23		prssi 3873	. . . . . . . . . . 11 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ -_e𝐵 ∈ ℝ^) → {-_e𝐴, -_e𝐵} ⊆ ℝ^)
24	21, 22, 23	syl2an 289	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {-_e𝐴, -_e𝐵} ⊆ ℝ^)
25		df-ss 3233	. . . . . . . . . 10 ⊢ ({-_e𝐴, -_e𝐵} ⊆ ℝ^* ↔ ({-_e𝐴, -_e𝐵} ∩ ℝ^*) = {-_e𝐴, -_e𝐵})
26	24, 25	sylib 122	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({-_e𝐴, -_e𝐵} ∩ ℝ^) = {-_e𝐴, -_e𝐵})
27	20, 26	eqtrid 2283	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ (𝑧 = -_e𝐴 ∨ 𝑧 = -_e𝐵)} = {-_e𝐴, -_e𝐵})
28	16, 27	eqtrd 2271	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}} = {-_e𝐴, -_e𝐵})
29	28	supeq1d 7334	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → sup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) = sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
30		xrmaxcl 12044	. . . . . . 7 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ -_e𝐵 ∈ ℝ^) → sup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^*)
31	21, 22, 30	syl2an 289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → sup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^*)
32	29, 31	eqeltrd 2315	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → sup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ∈ ℝ^)
33	32	xnegcld 10275	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ∈ ℝ^)
34		xnegeq 10247	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → -_e𝑦 = -_e𝐴)
35	34	adantl 277	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → -_e𝑦 = -_e𝐴)
36		xrmax1sup 12045	. . . . . . . . . 10 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ -_e𝐵 ∈ ℝ^) → -_e𝐴 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
37	21, 22, 36	syl2an 289	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → -_e𝐴 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
38	37	ad2antrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → -_e𝐴 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
39	35, 38	eqbrtrd 4153	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
40		simpll 531	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*))
41		simpr 110	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
42		simplll 539	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → 𝐴 ∈ ℝ^)
43	41, 42	eqeltrd 2315	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → 𝑦 ∈ ℝ^)
44		xnegeq 10247	. . . . . . . . . . . . 13 ⊢ (sup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) = sup({-_e𝐴, -_e𝐵}, ℝ^, < ) → -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) = -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ))
45	29, 44	syl 14	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) = -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ))
46	45	breq2d 4143	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ↔ 𝑦 < -_esup({-_e𝐴, -_e𝐵}, ℝ^, < )))
47	46	notbid 677	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ↔ ¬ 𝑦 < -_esup({-_e𝐴, -_e𝐵}, ℝ^, < )))
48	47	adantr 276	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (¬ 𝑦 < -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ↔ ¬ 𝑦 < -_esup({-_e𝐴, -_e𝐵}, ℝ^, < )))
49	31	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → sup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^)
50	49	xnegcld 10275	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^)
51		xrlenlt 8397	. . . . . . . . . 10 ⊢ ((-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^ ∧ 𝑦 ∈ ℝ^) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ≤ 𝑦 ↔ ¬ 𝑦 < -_esup({-_e𝐴, -_e𝐵}, ℝ^*, < )))
52	50, 51	sylancom 424	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ≤ 𝑦 ↔ ¬ 𝑦 < -_esup({-_e𝐴, -_e𝐵}, ℝ^, < )))
53		xleneg 10257	. . . . . . . . . . 11 ⊢ ((-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^ ∧ 𝑦 ∈ ℝ^) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ≤ 𝑦 ↔ -_e𝑦 ≤ -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^*, < )))
54	50, 53	sylancom 424	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ≤ 𝑦 ↔ -_e𝑦 ≤ -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^, < )))
55		xnegneg 10253	. . . . . . . . . . . 12 ⊢ (sup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^ → -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) = sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
56	49, 55	syl 14	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) = sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
57	56	breq2d 4143	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (-_e𝑦 ≤ -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ↔ -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < )))
58	54, 57	bitrd 188	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ≤ 𝑦 ↔ -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < )))
59	48, 52, 58	3bitr2d 216	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (¬ 𝑦 < -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ↔ -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < )))
60	40, 43, 59	syl2anc 415	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → (¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ↔ -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < )))
61	39, 60	mpbird 167	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐴) → ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ))
62		xnegeq 10247	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → -_e𝑦 = -_e𝐵)
63	62	adantl 277	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → -_e𝑦 = -_e𝐵)
64		xrmax2sup 12046	. . . . . . . . . 10 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ -_e𝐵 ∈ ℝ^) → -_e𝐵 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
65	21, 22, 64	syl2an 289	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → -_e𝐵 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
66	65	ad2antrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → -_e𝐵 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
67	63, 66	eqbrtrd 4153	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
68		simpll 531	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*))
69		simpr 110	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
70		simpllr 540	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → 𝐵 ∈ ℝ^)
71	69, 70	eqeltrd 2315	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ ℝ^)
72	68, 71, 59	syl2anc 415	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → (¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ↔ -_e𝑦 ≤ sup({-_e𝐴, -_e𝐵}, ℝ^, < )))
73	67, 72	mpbird 167	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) ∧ 𝑦 = 𝐵) → ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ))
74		elpri 3732	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
75	74	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
76	61, 73, 75	mpjaodan 810	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ {𝐴, 𝐵}) → ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ))
77	76	ralrimiva 2623	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ))
78	21	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_e𝐴 ∈ ℝ^)
79	22	ad3antlr 497	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_e𝐵 ∈ ℝ^)
80		simplr 533	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → 𝑦 ∈ ℝ^)
81	80	xnegcld 10275	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_e𝑦 ∈ ℝ^)
82		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦)
83	45	breq1d 4141	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (-_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦 ↔ -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) < 𝑦))
84	83	ad2antrr 492	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → (-_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦 ↔ -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) < 𝑦))
85	82, 84	mpbid 147	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) < 𝑦)
86	50	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^*)
87		xltneg 10256	. . . . . . . . . . . 12 ⊢ ((-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ∈ ℝ^ ∧ 𝑦 ∈ ℝ^) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) < 𝑦 ↔ -_e𝑦 < -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^*, < )))
88	86, 80, 87	syl2anc 415	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) < 𝑦 ↔ -_e𝑦 < -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^*, < )))
89	56	breq2d 4143	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (-_e𝑦 < -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ↔ -_e𝑦 < sup({-_e𝐴, -_e𝐵}, ℝ^, < )))
90	89	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → (-_e𝑦 < -_e-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) ↔ -_e𝑦 < sup({-_e𝐴, -_e𝐵}, ℝ^*, < )))
91	88, 90	bitrd 188	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → (-_esup({-_e𝐴, -_e𝐵}, ℝ^, < ) < 𝑦 ↔ -_e𝑦 < sup({-_e𝐴, -_e𝐵}, ℝ^*, < )))
92	85, 91	mpbid 147	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → -_e𝑦 < sup({-_e𝐴, -_e𝐵}, ℝ^, < ))
93		xrmaxleastlt 12048	. . . . . . . . 9 ⊢ (((-_e𝐴 ∈ ℝ^* ∧ -_e𝐵 ∈ ℝ^) ∧ (-_e𝑦 ∈ ℝ^ ∧ -_e𝑦 < sup({-_e𝐴, -_e𝐵}, ℝ^*, < ))) → (-_e𝑦 < -_e𝐴 ∨ -_e𝑦 < -_e𝐵))
94	78, 79, 81, 92, 93	syl22anc 1279	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → (-_e𝑦 < -_e𝐴 ∨ -_e𝑦 < -_e𝐵))
95		simplll 539	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → 𝐴 ∈ ℝ^)
96		xltneg 10256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝐴 < 𝑦 ↔ -_e𝑦 < -_e𝐴))
97	95, 80, 96	syl2anc 415	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → (𝐴 < 𝑦 ↔ -_e𝑦 < -_e𝐴))
98		simpllr 540	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦) → 𝐵 ∈ ℝ^)
99		xltneg 10256	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝐵 < 𝑦 ↔ -_e𝑦 < -_e𝐵))
100	98, 80, 99	syl2anc 415	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → (𝐵 < 𝑦 ↔ -_e𝑦 < -_e𝐵))
101	97, 100	orbi12d 805	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → ((𝐴 < 𝑦 ∨ 𝐵 < 𝑦) ↔ (-_e𝑦 < -_e𝐴 ∨ -_e𝑦 < -_e𝐵)))
102	94, 101	mpbird 167	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → (𝐴 < 𝑦 ∨ 𝐵 < 𝑦))
103		breq1 4134	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧 < 𝑦 ↔ 𝐴 < 𝑦))
104		breq1 4134	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (𝑧 < 𝑦 ↔ 𝐵 < 𝑦))
105	103, 104	rexprg 3761	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦 ↔ (𝐴 < 𝑦 ∨ 𝐵 < 𝑦)))
106	105	ad2antrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → (∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦 ↔ (𝐴 < 𝑦 ∨ 𝐵 < 𝑦)))
107	102, 106	mpbird 167	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) ∧ -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦) → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦)
108	107	ex 115	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (-_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦))
109	108	ralrimiva 2623	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ∀𝑦 ∈ ℝ^ (-_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦))
110		breq2 4135	. . . . . . . 8 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → (𝑦 < 𝑥 ↔ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < )))
111	110	notbid 677	. . . . . . 7 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < )))
112	111	ralbidv 2550	. . . . . 6 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → (∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < )))
113		breq1 4134	. . . . . . . 8 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → (𝑥 < 𝑦 ↔ -_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦))
114	113	imbi1d 231	. . . . . . 7 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → ((𝑥 < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦) ↔ (-_esup({𝑧 ∈ ℝ^ ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦)))
115	114	ralbidv 2550	. . . . . 6 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → (∀𝑦 ∈ ℝ^ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ^* (-_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦)))
116	112, 115	anbi12d 477	. . . . 5 ⊢ (𝑥 = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) → ((∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦)) ↔ (∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ∧ ∀𝑦 ∈ ℝ^ (-_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ) < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦))))
117	116	rspcev 2929	. . . 4 ⊢ ((-_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ∈ ℝ^ ∧ (∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) ∧ ∀𝑦 ∈ ℝ^ (-_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^, < ) < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ^ (∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦)))
118	33, 77, 109, 117	syl12anc 1276	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ∃𝑥 ∈ ℝ^ (∀𝑦 ∈ {𝐴, 𝐵} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ {𝐴, 𝐵}𝑧 < 𝑦)))
119		prssi 3873	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝐴, 𝐵} ⊆ ℝ^)
120	118, 119	infxrnegsupex 12055	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → inf({𝐴, 𝐵}, ℝ^, < ) = -_esup({𝑧 ∈ ℝ^* ∣ -_e𝑧 ∈ {𝐴, 𝐵}}, ℝ^*, < ))
121	120, 45	eqtrd 2271	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → inf({𝐴, 𝐵}, ℝ^, < ) = -_esup({-_e𝐴, -_e𝐵}, ℝ^*, < ))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 {cab 2224 ∀wral 2528 ∃wrex 2529 {crab 2532 ∩ cin 3219 ⊆ wss 3220 {cpr 3710 class class class wbr 4131 supcsup 7329 infcinf 7330 ℝ^*cxr 8366 < clt 8367 ≤ cle 8368 -_ecxne 10189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-rp 10073 df-xneg 10192 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788

This theorem is used by: xrmincl 12058 xrmin1inf 12059 xrmin2inf 12060 xrmineqinf 12061 xrltmininf 12062 xrlemininf 12063 xrminltinf 12064 xrminrecl 12065 xrminrpcl 12066 xrminadd 12067

W3C validator