xrbdtri - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrbdtri		GIF version

Theorem xrbdtri 11954

Description: Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)

**Assertion**
Ref	Expression
xrbdtri	⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))

**Proof of Theorem xrbdtri**
Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
2		simp1r 1049	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐴)
3	2	ad3antrrr 492	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐴)
4		simplr 529	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ)
5		simp2r 1051	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐵)
6	5	ad3antrrr 492	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐵)
7		simpllr 536	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ)
8		simp3r 1053	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < 𝐶)
9	8	ad3antrrr 492	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 < 𝐶)
10	7, 9	elrpd 10022	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ⁺)
11		bdtri 11918	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ⁺) → inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))
12	1, 3, 4, 6, 10, 11	syl221anc 1285	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))
13	1, 4	rexaddd 10183	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
14	13	preq1d 3773	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → {(𝐴 +_𝑒 𝐵), 𝐶} = {(𝐴 + 𝐵), 𝐶})
15	14	infeq1d 7302	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) = inf({(𝐴 + 𝐵), 𝐶}, ℝ^, < ))
16	1, 4	readdcld 8299	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
17		xrminrecl 11951	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({(𝐴 + 𝐵), 𝐶}, ℝ^*, < ) = inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ))
18	16, 7, 17	syl2anc 411	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({(𝐴 + 𝐵), 𝐶}, ℝ^*, < ) = inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ))
19	15, 18	eqtrd 2265	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ))
20		xrminrecl 11951	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ^*, < ) = inf({𝐴, 𝐶}, ℝ, < ))
21	1, 7, 20	syl2anc 411	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ^*, < ) = inf({𝐴, 𝐶}, ℝ, < ))
22		xrminrecl 11951	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ^*, < ) = inf({𝐵, 𝐶}, ℝ, < ))
23	4, 7, 22	syl2anc 411	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ^*, < ) = inf({𝐵, 𝐶}, ℝ, < ))
24	21, 23	oveq12d 6067	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^, < )) = (inf({𝐴, 𝐶}, ℝ, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ, < )))
25		mincl 11909	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ, < ) ∈ ℝ)
26	1, 7, 25	syl2anc 411	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ, < ) ∈ ℝ)
27		mincl 11909	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ, < ) ∈ ℝ)
28	4, 7, 27	syl2anc 411	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ, < ) ∈ ℝ)
29	26, 28	rexaddd 10183	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (inf({𝐴, 𝐶}, ℝ, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ, < )) = (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))
30	24, 29	eqtrd 2265	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^, < )) = (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))
31	12, 19, 30	3brtr4d 4140	. . . 4 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
32		simp3l 1052	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
33	32	xaddid1d 10193	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 +_𝑒 0) = 𝐶)
34	32	xrleidd 10130	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ 𝐶)
35		0xr 8316	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
36	35	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ∈ ℝ^*)
37	36, 32, 8	xrltled 10128	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐶)
38		simp2l 1050	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
39		xrlemininf 11949	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
40	36, 38, 32, 39	syl3anc 1274	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (0 ≤ inf({𝐵, 𝐶}, ℝ^*, < ) ↔ (0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
41	5, 37, 40	mpbir2and 953	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐵, 𝐶}, ℝ^*, < ))
42		xrmincl 11944	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
43	38, 32, 42	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^)
44		xle2add 10208	. . . . . . . . 9 ⊢ (((𝐶 ∈ ℝ^* ∧ 0 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^)) → ((𝐶 ≤ 𝐶 ∧ 0 ≤ inf({𝐵, 𝐶}, ℝ^, < )) → (𝐶 +_𝑒 0) ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^, < ))))
45	32, 36, 32, 43, 44	syl22anc 1275	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((𝐶 ≤ 𝐶 ∧ 0 ≤ inf({𝐵, 𝐶}, ℝ^, < )) → (𝐶 +_𝑒 0) ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^, < ))))
46	34, 41, 45	mp2and 433	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 +_𝑒 0) ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
47	33, 46	eqbrtrrd 4132	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
48	47	ad3antrrr 492	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
49		simp1l 1048	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
50	49, 38	xaddcld 10213	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
51	50	ad3antrrr 492	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
52	32	ad3antrrr 492	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ∈ ℝ^*)
53		pnfge 10118	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ^* → 𝐶 ≤ +∞)
54	52, 53	syl 14	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ +∞)
55		simpr 110	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐴 = +∞)
56	55	oveq1d 6064	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
57		simpl2l 1077	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
58	57	ad2antrr 488	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ^*)
59		simplr 529	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ)
60	59	renemnfd 8321	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ≠ -∞)
61		xaddpnf2 10176	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
62	58, 60, 61	syl2anc 411	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (+∞ +_𝑒 𝐵) = +∞)
63	56, 62	eqtrd 2265	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
64	54, 63	breqtrrd 4136	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ (𝐴 +_𝑒 𝐵))
65		xrmineqinf 11947	. . . . . 6 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ (𝐴 +_𝑒 𝐵)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
66	51, 52, 64, 65	syl3anc 1274	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
67	49	ad3antrrr 492	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐴 ∈ ℝ^*)
68	54, 55	breqtrrd 4136	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ 𝐴)
69		xrmineqinf 11947	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐴) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐶)
70	67, 52, 68, 69	syl3anc 1274	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐶)
71	70	oveq1d 6064	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^, < )) = (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
72	48, 66, 71	3brtr4d 4140	. . . 4 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
73		simpr 110	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 = -∞)
74		ge0nemnf 10153	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
75	49, 2, 74	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ≠ -∞)
76	75	ad3antrrr 492	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 ≠ -∞)
77	73, 76	pm2.21ddne 2495	. . . 4 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
78		elxr 10105	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
79	49, 78	sylib 122	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
80	79	ad2antrr 488	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
81	31, 72, 77, 80	mpjao3dan 1344	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
82		xrlemininf 11949	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶)))
83	36, 49, 32, 82	syl3anc 1274	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (0 ≤ inf({𝐴, 𝐶}, ℝ^*, < ) ↔ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶)))
84	2, 37, 83	mpbir2and 953	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐴, 𝐶}, ℝ^*, < ))
85		xrmincl 11944	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
86	49, 32, 85	syl2anc 411	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^)
87		xle2add 10208	. . . . . . 7 ⊢ (((0 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^)) → ((0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ∧ 𝐶 ≤ 𝐶) → (0 +_𝑒 𝐶) ≤ (inf({𝐴, 𝐶}, ℝ^*, < ) +_𝑒 𝐶)))
88	36, 32, 86, 32, 87	syl22anc 1275	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ∧ 𝐶 ≤ 𝐶) → (0 +_𝑒 𝐶) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 𝐶)))
89	84, 34, 88	mp2and 433	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (0 +_𝑒 𝐶) ≤ (inf({𝐴, 𝐶}, ℝ^*, < ) +_𝑒 𝐶))
90	89	ad2antrr 488	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (0 +_𝑒 𝐶) ≤ (inf({𝐴, 𝐶}, ℝ^*, < ) +_𝑒 𝐶))
91	50	ad2antrr 488	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
92	32	ad2antrr 488	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ∈ ℝ^*)
93	92, 53	syl 14	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ +∞)
94		simpr 110	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐵 = +∞)
95	94	oveq2d 6065	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
96		xaddpnf1 10175	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
97	49, 75, 96	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 +_𝑒 +∞) = +∞)
98	97	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 +∞) = +∞)
99	95, 98	eqtrd 2265	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
100	93, 99	breqtrrd 4136	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ (𝐴 +_𝑒 𝐵))
101	91, 92, 100, 65	syl3anc 1274	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
102		xaddid2 10192	. . . . . 6 ⊢ (𝐶 ∈ ℝ^* → (0 +_𝑒 𝐶) = 𝐶)
103	92, 102	syl 14	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (0 +_𝑒 𝐶) = 𝐶)
104	101, 103	eqtr4d 2268	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = (0 +_𝑒 𝐶))
105	57	adantr 276	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐵 ∈ ℝ^*)
106	93, 94	breqtrrd 4136	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ 𝐵)
107		xrmineqinf 11947	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
108	105, 92, 106, 107	syl3anc 1274	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
109	108	oveq2d 6065	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^, < )) = (inf({𝐴, 𝐶}, ℝ^*, < ) +_𝑒 𝐶))
110	90, 104, 109	3brtr4d 4140	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
111		simpr 110	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 = -∞)
112	57	adantr 276	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 ∈ ℝ^*)
113	5	ad2antrr 488	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 0 ≤ 𝐵)
114		ge0nemnf 10153	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) → 𝐵 ≠ -∞)
115	112, 113, 114	syl2anc 411	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 ≠ -∞)
116	111, 115	pm2.21ddne 2495	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
117		elxr 10105	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
118	57, 117	sylib 122	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
119	81, 110, 116, 118	mpjao3dan 1344	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
120	50	adantr 276	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
121	120	xrleidd 10130	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
122		prcom 3766	. . . . 5 ⊢ {𝐶, (𝐴 +_𝑒 𝐵)} = {(𝐴 +_𝑒 𝐵), 𝐶}
123	122	infeq1i 7303	. . . 4 ⊢ inf({𝐶, (𝐴 +_𝑒 𝐵)}, ℝ^, < ) = inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < )
124	32	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐶 ∈ ℝ^*)
125		pnfge 10118	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) ∈ ℝ^* → (𝐴 +_𝑒 𝐵) ≤ +∞)
126	120, 125	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ +∞)
127		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐶 = +∞)
128	126, 127	breqtrrd 4136	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ 𝐶)
129		xrmineqinf 11947	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ (𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ (𝐴 +_𝑒 𝐵) ≤ 𝐶) → inf({𝐶, (𝐴 +_𝑒 𝐵)}, ℝ^*, < ) = (𝐴 +_𝑒 𝐵))
130	124, 120, 128, 129	syl3anc 1274	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, (𝐴 +_𝑒 𝐵)}, ℝ^*, < ) = (𝐴 +_𝑒 𝐵))
131	123, 130	eqtr3id 2279	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = (𝐴 +_𝑒 𝐵))
132		prcom 3766	. . . . . 6 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
133	132	infeq1i 7303	. . . . 5 ⊢ inf({𝐶, 𝐴}, ℝ^, < ) = inf({𝐴, 𝐶}, ℝ^, < )
134	49	adantr 276	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ∈ ℝ^*)
135		pnfge 10118	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
136	134, 135	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ≤ +∞)
137	136, 127	breqtrrd 4136	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ≤ 𝐶)
138		xrmineqinf 11947	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
139	124, 134, 137, 138	syl3anc 1274	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
140	133, 139	eqtr3id 2279	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐴)
141		prcom 3766	. . . . . 6 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
142	141	infeq1i 7303	. . . . 5 ⊢ inf({𝐶, 𝐵}, ℝ^, < ) = inf({𝐵, 𝐶}, ℝ^, < )
143	38	adantr 276	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ∈ ℝ^*)
144		pnfge 10118	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
145	143, 144	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ≤ +∞)
146	145, 127	breqtrrd 4136	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ≤ 𝐶)
147		xrmineqinf 11947	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
148	124, 143, 146, 147	syl3anc 1274	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
149	142, 148	eqtr3id 2279	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐵)
150	140, 149	oveq12d 6067	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^, < )) = (𝐴 +_𝑒 𝐵))
151	121, 131, 150	3brtr4d 4140	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
152		simpl3r 1080	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → 0 < 𝐶)
153		nltmnf 10117	. . . . . 6 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
154	35, 153	ax-mp 5	. . . . 5 ⊢ ¬ 0 < -∞
155		breq2 4112	. . . . 5 ⊢ (𝐶 = -∞ → (0 < 𝐶 ↔ 0 < -∞))
156	154, 155	mtbiri 682	. . . 4 ⊢ (𝐶 = -∞ → ¬ 0 < 𝐶)
157	156	adantl 277	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → ¬ 0 < 𝐶)
158	152, 157	pm2.21dd 625	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
159		elxr 10105	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
160	32, 159	sylib 122	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
161	119, 151, 158, 160	mpjao3dan 1344	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < ) ≤ (inf({𝐴, 𝐶}, ℝ^, < ) +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ w3o 1004 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 ≠ wne 2412 {cpr 3689 class class class wbr 4108 (class class class)co 6049 infcinf 7273 ℝcr 8122 0cc0 8123 + caddc 8126 +∞cpnf 8301 -∞cmnf 8302 ℝ^*cxr 8303 < clt 8304 ≤ cle 8305 ℝ⁺crp 9982 +_𝑒 cxad 10099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-sup 7274 df-inf 7275 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-rp 9983 df-xneg 10101 df-xadd 10102 df-seqfrec 10806 df-exp 10897 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677

This theorem is referenced by: bdxmet 15353

W3C validator