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Theorem xrbdtri 11753

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 1027	. . . . . . 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 528	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ)
5		simp2r 1029	. . . . . . 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 534	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ)
8		simp3r 1031	. . . . . . . 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 9857	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ⁺)
11		bdtri 11717	. . . . . 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 1263	. . . . 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 10018	. . . . . . . 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 3729	. . . . . . 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 7147	. . . . . 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 8144	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
17		xrminrecl 11750	. . . . . . 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 2242	. . . . 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 11750	. . . . . . . 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 11750	. . . . . . . 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 5992	. . . . . 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 11708	. . . . . . . 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 11708	. . . . . . . 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 10018	. . . . . 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 2242	. . . . 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 4094	. . . 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 1030	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
33	32	xaddid1d 10028	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 +_𝑒 0) = 𝐶)
34	32	xrleidd 9965	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ 𝐶)
35		0xr 8161	. . . . . . . . . . 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 9963	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐶)
38		simp2l 1028	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
39		xrlemininf 11748	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
40	36, 38, 32, 39	syl3anc 1252	. . . . . . . . 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 949	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐵, 𝐶}, ℝ^*, < ))
42		xrmincl 11743	. . . . . . . . . 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 10043	. . . . . . . . 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 1253	. . . . . . . 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 4086	. . . . . 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 1026	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
50	49, 38	xaddcld 10048	. . . . . . 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 9953	. . . . . . . 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 5989	. . . . . . . 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 1055	. . . . . . . . . 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 528	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ)
60	59	renemnfd 8166	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ≠ -∞)
61		xaddpnf2 10011	. . . . . . . . 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 2242	. . . . . . 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 4090	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ (𝐴 +_𝑒 𝐵))
65		xrmineqinf 11746	. . . . . 6 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ (𝐴 +_𝑒 𝐵)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
66	51, 52, 64, 65	syl3anc 1252	. . . . 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 4090	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ 𝐴)
69		xrmineqinf 11746	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐴) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐶)
70	67, 52, 68, 69	syl3anc 1252	. . . . . 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 5989	. . . . 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 4094	. . . 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 9988	. . . . . . 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 2463	. . . 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 9940	. . . . . 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 1322	. . 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 11748	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶)))
83	36, 49, 32, 82	syl3anc 1252	. . . . . . 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 949	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐴, 𝐶}, ℝ^*, < ))
85		xrmincl 11743	. . . . . . . 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 10043	. . . . . . 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 1253	. . . . . 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 5990	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
96		xaddpnf1 10010	. . . . . . . . . 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 2242	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
100	93, 99	breqtrrd 4090	. . . . . 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 1252	. . . . 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 10027	. . . . . 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 2245	. . . 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 4090	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ 𝐵)
107		xrmineqinf 11746	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
108	105, 92, 106, 107	syl3anc 1252	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
109	108	oveq2d 5990	. . . 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 4094	. . 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 9988	. . . . 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 2463	. . 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 9940	. . . 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 1322	. 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 9965	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
122		prcom 3722	. . . . 5 ⊢ {𝐶, (𝐴 +_𝑒 𝐵)} = {(𝐴 +_𝑒 𝐵), 𝐶}
123	122	infeq1i 7148	. . . 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 9953	. . . . . . 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 4090	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ 𝐶)
129		xrmineqinf 11746	. . . . 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 1252	. . . 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 2256	. . 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 3722	. . . . . 6 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
133	132	infeq1i 7148	. . . . 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 9953	. . . . . . . 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 4090	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ≤ 𝐶)
138		xrmineqinf 11746	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
139	124, 134, 137, 138	syl3anc 1252	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
140	133, 139	eqtr3id 2256	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐴)
141		prcom 3722	. . . . . 6 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
142	141	infeq1i 7148	. . . . 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 9953	. . . . . . . 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 4090	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ≤ 𝐶)
147		xrmineqinf 11746	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
148	124, 143, 146, 147	syl3anc 1252	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
149	142, 148	eqtr3id 2256	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐵)
150	140, 149	oveq12d 5992	. . 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 4094	. 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 1058	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → 0 < 𝐶)
153		nltmnf 9952	. . . . . 6 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
154	35, 153	ax-mp 5	. . . . 5 ⊢ ¬ 0 < -∞
155		breq2 4066	. . . . 5 ⊢ (𝐶 = -∞ → (0 < 𝐶 ↔ 0 < -∞))
156	154, 155	mtbiri 679	. . . 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 623	. 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 9940	. . 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 1322	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 982 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ≠ wne 2380 {cpr 3647 class class class wbr 4062 (class class class)co 5974 infcinf 7118 ℝcr 7966 0cc0 7967 + caddc 7970 +∞cpnf 8146 -∞cmnf 8147 ℝ^*cxr 8148 < clt 8149 ≤ cle 8150 ℝ⁺crp 9817 +_𝑒 cxad 9934

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-frec 6507 df-sup 7119 df-inf 7120 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-rp 9818 df-xneg 9936 df-xadd 9937 df-seqfrec 10637 df-exp 10728 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476

This theorem is referenced by: bdxmet 15140

W3C validator