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Theorem xrbdtri 11239

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 109	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
2		simp1r 1017	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐴)
3	2	ad3antrrr 489	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐴)
4		simplr 525	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ)
5		simp2r 1019	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐵)
6	5	ad3antrrr 489	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐵)
7		simpllr 529	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ)
8		simp3r 1021	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < 𝐶)
9	8	ad3antrrr 489	. . . . . . 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 9650	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ⁺)
11		bdtri 11203	. . . . . 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 1244	. . . . 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 9811	. . . . . . . 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 3666	. . . . . . 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 6989	. . . . . 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 7949	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
17		xrminrecl 11236	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({(𝐴 + 𝐵), 𝐶}, ℝ^*, < ) = inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ))
18	16, 7, 17	syl2anc 409	. . . . . 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 2203	. . . . 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 11236	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ^*, < ) = inf({𝐴, 𝐶}, ℝ, < ))
21	1, 7, 20	syl2anc 409	. . . . . . 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 11236	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ^*, < ) = inf({𝐵, 𝐶}, ℝ, < ))
23	4, 7, 22	syl2anc 409	. . . . . . 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 5871	. . . . . 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 11194	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ, < ) ∈ ℝ)
26	1, 7, 25	syl2anc 409	. . . . . . 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 11194	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ, < ) ∈ ℝ)
28	4, 7, 27	syl2anc 409	. . . . . . 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 9811	. . . . . 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 2203	. . . . 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 4021	. . . 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 1020	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
33	32	xaddid1d 9821	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 +_𝑒 0) = 𝐶)
34	32	xrleidd 9758	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ 𝐶)
35		0xr 7966	. . . . . . . . . . 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 9756	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐶)
38		simp2l 1018	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
39		xrlemininf 11234	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
40	36, 38, 32, 39	syl3anc 1233	. . . . . . . . 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 939	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐵, 𝐶}, ℝ^*, < ))
42		xrmincl 11229	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
43	38, 32, 42	syl2anc 409	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^)
44		xle2add 9836	. . . . . . . . 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 1234	. . . . . . . 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 431	. . . . . . 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 4013	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
48	47	ad3antrrr 489	. . . . 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 1016	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
50	49, 38	xaddcld 9841	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
51	50	ad3antrrr 489	. . . . . 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 489	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ∈ ℝ^*)
53		pnfge 9746	. . . . . . . 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 109	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐴 = +∞)
56	55	oveq1d 5868	. . . . . . . 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 1045	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
58	57	ad2antrr 485	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ^*)
59		simplr 525	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ)
60	59	renemnfd 7971	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ≠ -∞)
61		xaddpnf2 9804	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
62	58, 60, 61	syl2anc 409	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (+∞ +_𝑒 𝐵) = +∞)
63	56, 62	eqtrd 2203	. . . . . . 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 4017	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ (𝐴 +_𝑒 𝐵))
65		xrmineqinf 11232	. . . . . 6 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ (𝐴 +_𝑒 𝐵)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
66	51, 52, 64, 65	syl3anc 1233	. . . . 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 489	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐴 ∈ ℝ^*)
68	54, 55	breqtrrd 4017	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ 𝐴)
69		xrmineqinf 11232	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐴) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐶)
70	67, 52, 68, 69	syl3anc 1233	. . . . . 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 5868	. . . . 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 4021	. . . 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 109	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 = -∞)
74		ge0nemnf 9781	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
75	49, 2, 74	syl2anc 409	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ≠ -∞)
76	75	ad3antrrr 489	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 ≠ -∞)
77	73, 76	pm2.21ddne 2423	. . . 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 9733	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
79	49, 78	sylib 121	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
80	79	ad2antrr 485	. . . 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 1302	. . 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 11234	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶)))
83	36, 49, 32, 82	syl3anc 1233	. . . . . . 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 939	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐴, 𝐶}, ℝ^*, < ))
85		xrmincl 11229	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
86	49, 32, 85	syl2anc 409	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^)
87		xle2add 9836	. . . . . . 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 1234	. . . . . 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 431	. . . . 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 485	. . . 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 485	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
92	32	ad2antrr 485	. . . . . 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 109	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐵 = +∞)
95	94	oveq2d 5869	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
96		xaddpnf1 9803	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
97	49, 75, 96	syl2anc 409	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 +_𝑒 +∞) = +∞)
98	97	ad2antrr 485	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 +∞) = +∞)
99	95, 98	eqtrd 2203	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
100	93, 99	breqtrrd 4017	. . . . . 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 1233	. . . . 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 9820	. . . . . 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 2206	. . . 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 274	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐵 ∈ ℝ^*)
106	93, 94	breqtrrd 4017	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ 𝐵)
107		xrmineqinf 11232	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
108	105, 92, 106, 107	syl3anc 1233	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
109	108	oveq2d 5869	. . . 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 4021	. . 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 109	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 = -∞)
112	57	adantr 274	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 ∈ ℝ^*)
113	5	ad2antrr 485	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 0 ≤ 𝐵)
114		ge0nemnf 9781	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) → 𝐵 ≠ -∞)
115	112, 113, 114	syl2anc 409	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 ≠ -∞)
116	111, 115	pm2.21ddne 2423	. . 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 9733	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
118	57, 117	sylib 121	. . 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 1302	. 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 274	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
121	120	xrleidd 9758	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
122		prcom 3659	. . . . 5 ⊢ {𝐶, (𝐴 +_𝑒 𝐵)} = {(𝐴 +_𝑒 𝐵), 𝐶}
123	122	infeq1i 6990	. . . 4 ⊢ inf({𝐶, (𝐴 +_𝑒 𝐵)}, ℝ^, < ) = inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^, < )
124	32	adantr 274	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐶 ∈ ℝ^*)
125		pnfge 9746	. . . . . . 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 109	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐶 = +∞)
128	126, 127	breqtrrd 4017	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ 𝐶)
129		xrmineqinf 11232	. . . . 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 1233	. . . 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 2217	. . 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 3659	. . . . . 6 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
133	132	infeq1i 6990	. . . . 5 ⊢ inf({𝐶, 𝐴}, ℝ^, < ) = inf({𝐴, 𝐶}, ℝ^, < )
134	49	adantr 274	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ∈ ℝ^*)
135		pnfge 9746	. . . . . . . 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 4017	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ≤ 𝐶)
138		xrmineqinf 11232	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
139	124, 134, 137, 138	syl3anc 1233	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
140	133, 139	eqtr3id 2217	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐴)
141		prcom 3659	. . . . . 6 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
142	141	infeq1i 6990	. . . . 5 ⊢ inf({𝐶, 𝐵}, ℝ^, < ) = inf({𝐵, 𝐶}, ℝ^, < )
143	38	adantr 274	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ∈ ℝ^*)
144		pnfge 9746	. . . . . . . 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 4017	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ≤ 𝐶)
147		xrmineqinf 11232	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
148	124, 143, 146, 147	syl3anc 1233	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
149	142, 148	eqtr3id 2217	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐵)
150	140, 149	oveq12d 5871	. . 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 4021	. 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 1048	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → 0 < 𝐶)
153		nltmnf 9745	. . . . . 6 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
154	35, 153	ax-mp 5	. . . . 5 ⊢ ¬ 0 < -∞
155		breq2 3993	. . . . 5 ⊢ (𝐶 = -∞ → (0 < 𝐶 ↔ 0 < -∞))
156	154, 155	mtbiri 670	. . . 4 ⊢ (𝐶 = -∞ → ¬ 0 < 𝐶)
157	156	adantl 275	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → ¬ 0 < 𝐶)
158	152, 157	pm2.21dd 615	. 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 9733	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
160	32, 159	sylib 121	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
161	119, 151, 158, 160	mpjao3dan 1302	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 103 ↔ wb 104 ∨ w3o 972 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 {cpr 3584 class class class wbr 3989 (class class class)co 5853 infcinf 6960 ℝcr 7773 0cc0 7774 + caddc 7777 +∞cpnf 7951 -∞cmnf 7952 ℝ^*cxr 7953 < clt 7954 ≤ cle 7955 ℝ⁺crp 9610 +_𝑒 cxad 9727

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-xneg 9729 df-xadd 9730 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: bdxmet 13295

W3C validator