xrbdtri - Intuitionistic Logic Explorer

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

Theorem xrbdtri 11038

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 1006	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐴)
3	2	ad3antrrr 483	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐴)
4		simplr 519	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ)
5		simp2r 1008	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐵)
6	5	ad3antrrr 483	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐵)
7		simpllr 523	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ)
8		simp3r 1010	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < 𝐶)
9	8	ad3antrrr 483	. . . . . . 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 9474	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ⁺)
11		bdtri 11004	. . . . . 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 1227	. . . . 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 9630	. . . . . . . 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 3601	. . . . . . 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 6892	. . . . . 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 7788	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
17		xrminrecl 11035	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({(𝐴 + 𝐵), 𝐶}, ℝ^*, < ) = inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ))
18	16, 7, 17	syl2anc 408	. . . . . 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 2170	. . . . 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 11035	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ^*, < ) = inf({𝐴, 𝐶}, ℝ, < ))
21	1, 7, 20	syl2anc 408	. . . . . . 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 11035	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ^*, < ) = inf({𝐵, 𝐶}, ℝ, < ))
23	4, 7, 22	syl2anc 408	. . . . . . 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 5785	. . . . . 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 10995	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐴, 𝐶}, ℝ, < ) ∈ ℝ)
26	1, 7, 25	syl2anc 408	. . . . . . 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 10995	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ, < ) ∈ ℝ)
28	4, 7, 27	syl2anc 408	. . . . . . 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 9630	. . . . . 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 2170	. . . . 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 3955	. . . 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 1009	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
33	32	xaddid1d 9640	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 +_𝑒 0) = 𝐶)
34	32	xrleidd 9580	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ 𝐶)
35		0xr 7805	. . . . . . . . . . 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 9578	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐶)
38		simp2l 1007	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
39		xrlemininf 11033	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
40	36, 38, 32, 39	syl3anc 1216	. . . . . . . . 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 928	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐵, 𝐶}, ℝ^*, < ))
42		xrmincl 11028	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^*)
43	38, 32, 42	syl2anc 408	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({𝐵, 𝐶}, ℝ^, < ) ∈ ℝ^)
44		xle2add 9655	. . . . . . . . 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 1217	. . . . . . . 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 429	. . . . . . 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 3947	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ (𝐶 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))
48	47	ad3antrrr 483	. . . . 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 1005	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
50	49, 38	xaddcld 9660	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
51	50	ad3antrrr 483	. . . . . 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 483	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ∈ ℝ^*)
53		pnfge 9568	. . . . . . . 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 5782	. . . . . . . 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 1034	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
58	57	ad2antrr 479	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ^*)
59		simplr 519	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ∈ ℝ)
60	59	renemnfd 7810	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ≠ -∞)
61		xaddpnf2 9623	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
62	58, 60, 61	syl2anc 408	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → (+∞ +_𝑒 𝐵) = +∞)
63	56, 62	eqtrd 2170	. . . . . . 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 3951	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ (𝐴 +_𝑒 𝐵))
65		xrmineqinf 11031	. . . . . 6 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ (𝐴 +_𝑒 𝐵)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
66	51, 52, 64, 65	syl3anc 1216	. . . . 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 483	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐴 ∈ ℝ^*)
68	54, 55	breqtrrd 3951	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ 𝐴)
69		xrmineqinf 11031	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐴) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐶)
70	67, 52, 68, 69	syl3anc 1216	. . . . . 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 5782	. . . . 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 3955	. . . 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 9600	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
75	49, 2, 74	syl2anc 408	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ≠ -∞)
76	75	ad3antrrr 483	. . . . 5 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 ≠ -∞)
77	73, 76	pm2.21ddne 2389	. . . 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 9556	. . . . . 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 479	. . . 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 1285	. . 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 11033	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶)))
83	36, 49, 32, 82	syl3anc 1216	. . . . . . 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 928	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐴, 𝐶}, ℝ^*, < ))
85		xrmincl 11028	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^*)
86	49, 32, 85	syl2anc 408	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → inf({𝐴, 𝐶}, ℝ^, < ) ∈ ℝ^)
87		xle2add 9655	. . . . . . 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 1217	. . . . . 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 429	. . . . 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 479	. . . 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 479	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
92	32	ad2antrr 479	. . . . . 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 5783	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
96		xaddpnf1 9622	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
97	49, 75, 96	syl2anc 408	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 +_𝑒 +∞) = +∞)
98	97	ad2antrr 479	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 +∞) = +∞)
99	95, 98	eqtrd 2170	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
100	93, 99	breqtrrd 3951	. . . . . 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 1216	. . . . 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 9639	. . . . . 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 2173	. . . 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 3951	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ 𝐵)
107		xrmineqinf 11031	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
108	105, 92, 106, 107	syl3anc 1216	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
109	108	oveq2d 5783	. . . 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 3955	. . 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 479	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 0 ≤ 𝐵)
114		ge0nemnf 9600	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) → 𝐵 ≠ -∞)
115	112, 113, 114	syl2anc 408	. . . 4 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → 𝐵 ≠ -∞)
116	111, 115	pm2.21ddne 2389	. . 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 9556	. . . 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 1285	. 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 9580	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
122		prcom 3594	. . . . 5 ⊢ {𝐶, (𝐴 +_𝑒 𝐵)} = {(𝐴 +_𝑒 𝐵), 𝐶}
123	122	infeq1i 6893	. . . 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 9568	. . . . . . 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 3951	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ 𝐶)
129		xrmineqinf 11031	. . . . 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 1216	. . . 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	syl5eqr 2184	. . 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 3594	. . . . . 6 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
133	132	infeq1i 6893	. . . . 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 9568	. . . . . . . 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 3951	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ≤ 𝐶)
138		xrmineqinf 11031	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
139	124, 134, 137, 138	syl3anc 1216	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
140	133, 139	syl5eqr 2184	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐴)
141		prcom 3594	. . . . . 6 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
142	141	infeq1i 6893	. . . . 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 9568	. . . . . . . 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 3951	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ≤ 𝐶)
147		xrmineqinf 11031	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
148	124, 143, 146, 147	syl3anc 1216	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
149	142, 148	syl5eqr 2184	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐵)
150	140, 149	oveq12d 5785	. . 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 3955	. 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 1037	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → 0 < 𝐶)
153		nltmnf 9567	. . . . . 6 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
154	35, 153	ax-mp 5	. . . . 5 ⊢ ¬ 0 < -∞
155		breq2 3928	. . . . 5 ⊢ (𝐶 = -∞ → (0 < 𝐶 ↔ 0 < -∞))
156	154, 155	mtbiri 664	. . . 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 609	. 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 9556	. . 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 1285	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 961 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 {cpr 3523 class class class wbr 3924 (class class class)co 5767 infcinf 6863 ℝcr 7612 0cc0 7613 + caddc 7616 +∞cpnf 7790 -∞cmnf 7791 ℝ^*cxr 7792 < clt 7793 ≤ cle 7794 ℝ⁺crp 9434 +_𝑒 cxad 9550

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-sup 6864 df-inf 6865 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-rp 9435 df-xneg 9552 df-xadd 9553 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764

This theorem is referenced by: bdxmet 12659

W3C validator