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Theorem xrbdtri 11284

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 1022	. . . . . . 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 1024	. . . . . . 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 1026	. . . . . . . 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 9693	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ⁺)
11		bdtri 11248	. . . . . 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 1249	. . . . 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 9854	. . . . . . . 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 3676	. . . . . . 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 7011	. . . . . 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 7987	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
17		xrminrecl 11281	. . . . . . 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 2210	. . . . 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 11281	. . . . . . . 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 11281	. . . . . . . 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 5893	. . . . . 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 11239	. . . . . . . 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 11239	. . . . . . . 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 9854	. . . . . 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 2210	. . . . 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 4036	. . . 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 1025	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
33	32	xaddid1d 9864	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 +_𝑒 0) = 𝐶)
34	32	xrleidd 9801	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ≤ 𝐶)
35		0xr 8004	. . . . . . . . . . 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 9799	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ 𝐶)
38		simp2l 1023	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
39		xrlemininf 11279	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
40	36, 38, 32, 39	syl3anc 1238	. . . . . . . . 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 944	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐵, 𝐶}, ℝ^*, < ))
42		xrmincl 11274	. . . . . . . . . 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 9879	. . . . . . . . 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 1239	. . . . . . . 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 4028	. . . . . 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 1021	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
50	49, 38	xaddcld 9884	. . . . . . 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 9789	. . . . . . . 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 5890	. . . . . . . 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 1050	. . . . . . . . . 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 8009	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐵 ≠ -∞)
61		xaddpnf2 9847	. . . . . . . . 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 2210	. . . . . . 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 4032	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ (𝐴 +_𝑒 𝐵))
65		xrmineqinf 11277	. . . . . 6 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ (𝐴 +_𝑒 𝐵)) → inf({(𝐴 +_𝑒 𝐵), 𝐶}, ℝ^*, < ) = 𝐶)
66	51, 52, 64, 65	syl3anc 1238	. . . . 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 4032	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = +∞) → 𝐶 ≤ 𝐴)
69		xrmineqinf 11277	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐴) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐶)
70	67, 52, 68, 69	syl3anc 1238	. . . . . 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 5890	. . . . 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 4036	. . . 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 9824	. . . . . . 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 2430	. . . 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 9776	. . . . . 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 1307	. . 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 11279	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (0 ≤ inf({𝐴, 𝐶}, ℝ^, < ) ↔ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶)))
83	36, 49, 32, 82	syl3anc 1238	. . . . . . 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 944	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 ≤ inf({𝐴, 𝐶}, ℝ^*, < ))
85		xrmincl 11274	. . . . . . . 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 9879	. . . . . . 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 1239	. . . . . 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 5891	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
96		xaddpnf1 9846	. . . . . . . . . 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 2210	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
100	93, 99	breqtrrd 4032	. . . . . 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 1238	. . . . 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 9863	. . . . . 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 2213	. . . 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 4032	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → 𝐶 ≤ 𝐵)
107		xrmineqinf 11277	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
108	105, 92, 106, 107	syl3anc 1238	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐶)
109	108	oveq2d 5891	. . . 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 4036	. . 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 9824	. . . . 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 2430	. . 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 9776	. . . 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 1307	. 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 9801	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
122		prcom 3669	. . . . 5 ⊢ {𝐶, (𝐴 +_𝑒 𝐵)} = {(𝐴 +_𝑒 𝐵), 𝐶}
123	122	infeq1i 7012	. . . 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 9789	. . . . . . 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 4032	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 +_𝑒 𝐵) ≤ 𝐶)
129		xrmineqinf 11277	. . . . 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 1238	. . . 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 2224	. . 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 3669	. . . . . 6 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
133	132	infeq1i 7012	. . . . 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 9789	. . . . . . . 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 4032	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐴 ≤ 𝐶)
138		xrmineqinf 11277	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
139	124, 134, 137, 138	syl3anc 1238	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐴}, ℝ^*, < ) = 𝐴)
140	133, 139	eqtr3id 2224	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐴, 𝐶}, ℝ^*, < ) = 𝐴)
141		prcom 3669	. . . . . 6 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
142	141	infeq1i 7012	. . . . 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 9789	. . . . . . . 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 4032	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → 𝐵 ≤ 𝐶)
147		xrmineqinf 11277	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
148	124, 143, 146, 147	syl3anc 1238	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐶, 𝐵}, ℝ^*, < ) = 𝐵)
149	142, 148	eqtr3id 2224	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → inf({𝐵, 𝐶}, ℝ^*, < ) = 𝐵)
150	140, 149	oveq12d 5893	. . 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 4036	. 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 1053	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = -∞) → 0 < 𝐶)
153		nltmnf 9788	. . . . . 6 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
154	35, 153	ax-mp 5	. . . . 5 ⊢ ¬ 0 < -∞
155		breq2 4008	. . . . 5 ⊢ (𝐶 = -∞ → (0 < 𝐶 ↔ 0 < -∞))
156	154, 155	mtbiri 675	. . . 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 620	. 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 9776	. . 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 1307	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 977 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 {cpr 3594 class class class wbr 4004 (class class class)co 5875 infcinf 6982 ℝcr 7810 0cc0 7811 + caddc 7814 +∞cpnf 7989 -∞cmnf 7990 ℝ^*cxr 7991 < clt 7992 ≤ cle 7993 ℝ⁺crp 9653 +_𝑒 cxad 9770

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-rp 9654 df-xneg 9772 df-xadd 9773 df-seqfrec 10446 df-exp 10520 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008

This theorem is referenced by: bdxmet 14004

W3C validator