Theorem metdscnlem

Description: Lemma for metdscn . (Contributed by Mario Carneiro, 4-Sep-2015)

	Ref	Expression
Hypotheses	metdscn.f	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) )
	metdscn.j	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
	metdscn.c	⊢ 𝐶 = ( dist ‘ ℝ*𝑠 )
	metdscn.k	⊢ 𝐾 = ( MetOpen ‘ 𝐶 )
	metdscnlem.1	⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
	metdscnlem.2	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )
	metdscnlem.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	metdscnlem.4	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
	metdscnlem.5	⊢ ( 𝜑 → 𝑅 ∈ ℝ+ )
	metdscnlem.6	⊢ ( 𝜑 → ( 𝐴 𝐷 𝐵 ) < 𝑅 )
Assertion	metdscnlem	⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) < 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		metdscn.f	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) )
2		metdscn.j	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
3		metdscn.c	⊢ 𝐶 = ( dist ‘ ℝ*𝑠 )
4		metdscn.k	⊢ 𝐾 = ( MetOpen ‘ 𝐶 )
5		metdscnlem.1	⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
6		metdscnlem.2	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )
7		metdscnlem.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
8		metdscnlem.4	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
9		metdscnlem.5	⊢ ( 𝜑 → 𝑅 ∈ ℝ+ )
10		metdscnlem.6	⊢ ( 𝜑 → ( 𝐴 𝐷 𝐵 ) < 𝑅 )
11	1	metdsf	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
12	5 6 11	syl2anc	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
13	12 7	ffvelrnd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
14		eliccxr	⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* )
15	13 14	syl	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* )
16	12 8	ffvelrnd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) )
17		eliccxr	⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝐵 ) ∈ ℝ* )
18	16 17	syl	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ℝ* )
19	18	xnegcld	⊢ ( 𝜑 → -𝑒 ( 𝐹 ‘ 𝐵 ) ∈ ℝ* )
20	15 19	xaddcld	⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ∈ ℝ* )
21		xmetcl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
22	5 7 8 21	syl3anc	⊢ ( 𝜑 → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
23	9	rpxrd	⊢ ( 𝜑 → 𝑅 ∈ ℝ* )
24	1	metdstri	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) )
25	5 6 7 8 24	syl22anc	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) )
26		elxrge0	⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐴 ) ) )
27	26	simprbi	⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝐴 ) )
28	13 27	syl	⊢ ( 𝜑 → 0 ≤ ( 𝐹 ‘ 𝐴 ) )
29		elxrge0	⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐵 ) ) )
30	29	simprbi	⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝐵 ) )
31	16 30	syl	⊢ ( 𝜑 → 0 ≤ ( 𝐹 ‘ 𝐵 ) )
32		ge0nemnf	⊢ ( ( ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ≠ -∞ )
33	18 31 32	syl2anc	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ≠ -∞ )
34		xmetge0	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 0 ≤ ( 𝐴 𝐷 𝐵 ) )
35	5 7 8 34	syl3anc	⊢ ( 𝜑 → 0 ≤ ( 𝐴 𝐷 𝐵 ) )
36		xlesubadd	⊢ ( ( ( ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ∧ ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ∧ ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ) ∧ ( 0 ≤ ( 𝐹 ‘ 𝐴 ) ∧ ( 𝐹 ‘ 𝐵 ) ≠ -∞ ∧ 0 ≤ ( 𝐴 𝐷 𝐵 ) ) ) → ( ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ≤ ( 𝐴 𝐷 𝐵 ) ↔ ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) ) )
37	15 18 22 28 33 35 36	syl33anc	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ≤ ( 𝐴 𝐷 𝐵 ) ↔ ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) ) )
38	25 37	mpbird	⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ≤ ( 𝐴 𝐷 𝐵 ) )
39	20 22 23 38 10	xrlelttrd	⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) < 𝑅 )