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Theorem stdbdxmet 24244

Description: The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)

**Hypothesis**
Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))

**Assertion**
Ref	Expression
stdbdxmet	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))

Distinct variable groups: 𝑥,𝑦,𝐶 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦)

Proof of Theorem stdbdxmet Dummy variables 𝑎 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1134	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
2		xmetcl 24057	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ ℝ^*)
3		xmetge0 24070	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑦))
4		elxrge0 13438	. . . . . . 7 ⊢ ((𝑥𝐶𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐶𝑦) ∈ ℝ^* ∧ 0 ≤ (𝑥𝐶𝑦)))
5	2, 3, 4	sylanbrc 581	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
6	5	3expb 1118	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
7	1, 6	sylan 578	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
8		xmetf 24055	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
9	8	3ad2ant1 1131	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
10	9	ffnd 6717	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 Fn (𝑋 × 𝑋))
11		fnov 7542	. . . . 5 ⊢ (𝐶 Fn (𝑋 × 𝑋) ↔ 𝐶 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐶𝑦)))
12	10, 11	sylib 217	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐶𝑦)))
13		eqidd 2731	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)) = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)))
14		breq1 5150	. . . . 5 ⊢ (𝑧 = (𝑥𝐶𝑦) → (𝑧 ≤ 𝑅 ↔ (𝑥𝐶𝑦) ≤ 𝑅))
15		id 22	. . . . 5 ⊢ (𝑧 = (𝑥𝐶𝑦) → 𝑧 = (𝑥𝐶𝑦))
16	14, 15	ifbieq1d 4551	. . . 4 ⊢ (𝑧 = (𝑥𝐶𝑦) → if(𝑧 ≤ 𝑅, 𝑧, 𝑅) = if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
17	7, 12, 13, 16	fmpoco 8083	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)) ∘ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅)))
18		stdbdmet.1	. . 3 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
19	17, 18	eqtr4di 2788	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)) ∘ 𝐶) = 𝐷)
20		eliccxr 13416	. . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ^*)
21		simp2 1135	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝑅 ∈ ℝ^*)
22		ifcl 4572	. . . . 5 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → if(𝑧 ≤ 𝑅, 𝑧, 𝑅) ∈ ℝ^)
23	20, 21, 22	syl2anr 595	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑧 ∈ (0[,]+∞)) → if(𝑧 ≤ 𝑅, 𝑧, 𝑅) ∈ ℝ^*)
24	23	fmpttd 7115	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)):(0[,]+∞)⟶ℝ^*)
25		id 22	. . . . . 6 ⊢ (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ (0[,]+∞))
26		vex 3476	. . . . . . 7 ⊢ 𝑎 ∈ V
27		ifexg 4576	. . . . . . 7 ⊢ ((𝑎 ∈ V ∧ 𝑅 ∈ ℝ^*) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ∈ V)
28	26, 21, 27	sylancr 585	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ∈ V)
29		breq1 5150	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (𝑧 ≤ 𝑅 ↔ 𝑎 ≤ 𝑅))
30		id 22	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → 𝑧 = 𝑎)
31	29, 30	ifbieq1d 4551	. . . . . . 7 ⊢ (𝑧 = 𝑎 → if(𝑧 ≤ 𝑅, 𝑧, 𝑅) = if(𝑎 ≤ 𝑅, 𝑎, 𝑅))
32		eqid 2730	. . . . . . 7 ⊢ (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)) = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))
33	31, 32	fvmptg 6995	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ∈ V) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) = if(𝑎 ≤ 𝑅, 𝑎, 𝑅))
34	25, 28, 33	syl2anr 595	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) = if(𝑎 ≤ 𝑅, 𝑎, 𝑅))
35	34	eqeq1d 2732	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) = 0 ↔ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) = 0))
36		eqeq1 2734	. . . . . 6 ⊢ (𝑎 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → (𝑎 = 0 ↔ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) = 0))
37	36	bibi1d 342	. . . . 5 ⊢ (𝑎 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → ((𝑎 = 0 ↔ 𝑎 = 0) ↔ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) = 0 ↔ 𝑎 = 0)))
38		eqeq1 2734	. . . . . 6 ⊢ (𝑅 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → (𝑅 = 0 ↔ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) = 0))
39	38	bibi1d 342	. . . . 5 ⊢ (𝑅 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → ((𝑅 = 0 ↔ 𝑎 = 0) ↔ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) = 0 ↔ 𝑎 = 0)))
40		biidd 261	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) ∧ 𝑎 ≤ 𝑅) → (𝑎 = 0 ↔ 𝑎 = 0))
41		simp3 1136	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 0 < 𝑅)
42	41	gt0ne0d 11782	. . . . . . . 8 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝑅 ≠ 0)
43	42	neneqd 2943	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ¬ 𝑅 = 0)
44	43	ad2antrr 722	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) ∧ ¬ 𝑎 ≤ 𝑅) → ¬ 𝑅 = 0)
45		0xr 11265	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
46		xrltle 13132	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 < 𝑅 → 0 ≤ 𝑅))
47	45, 21, 46	sylancr 585	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (0 < 𝑅 → 0 ≤ 𝑅))
48	41, 47	mpd 15	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 0 ≤ 𝑅)
49	48	adantr 479	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ 𝑅)
50		breq1 5150	. . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎 ≤ 𝑅 ↔ 0 ≤ 𝑅))
51	49, 50	syl5ibrcom 246	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 = 0 → 𝑎 ≤ 𝑅))
52	51	con3dimp 407	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) ∧ ¬ 𝑎 ≤ 𝑅) → ¬ 𝑎 = 0)
53	44, 52	2falsed 375	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) ∧ ¬ 𝑎 ≤ 𝑅) → (𝑅 = 0 ↔ 𝑎 = 0))
54	37, 39, 40, 53	ifbothda 4565	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) = 0 ↔ 𝑎 = 0))
55	35, 54	bitrd 278	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) = 0 ↔ 𝑎 = 0))
56		eliccxr 13416	. . . . . . . . 9 ⊢ (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ ℝ^*)
57	56	ad2antrl 724	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑎 ∈ ℝ^*)
58	21	adantr 479	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑅 ∈ ℝ^*)
59		xrmin1 13160	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑎)
60	57, 58, 59	syl2anc 582	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑎)
61	57, 58	ifcld 4573	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ∈ ℝ^*)
62		eliccxr 13416	. . . . . . . . 9 ⊢ (𝑏 ∈ (0[,]+∞) → 𝑏 ∈ ℝ^*)
63	62	ad2antll 725	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ ℝ^*)
64		xrletr 13141	. . . . . . . 8 ⊢ ((if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → ((if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑎 ∧ 𝑎 ≤ 𝑏) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑏))
65	61, 57, 63, 64	syl3anc 1369	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑎 ∧ 𝑎 ≤ 𝑏) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑏))
66	60, 65	mpand 691	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑏))
67		xrmin2 13161	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑅)
68	57, 58, 67	syl2anc 582	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑅)
69	66, 68	jctird 525	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑏 ∧ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑅)))
70		xrlemin 13167	. . . . . 6 ⊢ ((if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ↔ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑏 ∧ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑅)))
71	61, 63, 58, 70	syl3anc 1369	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ↔ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑏 ∧ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ 𝑅)))
72	69, 71	sylibrd 258	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
73	34	adantrr 713	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) = if(𝑎 ≤ 𝑅, 𝑎, 𝑅))
74		simpr 483	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → 𝑏 ∈ (0[,]+∞))
75		vex 3476	. . . . . . 7 ⊢ 𝑏 ∈ V
76		ifexg 4576	. . . . . . 7 ⊢ ((𝑏 ∈ V ∧ 𝑅 ∈ ℝ^*) → if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ∈ V)
77	75, 21, 76	sylancr 585	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ∈ V)
78		breq1 5150	. . . . . . . 8 ⊢ (𝑧 = 𝑏 → (𝑧 ≤ 𝑅 ↔ 𝑏 ≤ 𝑅))
79		id 22	. . . . . . . 8 ⊢ (𝑧 = 𝑏 → 𝑧 = 𝑏)
80	78, 79	ifbieq1d 4551	. . . . . . 7 ⊢ (𝑧 = 𝑏 → if(𝑧 ≤ 𝑅, 𝑧, 𝑅) = if(𝑏 ≤ 𝑅, 𝑏, 𝑅))
81	80, 32	fvmptg 6995	. . . . . 6 ⊢ ((𝑏 ∈ (0[,]+∞) ∧ if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ∈ V) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑏) = if(𝑏 ≤ 𝑅, 𝑏, 𝑅))
82	74, 77, 81	syl2anr 595	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑏) = if(𝑏 ≤ 𝑅, 𝑏, 𝑅))
83	73, 82	breq12d 5160	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) ≤ ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑏) ↔ if(𝑎 ≤ 𝑅, 𝑎, 𝑅) ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
84	72, 83	sylibrd 258	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) ≤ ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑏)))
85	57, 63	xaddcld 13284	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ ℝ^*)
86		xrmin1 13160	. . . . . . 7 ⊢ (((𝑎 +_𝑒 𝑏) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑏))
87	85, 58, 86	syl2anc 582	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑏))
88	85, 58	ifcld 4573	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ∈ ℝ^*)
89	57, 58	xaddcld 13284	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑅) ∈ ℝ^*)
90		xrmin2 13161	. . . . . . . 8 ⊢ (((𝑎 +_𝑒 𝑏) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ 𝑅)
91	85, 58, 90	syl2anc 582	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ 𝑅)
92		xaddlid 13225	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ^* → (0 +_𝑒 𝑅) = 𝑅)
93	58, 92	syl 17	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (0 +_𝑒 𝑅) = 𝑅)
94	45	a1i 11	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ∈ ℝ^*)
95		elxrge0 13438	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (0[,]+∞) ↔ (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
96	95	simprbi 495	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0[,]+∞) → 0 ≤ 𝑎)
97	96	ad2antrl 724	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑎)
98		xleadd1a 13236	. . . . . . . . 9 ⊢ (((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ 0 ≤ 𝑎) → (0 +_𝑒 𝑅) ≤ (𝑎 +_𝑒 𝑅))
99	94, 57, 58, 97, 98	syl31anc 1371	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (0 +_𝑒 𝑅) ≤ (𝑎 +_𝑒 𝑅))
100	93, 99	eqbrtrrd 5171	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑅 ≤ (𝑎 +_𝑒 𝑅))
101	88, 58, 89, 91, 100	xrletrd 13145	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑅))
102		oveq2 7419	. . . . . . . 8 ⊢ (𝑏 = if(𝑏 ≤ 𝑅, 𝑏, 𝑅) → (𝑎 +_𝑒 𝑏) = (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
103	102	breq2d 5159	. . . . . . 7 ⊢ (𝑏 = if(𝑏 ≤ 𝑅, 𝑏, 𝑅) → (if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑏) ↔ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅))))
104		oveq2 7419	. . . . . . . 8 ⊢ (𝑅 = if(𝑏 ≤ 𝑅, 𝑏, 𝑅) → (𝑎 +_𝑒 𝑅) = (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
105	104	breq2d 5159	. . . . . . 7 ⊢ (𝑅 = if(𝑏 ≤ 𝑅, 𝑏, 𝑅) → (if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑅) ↔ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅))))
106	103, 105	ifboth 4566	. . . . . 6 ⊢ ((if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑏) ∧ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 𝑅)) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
107	87, 101, 106	syl2anc 582	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
108	63, 58	ifcld 4573	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ∈ ℝ^*)
109	58, 108	xaddcld 13284	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) ∈ ℝ^*)
110	58	xaddridd 13226	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑅 +_𝑒 0) = 𝑅)
111		elxrge0 13438	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (0[,]+∞) ↔ (𝑏 ∈ ℝ^* ∧ 0 ≤ 𝑏))
112	111	simprbi 495	. . . . . . . . . 10 ⊢ (𝑏 ∈ (0[,]+∞) → 0 ≤ 𝑏)
113	112	ad2antll 725	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑏)
114	48	adantr 479	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑅)
115		breq2 5151	. . . . . . . . . 10 ⊢ (𝑏 = if(𝑏 ≤ 𝑅, 𝑏, 𝑅) → (0 ≤ 𝑏 ↔ 0 ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
116		breq2 5151	. . . . . . . . . 10 ⊢ (𝑅 = if(𝑏 ≤ 𝑅, 𝑏, 𝑅) → (0 ≤ 𝑅 ↔ 0 ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
117	115, 116	ifboth 4566	. . . . . . . . 9 ⊢ ((0 ≤ 𝑏 ∧ 0 ≤ 𝑅) → 0 ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅))
118	113, 114, 117	syl2anc 582	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅))
119		xleadd2a 13237	. . . . . . . 8 ⊢ (((0 ∈ ℝ^* ∧ if(𝑏 ≤ 𝑅, 𝑏, 𝑅) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ 0 ≤ if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) → (𝑅 +_𝑒 0) ≤ (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
120	94, 108, 58, 118, 119	syl31anc 1371	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑅 +_𝑒 0) ≤ (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
121	110, 120	eqbrtrrd 5171	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑅 ≤ (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
122	88, 58, 109, 91, 121	xrletrd 13145	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
123		oveq1 7418	. . . . . . 7 ⊢ (𝑎 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) = (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
124	123	breq2d 5159	. . . . . 6 ⊢ (𝑎 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → (if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) ↔ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅))))
125		oveq1 7418	. . . . . . 7 ⊢ (𝑅 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) = (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
126	125	breq2d 5159	. . . . . 6 ⊢ (𝑅 = if(𝑎 ≤ 𝑅, 𝑎, 𝑅) → (if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) ↔ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅))))
127	124, 126	ifboth 4566	. . . . 5 ⊢ ((if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑎 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)) ∧ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (𝑅 +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
128	107, 122, 127	syl2anc 582	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ≤ (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
129		ge0xaddcl 13443	. . . . 5 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
130		ovex 7444	. . . . . 6 ⊢ (𝑎 +_𝑒 𝑏) ∈ V
131		ifexg 4576	. . . . . 6 ⊢ (((𝑎 +_𝑒 𝑏) ∈ V ∧ 𝑅 ∈ ℝ^*) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ∈ V)
132	130, 21, 131	sylancr 585	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ∈ V)
133		breq1 5150	. . . . . . 7 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → (𝑧 ≤ 𝑅 ↔ (𝑎 +_𝑒 𝑏) ≤ 𝑅))
134		id 22	. . . . . . 7 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → 𝑧 = (𝑎 +_𝑒 𝑏))
135	133, 134	ifbieq1d 4551	. . . . . 6 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → if(𝑧 ≤ 𝑅, 𝑧, 𝑅) = if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅))
136	135, 32	fvmptg 6995	. . . . 5 ⊢ (((𝑎 +_𝑒 𝑏) ∈ (0[,]+∞) ∧ if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅) ∈ V) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘(𝑎 +_𝑒 𝑏)) = if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅))
137	129, 132, 136	syl2anr 595	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘(𝑎 +_𝑒 𝑏)) = if((𝑎 +_𝑒 𝑏) ≤ 𝑅, (𝑎 +_𝑒 𝑏), 𝑅))
138	73, 82	oveq12d 7429	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) +_𝑒 ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑏)) = (if(𝑎 ≤ 𝑅, 𝑎, 𝑅) +_𝑒 if(𝑏 ≤ 𝑅, 𝑏, 𝑅)))
139	128, 137, 138	3brtr4d 5179	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘(𝑎 +_𝑒 𝑏)) ≤ (((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑎) +_𝑒 ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅))‘𝑏)))
140	1, 24, 55, 84, 139	comet 24242	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ if(𝑧 ≤ 𝑅, 𝑧, 𝑅)) ∘ 𝐶) ∈ (∞Met‘𝑋))
141	19, 140	eqeltrrd 2832	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ∘ ccom 5679 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 0cc0 11112 +∞cpnf 11249 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 +_𝑒 cxad 13094 [,]cicc 13331 ∞Metcxmet 21129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-icc 13335 df-xmet 21137

This theorem is referenced by: stdbdmet 24245 stdbdbl 24246 stdbdmopn 24247

W3C validator