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Theorem xrsxmet 23416

Description: The metric on the extended reals is a proper extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

**Hypothesis**
Ref	Expression
xrsxmet.1	⊢ 𝐷 = (dist‘ℝ^*_𝑠)

**Assertion**
Ref	Expression
xrsxmet	⊢ 𝐷 ∈ (∞Met‘ℝ^*)

Proof of Theorem xrsxmet Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrex 12385	. . . 4 ⊢ ℝ^* ∈ V
2	1	a1i 11	. . 3 ⊢ (⊤ → ℝ^* ∈ V)
3		id 22	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ∈ ℝ^*)
4		xnegcl 12605	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* → -_𝑒𝑥 ∈ ℝ^*)
5		xaddcl 12631	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑥 ∈ ℝ^) → (𝑦 +_𝑒 -_𝑒𝑥) ∈ ℝ^)
6	3, 4, 5	syl2anr 598	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 +_𝑒 -_𝑒𝑥) ∈ ℝ^)
7		xnegcl 12605	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → -_𝑒𝑦 ∈ ℝ^*)
8		xaddcl 12631	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ -_𝑒𝑦 ∈ ℝ^) → (𝑥 +_𝑒 -_𝑒𝑦) ∈ ℝ^)
9	7, 8	sylan2 594	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 +_𝑒 -_𝑒𝑦) ∈ ℝ^)
10	6, 9	ifcld 4511	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) ∈ ℝ^)
11	10	rgen2 3203	. . . . 5 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) ∈ ℝ^*
12		xrsxmet.1	. . . . . . 7 ⊢ 𝐷 = (dist‘ℝ^*_𝑠)
13	12	xrsds 20587	. . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)))
14	13	fmpo 7765	. . . . 5 ⊢ (∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) ∈ ℝ^* ↔ 𝐷:(ℝ^* × ℝ^)⟶ℝ^)
15	11, 14	mpbi 232	. . . 4 ⊢ 𝐷:(ℝ^* × ℝ^)⟶ℝ^
16	15	a1i 11	. . 3 ⊢ (⊤ → 𝐷:(ℝ^* × ℝ^)⟶ℝ^)
17		breq2 5069	. . . . . 6 ⊢ ((𝑦 +_𝑒 -_𝑒𝑥) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) → (0 ≤ (𝑦 +_𝑒 -_𝑒𝑥) ↔ 0 ≤ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦))))
18		breq2 5069	. . . . . 6 ⊢ ((𝑥 +_𝑒 -_𝑒𝑦) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) → (0 ≤ (𝑥 +_𝑒 -_𝑒𝑦) ↔ 0 ≤ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦))))
19		xsubge0 12653	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (0 ≤ (𝑦 +_𝑒 -_𝑒𝑥) ↔ 𝑥 ≤ 𝑦))
20	19	ancoms 461	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (0 ≤ (𝑦 +_𝑒 -_𝑒𝑥) ↔ 𝑥 ≤ 𝑦))
21	20	biimpar 480	. . . . . 6 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≤ 𝑦) → 0 ≤ (𝑦 +_𝑒 -_𝑒𝑥))
22		xrletri 12545	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
23	22	orcanai 999	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
24		xsubge0 12653	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (0 ≤ (𝑥 +_𝑒 -_𝑒𝑦) ↔ 𝑦 ≤ 𝑥))
25	24	biimpar 480	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑦 ≤ 𝑥) → 0 ≤ (𝑥 +_𝑒 -_𝑒𝑦))
26	23, 25	syldan 593	. . . . . 6 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ ¬ 𝑥 ≤ 𝑦) → 0 ≤ (𝑥 +_𝑒 -_𝑒𝑦))
27	17, 18, 21, 26	ifbothda 4503	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 0 ≤ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)))
28	12	xrsdsval 20588	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥𝐷𝑦) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)))
29	27, 28	breqtrrd 5093	. . . 4 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → 0 ≤ (𝑥𝐷𝑦))
30	29	adantl 484	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 0 ≤ (𝑥𝐷𝑦))
31	29	biantrud 534	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥𝐷𝑦) ≤ 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
32	28, 10	eqeltrd 2913	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥𝐷𝑦) ∈ ℝ^)
33		0xr 10687	. . . . . 6 ⊢ 0 ∈ ℝ^*
34		xrletri3 12546	. . . . . 6 ⊢ (((𝑥𝐷𝑦) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
35	32, 33, 34	sylancl 588	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
36		simpr 487	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
37		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (𝑥𝐷𝑦) = 0)
38		0re 10642	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
39	37, 38	eqeltrdi 2921	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (𝑥𝐷𝑦) ∈ ℝ)
40	12	xrsdsreclb 20591	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≠ 𝑦) → ((𝑥𝐷𝑦) ∈ ℝ ↔ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
41	40	ad4ant124 1169	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → ((𝑥𝐷𝑦) ∈ ℝ ↔ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
42	39, 41	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
43	42	simpld 497	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ ℝ)
44	43	recnd 10668	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ ℂ)
45	42	simprd 498	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ ℝ)
46	45	recnd 10668	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ ℂ)
47		rexsub 12625	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 +_𝑒 -_𝑒𝑦) = (𝑥 − 𝑦))
48	42, 47	syl 17	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (𝑥 +_𝑒 -_𝑒𝑦) = (𝑥 − 𝑦))
49	28	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥𝐷𝑦) = 0 ↔ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0))
50	49	biimpa 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) → if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0)
51	50	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0)
52		xneg11 12607	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 +_𝑒 -_𝑒𝑥) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (-_𝑒(𝑦 +_𝑒 -_𝑒𝑥) = -_𝑒0 ↔ (𝑦 +_𝑒 -_𝑒𝑥) = 0))
53	6, 33, 52	sylancl 588	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (-_𝑒(𝑦 +_𝑒 -_𝑒𝑥) = -_𝑒0 ↔ (𝑦 +_𝑒 -_𝑒𝑥) = 0))
54		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
55	4	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → -_𝑒𝑥 ∈ ℝ^)
56		xnegdi 12640	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ -_𝑒𝑥 ∈ ℝ^*) → -_𝑒(𝑦 +_𝑒 -_𝑒𝑥) = (-_𝑒𝑦 +_𝑒 -_𝑒-_𝑒𝑥))
57	54, 55, 56	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → -_𝑒(𝑦 +_𝑒 -_𝑒𝑥) = (-_𝑒𝑦 +_𝑒 -_𝑒-_𝑒𝑥))
58		xnegneg 12606	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ^* → -_𝑒-_𝑒𝑥 = 𝑥)
59	58	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → -_𝑒-_𝑒𝑥 = 𝑥)
60	59	oveq2d 7171	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (-_𝑒𝑦 +_𝑒 -_𝑒-_𝑒𝑥) = (-_𝑒𝑦 +_𝑒 𝑥))
61	7	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → -_𝑒𝑦 ∈ ℝ^)
62		simpl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
63		xaddcom 12632	. . . . . . . . . . . . . . . . 17 ⊢ ((-_𝑒𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (-_𝑒𝑦 +_𝑒 𝑥) = (𝑥 +_𝑒 -_𝑒𝑦))
64	61, 62, 63	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (-_𝑒𝑦 +_𝑒 𝑥) = (𝑥 +_𝑒 -_𝑒𝑦))
65	57, 60, 64	3eqtrd 2860	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → -_𝑒(𝑦 +_𝑒 -_𝑒𝑥) = (𝑥 +_𝑒 -_𝑒𝑦))
66		xneg0 12604	. . . . . . . . . . . . . . . 16 ⊢ -_𝑒0 = 0
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → -_𝑒0 = 0)
68	65, 67	eqeq12d 2837	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (-_𝑒(𝑦 +_𝑒 -_𝑒𝑥) = -_𝑒0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0))
69	53, 68	bitr3d 283	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑦 +_𝑒 -_𝑒𝑥) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0))
70	69	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → ((𝑦 +_𝑒 -_𝑒𝑥) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0))
71		biidd 264	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → ((𝑥 +_𝑒 -_𝑒𝑦) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0))
72		eqeq1 2825	. . . . . . . . . . . . . 14 ⊢ ((𝑦 +_𝑒 -_𝑒𝑥) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) → ((𝑦 +_𝑒 -_𝑒𝑥) = 0 ↔ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0))
73	72	bibi1d 346	. . . . . . . . . . . . 13 ⊢ ((𝑦 +_𝑒 -_𝑒𝑥) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) → (((𝑦 +_𝑒 -_𝑒𝑥) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0) ↔ (if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0)))
74		eqeq1 2825	. . . . . . . . . . . . . 14 ⊢ ((𝑥 +_𝑒 -_𝑒𝑦) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) → ((𝑥 +_𝑒 -_𝑒𝑦) = 0 ↔ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0))
75	74	bibi1d 346	. . . . . . . . . . . . 13 ⊢ ((𝑥 +_𝑒 -_𝑒𝑦) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) → (((𝑥 +_𝑒 -_𝑒𝑦) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0) ↔ (if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0)))
76	73, 75	ifboth 4504	. . . . . . . . . . . 12 ⊢ ((((𝑦 +_𝑒 -_𝑒𝑥) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0) ∧ ((𝑥 +_𝑒 -_𝑒𝑦) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0)) → (if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0))
77	70, 71, 76	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = 0 ↔ (𝑥 +_𝑒 -_𝑒𝑦) = 0))
78	51, 77	mpbid 234	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (𝑥 +_𝑒 -_𝑒𝑦) = 0)
79	48, 78	eqtr3d 2858	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → (𝑥 − 𝑦) = 0)
80	44, 46, 79	subeq0d 11004	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) ∧ 𝑥 ≠ 𝑦) → 𝑥 = 𝑦)
81	36, 80	pm2.61dane 3104	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥𝐷𝑦) = 0) → 𝑥 = 𝑦)
82	81	ex 415	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥𝐷𝑦) = 0 → 𝑥 = 𝑦))
83	12	xrsdsval 20588	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦𝐷𝑦) = if(𝑦 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑦)))
84	83	anidms 569	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → (𝑦𝐷𝑦) = if(𝑦 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑦)))
85		xrleid 12543	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ 𝑦)
86	85	iftrued 4474	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → if(𝑦 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑦)) = (𝑦 +_𝑒 -_𝑒𝑦))
87		xnegid 12630	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → (𝑦 +_𝑒 -_𝑒𝑦) = 0)
88	84, 86, 87	3eqtrd 2860	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (𝑦𝐷𝑦) = 0)
89	88	adantl 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦𝐷𝑦) = 0)
90		oveq1 7162	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥𝐷𝑦) = (𝑦𝐷𝑦))
91	90	eqeq1d 2823	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥𝐷𝑦) = 0 ↔ (𝑦𝐷𝑦) = 0))
92	89, 91	syl5ibrcom 249	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 = 𝑦 → (𝑥𝐷𝑦) = 0))
93	82, 92	impbid 214	. . . . 5 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
94	31, 35, 93	3bitr2d 309	. . . 4 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))
95	94	adantl 484	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))
96		simplrr 776	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑦) ∈ ℝ)
97	96	leidd 11205	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑦) ≤ (𝑧𝐷𝑦))
98		simpr 487	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → 𝑧 = 𝑥)
99	98	oveq1d 7170	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑦) = (𝑥𝐷𝑦))
100	98	oveq1d 7170	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑥) = (𝑥𝐷𝑥))
101		simpll1 1208	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → 𝑥 ∈ ℝ^)
102		oveq12 7164	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑥 ∧ 𝑦 = 𝑥) → (𝑦𝐷𝑦) = (𝑥𝐷𝑥))
103	102	anidms 569	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦𝐷𝑦) = (𝑥𝐷𝑥))
104	103	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝑦𝐷𝑦) = 0 ↔ (𝑥𝐷𝑥) = 0))
105	104, 88	vtoclga 3573	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (𝑥𝐷𝑥) = 0)
106	101, 105	syl 17	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑥𝐷𝑥) = 0)
107	100, 106	eqtrd 2856	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑥) = 0)
108	107	oveq1d 7170	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = (0 + (𝑧𝐷𝑦)))
109	96	recnd 10668	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑦) ∈ ℂ)
110	109	addid2d 10840	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (0 + (𝑧𝐷𝑦)) = (𝑧𝐷𝑦))
111	108, 110	eqtr2d 2857	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑧𝐷𝑦) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
112	97, 99, 111	3brtr3d 5096	. . . . 5 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑥) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
113		simpr 487	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → 𝑧 = 𝑦)
114	113	oveq1d 7170	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑧𝐷𝑥) = (𝑦𝐷𝑥))
115		simplrl 775	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑧𝐷𝑥) ∈ ℝ)
116	114, 115	eqeltrrd 2914	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑦𝐷𝑥) ∈ ℝ)
117	116	leidd 11205	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑦𝐷𝑥) ≤ (𝑦𝐷𝑥))
118		simpll1 1208	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → 𝑥 ∈ ℝ^)
119		simpll2 1209	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → 𝑦 ∈ ℝ^)
120		oveq2 7163	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑦𝐷𝑥) = (𝑦𝐷𝑦))
121	90, 120	eqtr4d 2859	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
122	121	adantl 484	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 = 𝑦) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
123		eqeq2 2833	. . . . . . . . . 10 ⊢ ((𝑥 +_𝑒 -_𝑒𝑦) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥)) → (if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = (𝑥 +_𝑒 -_𝑒𝑦) ↔ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥))))
124		eqeq2 2833	. . . . . . . . . 10 ⊢ ((𝑦 +_𝑒 -_𝑒𝑥) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥)) → (if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = (𝑦 +_𝑒 -_𝑒𝑥) ↔ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥))))
125		xrleloe 12536	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
126	125	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
127		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
128	127	neneqd 3021	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → ¬ 𝑥 = 𝑦)
129		biorf 933	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 < 𝑦 ↔ (𝑥 = 𝑦 ∨ 𝑥 < 𝑦)))
130		orcom 866	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑦 ∨ 𝑥 < 𝑦) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦))
131	129, 130	syl6bb 289	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
132	128, 131	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
133		xrltnle 10707	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
134	133	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
135	126, 132, 134	3bitr2d 309	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
136	135	con2bid 357	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑦 ≤ 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
137	136	biimpa 479	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) ∧ 𝑦 ≤ 𝑥) → ¬ 𝑥 ≤ 𝑦)
138	137	iffalsed 4477	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) ∧ 𝑦 ≤ 𝑥) → if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = (𝑥 +_𝑒 -_𝑒𝑦))
139	135	biimpar 480	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) ∧ ¬ 𝑦 ≤ 𝑥) → 𝑥 ≤ 𝑦)
140	139	iftrued 4474	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) ∧ ¬ 𝑦 ≤ 𝑥) → if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = (𝑦 +_𝑒 -_𝑒𝑥))
141	123, 124, 138, 140	ifbothda 4503	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥)))
142	28	adantr 483	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑥𝐷𝑦) = if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)))
143	12	xrsdsval 20588	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦𝐷𝑥) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥)))
144	143	ancoms 461	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦𝐷𝑥) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥)))
145	144	adantr 483	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑦𝐷𝑥) = if(𝑦 ≤ 𝑥, (𝑥 +_𝑒 -_𝑒𝑦), (𝑦 +_𝑒 -_𝑒𝑥)))
146	141, 142, 145	3eqtr4d 2866	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ 𝑥 ≠ 𝑦) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
147	122, 146	pm2.61dane 3104	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
148	118, 119, 147	syl2anc 586	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑥𝐷𝑦) = (𝑦𝐷𝑥))
149	113	oveq1d 7170	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑧𝐷𝑦) = (𝑦𝐷𝑦))
150	119, 88	syl 17	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑦𝐷𝑦) = 0)
151	149, 150	eqtrd 2856	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑧𝐷𝑦) = 0)
152	114, 151	oveq12d 7173	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((𝑦𝐷𝑥) + 0))
153	116	recnd 10668	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑦𝐷𝑥) ∈ ℂ)
154	153	addid1d 10839	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → ((𝑦𝐷𝑥) + 0) = (𝑦𝐷𝑥))
155	152, 154	eqtrd 2856	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = (𝑦𝐷𝑥))
156	117, 148, 155	3brtr4d 5097	. . . . 5 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ 𝑧 = 𝑦) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
157		simplrl 775	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧𝐷𝑥) ∈ ℝ)
158		simpll3 1210	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑧 ∈ ℝ^)
159		simpll1 1208	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑥 ∈ ℝ^)
160		simprl 769	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑧 ≠ 𝑥)
161	12	xrsdsreclb 20591	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑧 ≠ 𝑥) → ((𝑧𝐷𝑥) ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ)))
162	158, 159, 160, 161	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → ((𝑧𝐷𝑥) ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ)))
163	157, 162	mpbid 234	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ))
164	163	simprd 498	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑥 ∈ ℝ)
165	164	recnd 10668	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑥 ∈ ℂ)
166		simplrr 776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧𝐷𝑦) ∈ ℝ)
167		simpll2 1209	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑦 ∈ ℝ^)
168		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑧 ≠ 𝑦)
169	12	xrsdsreclb 20591	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ≠ 𝑦) → ((𝑧𝐷𝑦) ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
170	158, 167, 168, 169	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → ((𝑧𝐷𝑦) ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
171	166, 170	mpbid 234	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ))
172	171	simprd 498	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑦 ∈ ℝ)
173	172	recnd 10668	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑦 ∈ ℂ)
174	163	simpld 497	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑧 ∈ ℝ)
175	174	recnd 10668	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → 𝑧 ∈ ℂ)
176	165, 173, 175	abs3difd 14819	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (abs‘(𝑥 − 𝑦)) ≤ ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦))))
177	12	xrsdsreval 20589	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝐷𝑦) = (abs‘(𝑥 − 𝑦)))
178	164, 172, 177	syl2anc 586	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑥𝐷𝑦) = (abs‘(𝑥 − 𝑦)))
179	12	xrsdsreval 20589	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧𝐷𝑥) = (abs‘(𝑧 − 𝑥)))
180	163, 179	syl 17	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧𝐷𝑥) = (abs‘(𝑧 − 𝑥)))
181	175, 165	abssubd 14812	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (abs‘(𝑧 − 𝑥)) = (abs‘(𝑥 − 𝑧)))
182	180, 181	eqtrd 2856	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧𝐷𝑥) = (abs‘(𝑥 − 𝑧)))
183	12	xrsdsreval 20589	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧𝐷𝑦) = (abs‘(𝑧 − 𝑦)))
184	171, 183	syl 17	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑧𝐷𝑦) = (abs‘(𝑧 − 𝑦)))
185	182, 184	oveq12d 7173	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦))))
186	176, 178, 185	3brtr4d 5097	. . . . 5 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) ∧ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
187	112, 156, 186	pm2.61da2ne 3105	. . . 4 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
188	187	3adant1 1126	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
189	2, 16, 30, 95, 188	isxmet2d 22936	. 2 ⊢ (⊤ → 𝐷 ∈ (∞Met‘ℝ^*))
190	189	mptru 1540	1 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ifcif 4466 class class class wbr 5065 × cxp 5552 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 0cc0 10536 + caddc 10539 ℝ^*cxr 10673 < clt 10674 ≤ cle 10675 − cmin 10869 -_𝑒cxne 12503 +_𝑒 cxad 12504 abscabs 14592 distcds 16573 ℝ^*_𝑠cxrs 16772 ∞Metcxmet 20529

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-xneg 12506 df-xadd 12507 df-icc 12744 df-fz 12892 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-mulr 16578 df-tset 16583 df-ple 16584 df-ds 16586 df-xrs 16774 df-xmet 20537

This theorem is referenced by: xrsdsre 23417 xrsblre 23418 xrsmopn 23419 metdcnlem 23443 xmetdcn2 23444 xmetdcn 23445 metdscn 23463 metdscn2 23464

W3C validator