Theorem metideq 31843

Description: Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
metideq	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))

Proof of Theorem metideq

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
2		metidss 31841	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
3		dmss 5811	. . . . . . . . 9 ⊢ ((~Met‘𝐷) ⊆ (𝑋 × 𝑋) → dom (~Met‘𝐷) ⊆ dom (𝑋 × 𝑋))
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom (~Met‘𝐷) ⊆ dom (𝑋 × 𝑋))
5		dmxpid 5839	. . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋
6	4, 5	sseqtrdi 3971	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom (~Met‘𝐷) ⊆ 𝑋)
7	1, 6	syl 17	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → dom (~Met‘𝐷) ⊆ 𝑋)
8		xpss 5605	. . . . . . . . . 10 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
9	2, 8	sstrdi 3933	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (V × V))
10		df-rel 5596	. . . . . . . . 9 ⊢ (Rel (~Met‘𝐷) ↔ (~Met‘𝐷) ⊆ (V × V))
11	9, 10	sylibr 233	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → Rel (~Met‘𝐷))
12	1, 11	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → Rel (~Met‘𝐷))
13		simprl 768	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐴(~Met‘𝐷)𝐵)
14		releldm 5853	. . . . . . 7 ⊢ ((Rel (~Met‘𝐷) ∧ 𝐴(~Met‘𝐷)𝐵) → 𝐴 ∈ dom (~Met‘𝐷))
15	12, 13, 14	syl2anc 584	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐴 ∈ dom (~Met‘𝐷))
16	7, 15	sseldd 3922	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐴 ∈ 𝑋)
17		simprr 770	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐸(~Met‘𝐷)𝐹)
18		releldm 5853	. . . . . . 7 ⊢ ((Rel (~Met‘𝐷) ∧ 𝐸(~Met‘𝐷)𝐹) → 𝐸 ∈ dom (~Met‘𝐷))
19	12, 17, 18	syl2anc 584	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐸 ∈ dom (~Met‘𝐷))
20	7, 19	sseldd 3922	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐸 ∈ 𝑋)
21		psmetsym 23463	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
22	1, 16, 20, 21	syl3anc 1370	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23		psmetf 23459	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
24	23	fovrnda 7443	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐸𝐷𝐴) ∈ ℝ*)
25	1, 20, 16, 24	syl12anc 834	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐸𝐷𝐴) ∈ ℝ*)
26	22, 25	eqeltrd 2839	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27		rnss 5848	. . . . . . . 8 ⊢ ((~Met‘𝐷) ⊆ (𝑋 × 𝑋) → ran (~Met‘𝐷) ⊆ ran (𝑋 × 𝑋))
28	2, 27	syl 17	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran (~Met‘𝐷) ⊆ ran (𝑋 × 𝑋))
29		rnxpid 6076	. . . . . . 7 ⊢ ran (𝑋 × 𝑋) = 𝑋
30	28, 29	sseqtrdi 3971	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran (~Met‘𝐷) ⊆ 𝑋)
31	1, 30	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ran (~Met‘𝐷) ⊆ 𝑋)
32		relelrn 5854	. . . . . 6 ⊢ ((Rel (~Met‘𝐷) ∧ 𝐴(~Met‘𝐷)𝐵) → 𝐵 ∈ ran (~Met‘𝐷))
33	12, 13, 32	syl2anc 584	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐵 ∈ ran (~Met‘𝐷))
34	31, 33	sseldd 3922	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐵 ∈ 𝑋)
35	23	fovrnda 7443	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐵𝐷𝐸) ∈ ℝ*)
36	1, 34, 20, 35	syl12anc 834	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ∈ ℝ*)
37		relelrn 5854	. . . . . . 7 ⊢ ((Rel (~Met‘𝐷) ∧ 𝐸(~Met‘𝐷)𝐹) → 𝐹 ∈ ran (~Met‘𝐷))
38	12, 17, 37	syl2anc 584	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐹 ∈ ran (~Met‘𝐷))
39	31, 38	sseldd 3922	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐹 ∈ 𝑋)
40		psmetsym 23463	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
41	1, 39, 34, 40	syl3anc 1370	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
42	23	fovrnda 7443	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹𝐷𝐵) ∈ ℝ*)
43	1, 39, 34, 42	syl12anc 834	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐵) ∈ ℝ*)
44	41, 43	eqeltrrd 2840	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45		psmettri2 23462	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
46	1, 34, 16, 20, 45	syl13anc 1371	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47		psmetsym 23463	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
48	1, 16, 34, 47	syl3anc 1370	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
49	16, 34	jca 512	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
50		metidv 31842	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
51	50	biimpa 477	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴(~Met‘𝐷)𝐵) → (𝐴𝐷𝐵) = 0)
52	1, 49, 13, 51	syl21anc 835	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐵) = 0)
53	48, 52	eqtr3d 2780	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
54	53	oveq1d 7290	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55		xaddid2 12976	. . . . . 6 ⊢ ((𝐵𝐷𝐸) ∈ ℝ* → (0 +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
56	36, 55	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (0 +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
57	54, 56	eqtrd 2778	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
58	46, 57	breqtrd 5100	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59		psmettri2 23462	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
60	1, 39, 34, 20, 59	syl13anc 1371	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61		psmetsym 23463	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
62	1, 39, 20, 61	syl3anc 1370	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
63	20, 39	jca 512	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
64		metidv 31842	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐸(~Met‘𝐷)𝐹 ↔ (𝐸𝐷𝐹) = 0))
65	64	biimpa 477	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ∧ 𝐸(~Met‘𝐷)𝐹) → (𝐸𝐷𝐹) = 0)
66	1, 63, 17, 65	syl21anc 835	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐸𝐷𝐹) = 0)
67	62, 66	eqtrd 2778	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
68	67	oveq2d 7291	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69		xaddid1 12975	. . . . . 6 ⊢ ((𝐹𝐷𝐵) ∈ ℝ* → ((𝐹𝐷𝐵) +𝑒 0) = (𝐹𝐷𝐵))
70	43, 69	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 0) = (𝐹𝐷𝐵))
71	68, 70, 41	3eqtrd 2782	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
72	60, 71	breqtrd 5100	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
73	26, 36, 44, 58, 72	xrletrd 12896	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
74	23	fovrnda 7443	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐴𝐷𝐹) ∈ ℝ*)
75	1, 16, 39, 74	syl12anc 834	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ∈ ℝ*)
76		psmettri2 23462	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
77	1, 16, 34, 39, 76	syl13anc 1371	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
78	52	oveq1d 7290	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79		xaddid2 12976	. . . . . 6 ⊢ ((𝐴𝐷𝐹) ∈ ℝ* → (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
80	75, 79	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (0 +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
81	78, 80	eqtrd 2778	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
82	77, 81	breqtrd 5100	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83		psmettri2 23462	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
84	1, 20, 16, 39, 83	syl13anc 1371	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85		xaddid1 12975	. . . . . 6 ⊢ ((𝐸𝐷𝐴) ∈ ℝ* → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
86	25, 85	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 0) = (𝐸𝐷𝐴))
87	66	oveq2d 7291	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
88	86, 87, 22	3eqtr4d 2788	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
89	84, 88	breqtrd 5100	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
90	44, 75, 26, 82, 89	xrletrd 12896	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91		xrletri3 12888	. . 3 ⊢ (((𝐴𝐷𝐸) ∈ ℝ* ∧ (𝐵𝐷𝐹) ∈ ℝ*) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
92	26, 44, 91	syl2anc 584	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐸) = (𝐵𝐷𝐹) ↔ ((𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹) ∧ (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))))
93	73, 90, 92	mpbir2and 710	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   × cxp 5587  dom cdm 5589  ran crn 5590  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275  0cc0 10871  ℝ^*cxr 11008   ≤ cle 11010   +_𝑒 cxad 12846  PsMetcpsmet 20581  ~_Metcmetid 31836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-xadd 12849  df-psmet 20589  df-metid 31838

This theorem is referenced by:  pstmfval  31846