Theorem metideq 33853

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 482	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
2		metidss 33851	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
3		dmss 5880	. . . . . . . . 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 5908	. . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋
6	4, 5	sseqtrdi 3997	. . . . . . 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 5668	. . . . . . . . . 10 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
9	2, 8	sstrdi 3969	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (V × V))
10		df-rel 5659	. . . . . . . . 9 ⊢ (Rel (~Met‘𝐷) ↔ (~Met‘𝐷) ⊆ (V × V))
11	9, 10	sylibr 234	. . . . . . . 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 770	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐴(~Met‘𝐷)𝐵)
14		releldm 5922	. . . . . . 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 3957	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐴 ∈ 𝑋)
17		simprr 772	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐸(~Met‘𝐷)𝐹)
18		releldm 5922	. . . . . . 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 3957	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐸 ∈ 𝑋)
21		psmetsym 24236	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
22	1, 16, 20, 21	syl3anc 1372	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23		psmetf 24232	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
24	23	fovcdmda 7573	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐸𝐷𝐴) ∈ ℝ*)
25	1, 20, 16, 24	syl12anc 836	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐸𝐷𝐴) ∈ ℝ*)
26	22, 25	eqeltrd 2833	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27		rnss 5917	. . . . . . . 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 6160	. . . . . . 7 ⊢ ran (𝑋 × 𝑋) = 𝑋
30	28, 29	sseqtrdi 3997	. . . . . 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 5923	. . . . . 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 3957	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐵 ∈ 𝑋)
35	23	fovcdmda 7573	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐵𝐷𝐸) ∈ ℝ*)
36	1, 34, 20, 35	syl12anc 836	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ∈ ℝ*)
37		relelrn 5923	. . . . . . 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 3957	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐹 ∈ 𝑋)
40		psmetsym 24236	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
41	1, 39, 34, 40	syl3anc 1372	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
42	23	fovcdmda 7573	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹𝐷𝐵) ∈ ℝ*)
43	1, 39, 34, 42	syl12anc 836	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐵) ∈ ℝ*)
44	41, 43	eqeltrrd 2834	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45		psmettri2 24235	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
46	1, 34, 16, 20, 45	syl13anc 1373	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47		psmetsym 24236	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
48	1, 16, 34, 47	syl3anc 1372	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
49	16, 34	jca 511	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
50		metidv 33852	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
51	50	biimpa 476	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴(~Met‘𝐷)𝐵) → (𝐴𝐷𝐵) = 0)
52	1, 49, 13, 51	syl21anc 837	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐵) = 0)
53	48, 52	eqtr3d 2771	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
54	53	oveq1d 7415	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55		xaddlid 13251	. . . . . 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 2769	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
58	46, 57	breqtrd 5143	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59		psmettri2 24235	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
60	1, 39, 34, 20, 59	syl13anc 1373	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61		psmetsym 24236	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
62	1, 39, 20, 61	syl3anc 1372	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
63	20, 39	jca 511	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
64		metidv 33852	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐸(~Met‘𝐷)𝐹 ↔ (𝐸𝐷𝐹) = 0))
65	64	biimpa 476	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ∧ 𝐸(~Met‘𝐷)𝐹) → (𝐸𝐷𝐹) = 0)
66	1, 63, 17, 65	syl21anc 837	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐸𝐷𝐹) = 0)
67	62, 66	eqtrd 2769	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
68	67	oveq2d 7416	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69		xaddrid 13250	. . . . . 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 2773	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
72	60, 71	breqtrd 5143	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
73	26, 36, 44, 58, 72	xrletrd 13171	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
74	23	fovcdmda 7573	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐴𝐷𝐹) ∈ ℝ*)
75	1, 16, 39, 74	syl12anc 836	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ∈ ℝ*)
76		psmettri2 24235	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
77	1, 16, 34, 39, 76	syl13anc 1373	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
78	52	oveq1d 7415	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79		xaddlid 13251	. . . . . 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 2769	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
82	77, 81	breqtrd 5143	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83		psmettri2 24235	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
84	1, 20, 16, 39, 83	syl13anc 1373	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85		xaddrid 13250	. . . . . 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 7416	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
88	86, 87, 22	3eqtr4d 2779	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
89	84, 88	breqtrd 5143	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
90	44, 75, 26, 82, 89	xrletrd 13171	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91		xrletri3 13163	. . 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 713	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457   ⊆ wss 3924   class class class wbr 5117   × cxp 5650  dom cdm 5652  ran crn 5653  Rel wrel 5657  ‘cfv 6528  (class class class)co 7400  0cc0 11122  ℝ^*cxr 11261   ≤ cle 11263   +_𝑒 cxad 13119  PsMetcpsmet 21286  ~_Metcmetid 33846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724  ax-cnex 11178  ax-resscn 11179  ax-1cn 11180  ax-icn 11181  ax-addcl 11182  ax-addrcl 11183  ax-mulcl 11184  ax-mulrcl 11185  ax-mulcom 11186  ax-addass 11187  ax-mulass 11188  ax-distr 11189  ax-i2m1 11190  ax-1ne0 11191  ax-1rid 11192  ax-rnegex 11193  ax-rrecex 11194  ax-cnre 11195  ax-pre-lttri 11196  ax-pre-lttrn 11197  ax-pre-ltadd 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-po 5559  df-so 5560  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-er 8714  df-map 8837  df-en 8955  df-dom 8956  df-sdom 8957  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-xadd 13122  df-psmet 21294  df-metid 33848

This theorem is referenced by:  pstmfval  33856