Theorem metideq 33924

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 33922	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
3		dmss 5882	. . . . . . . . 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 5910	. . . . . . . 8 ⊢ dom (𝑋 × 𝑋) = 𝑋
6	4, 5	sseqtrdi 3999	. . . . . . 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 5670	. . . . . . . . . 10 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
9	2, 8	sstrdi 3971	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (V × V))
10		df-rel 5661	. . . . . . . . 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 5924	. . . . . . 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 3959	. . . . 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 5924	. . . . . . 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 3959	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐸 ∈ 𝑋)
21		psmetsym 24249	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
22	1, 16, 20, 21	syl3anc 1373	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐸𝐷𝐴))
23		psmetf 24245	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
24	23	fovcdmda 7578	. . . . 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 2834	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ∈ ℝ*)
27		rnss 5919	. . . . . . . 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 6162	. . . . . . 7 ⊢ ran (𝑋 × 𝑋) = 𝑋
30	28, 29	sseqtrdi 3999	. . . . . 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 5925	. . . . . 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 3959	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐵 ∈ 𝑋)
35	23	fovcdmda 7578	. . . 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 5925	. . . . . . 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 3959	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → 𝐹 ∈ 𝑋)
40		psmetsym 24249	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
41	1, 39, 34, 40	syl3anc 1373	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐵) = (𝐵𝐷𝐹))
42	23	fovcdmda 7578	. . . . 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 2835	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ∈ ℝ*)
45		psmettri2 24248	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
46	1, 34, 16, 20, 45	syl13anc 1374	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)))
47		psmetsym 24249	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
48	1, 16, 34, 47	syl3anc 1373	. . . . . . 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 33923	. . . . . . . . 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 2772	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐴) = 0)
54	53	oveq1d 7420	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (0 +𝑒 (𝐵𝐷𝐸)))
55		xaddlid 13258	. . . . . 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 2770	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐵𝐷𝐴) +𝑒 (𝐵𝐷𝐸)) = (𝐵𝐷𝐸))
58	46, 57	breqtrd 5145	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐸))
59		psmettri2 24248	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
60	1, 39, 34, 20, 59	syl13anc 1374	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)))
61		psmetsym 24249	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋) → (𝐹𝐷𝐸) = (𝐸𝐷𝐹))
62	1, 39, 20, 61	syl3anc 1373	. . . . . . 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 33923	. . . . . . . . 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 2770	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐹𝐷𝐸) = 0)
68	67	oveq2d 7421	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = ((𝐹𝐷𝐵) +𝑒 0))
69		xaddrid 13257	. . . . . 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 2774	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐹𝐷𝐵) +𝑒 (𝐹𝐷𝐸)) = (𝐵𝐷𝐹))
72	60, 71	breqtrd 5145	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐸) ≤ (𝐵𝐷𝐹))
73	26, 36, 44, 58, 72	xrletrd 13178	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) ≤ (𝐵𝐷𝐹))
74	23	fovcdmda 7578	. . . 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 24248	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
77	1, 16, 34, 39, 76	syl13anc 1374	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)))
78	52	oveq1d 7420	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (0 +𝑒 (𝐴𝐷𝐹)))
79		xaddlid 13258	. . . . . 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 2770	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐹)) = (𝐴𝐷𝐹))
82	77, 81	breqtrd 5145	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐹))
83		psmettri2 24248	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
84	1, 20, 16, 39, 83	syl13anc 1374	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)))
85		xaddrid 13257	. . . . . 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 7421	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = ((𝐸𝐷𝐴) +𝑒 0))
88	86, 87, 22	3eqtr4d 2780	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → ((𝐸𝐷𝐴) +𝑒 (𝐸𝐷𝐹)) = (𝐴𝐷𝐸))
89	84, 88	breqtrd 5145	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐹) ≤ (𝐴𝐷𝐸))
90	44, 75, 26, 82, 89	xrletrd 13178	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐵𝐷𝐹) ≤ (𝐴𝐷𝐸))
91		xrletri3 13170	. . 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 1540   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926   class class class wbr 5119   × cxp 5652  dom cdm 5654  ran crn 5655  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405  0cc0 11129  ℝ^*cxr 11268   ≤ cle 11270   +_𝑒 cxad 13126  PsMetcpsmet 21299  ~_Metcmetid 33917

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-xadd 13129  df-psmet 21307  df-metid 33919

This theorem is referenced by:  pstmfval  33927