Theorem comet 24862

Description: The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)

Hypotheses

Ref	Expression
comet.1	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
comet.2	⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ*)
comet.3	⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
comet.4	⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
comet.5	⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +e 𝑦)) ≤ ((𝐹‘𝑥) +e (𝐹‘𝑦)))

Assertion

Ref	Expression
comet	⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))

Distinct variable groups:   𝑥,𝑦,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑋(𝑥, 𝑦)

Proof of Theorem comet

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		comet.1	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
2	1	elfvexd 6944	. 2 ⊢ (𝜑 → 𝑋 ∈ V)
3		comet.2	. . 3 ⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ*)
4		xmetf 24678	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
5	1, 4	syl 18	. . . . 5 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	ffnd 6732	. . . 4 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
7		xmetcl 24680	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎𝐷𝑏) ∈ ℝ*)
8		xmetge0 24693	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → 0 ≤ (𝑎𝐷𝑏))
9		elxrge0 13615	. . . . . . . 8 ⊢ ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ* ∧ 0 ≤ (𝑎𝐷𝑏)))
10	7, 8, 9	sylanbrc 595	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
11	10	3expb 1138	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
12	1, 11	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
13	12	ralrimivva 3209	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
14		ffnov 7569	. . . 4 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
15	6, 13, 14	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
16	3, 15	fcod 6757	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐷):(𝑋 × 𝑋)⟶ℝ*)
17		opelxpi 5692	. . . . . 6 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
18		fvco3 7008	. . . . . 6 ⊢ ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
19	5, 17, 18	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
20		df-ov 7446	. . . . 5 ⊢ (𝑎(𝐹 ∘ 𝐷)𝑏) = ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩)
21		df-ov 7446	. . . . . 6 ⊢ (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
22	21	fveq2i 6911	. . . . 5 ⊢ (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
23	19, 20, 22	3eqtr4g 2823	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
24	23	eqeq1d 2765	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎(𝐹 ∘ 𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
25		fveq2 6908	. . . . . 6 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝐹‘𝑥) = (𝐹‘(𝑎𝐷𝑏)))
26	25	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
27		eqeq1 2767	. . . . 5 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝑥 = 0 ↔ (𝑎𝐷𝑏) = 0))
28	26, 27	bibi12d 348	. . . 4 ⊢ (𝑥 = (𝑎𝐷𝑏) → (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0) ↔ ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0)))
29		comet.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
30	29	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
31	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
32	28, 31, 12	rspcdva 3581	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
33		xmeteq0 24687	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
34	33	3expb 1138	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
35	1, 34	sylan 592	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
36	24, 32, 35	3bitrd 308	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎(𝐹 ∘ 𝐷)𝑏) = 0 ↔ 𝑎 = 𝑏))
37	3	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐹:(0[,]+∞)⟶ℝ*)
38	12	3adantr3 1190	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
39	37, 38	ffvelcdmd 7109	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ*)
40	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
41		simpr3 1215	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑐 ∈ 𝑋)
42		simpr1 1213	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
43	40, 41, 42	fovcdmd 7617	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
44		simpr2 1214	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
45	40, 41, 44	fovcdmd 7617	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
46		ge0xaddcl 13620	. . . . . 6 ⊢ (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) ∈ (0[,]+∞))
47	43, 45, 46	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) ∈ (0[,]+∞))
48	37, 47	ffvelcdmd 7109	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))) ∈ ℝ*)
49	37, 43	ffvelcdmd 7109	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ*)
50	37, 45	ffvelcdmd 7109	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ*)
51	49, 50	xaddcld 13458	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ*)
52		3anrot 1117	. . . . . . 7 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋))
53		xmettri2 24689	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
54	52, 53	sylan2br 607	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
55	1, 54	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
56		comet.4	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
57	56	ralrimivva 3209	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
58	57	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
59		breq1 5109	. . . . . . . 8 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝑥 ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
60	25	breq1d 5116	. . . . . . . 8 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦)))
61	59, 60	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦))))
62		breq2 5110	. . . . . . . 8 ⊢ (𝑦 = ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))))
63		fveq2 6908	. . . . . . . . 9 ⊢ (𝑦 = ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → (𝐹‘𝑦) = (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))))
64	63	breq2d 5118	. . . . . . . 8 ⊢ (𝑦 = ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → ((𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))))
65	62, 64	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → (((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦)) ↔ ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))))))
66	61, 65	rspc2va 3591	. . . . . 6 ⊢ ((((𝑎𝐷𝑏) ∈ (0[,]+∞) ∧ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) ∈ (0[,]+∞)) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))))
67	38, 47, 58, 66	syl21anc 851	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))))
68	55, 67	mpd 16	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))))
69		comet.5	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +e 𝑦)) ≤ ((𝐹‘𝑥) +e (𝐹‘𝑦)))
70	69	ralrimivva 3209	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +e 𝑦)) ≤ ((𝐹‘𝑥) +e (𝐹‘𝑦)))
71	70	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +e 𝑦)) ≤ ((𝐹‘𝑥) +e (𝐹‘𝑦)))
72		fvoveq1 7466	. . . . . . 7 ⊢ (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +e 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +e 𝑦)))
73		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = (𝑐𝐷𝑎) → (𝐹‘𝑥) = (𝐹‘(𝑐𝐷𝑎)))
74	73	oveq1d 7458	. . . . . . 7 ⊢ (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘𝑥) +e (𝐹‘𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘𝑦)))
75	72, 74	breq12d 5119	. . . . . 6 ⊢ (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +e 𝑦)) ≤ ((𝐹‘𝑥) +e (𝐹‘𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +e 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘𝑦))))
76		oveq2 7451	. . . . . . . 8 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +e 𝑦) = ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
77	76	fveq2d 6912	. . . . . . 7 ⊢ (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +e 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))))
78		fveq2 6908	. . . . . . . 8 ⊢ (𝑦 = (𝑐𝐷𝑏) → (𝐹‘𝑦) = (𝐹‘(𝑐𝐷𝑏)))
79	78	oveq2d 7459	. . . . . . 7 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏))))
80	77, 79	breq12d 5119	. . . . . 6 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘((𝑐𝐷𝑎) +e 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏)))))
81	75, 80	rspc2va 3591	. . . . 5 ⊢ ((((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +e 𝑦)) ≤ ((𝐹‘𝑥) +e (𝐹‘𝑦))) → (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏))))
82	43, 45, 71, 81	syl21anc 851	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘((𝑐𝐷𝑎) +e (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏))))
83	39, 48, 51, 68, 82	xrletrd 13318	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏))))
84	23	3adantr3 1190	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
85	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
86	41, 42	opelxpd 5694	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
87	85, 86	fvco3d 7009	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
88		df-ov 7446	. . . . 5 ⊢ (𝑐(𝐹 ∘ 𝐷)𝑎) = ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑎⟩)
89		df-ov 7446	. . . . . 6 ⊢ (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
90	89	fveq2i 6911	. . . . 5 ⊢ (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
91	87, 88, 90	3eqtr4g 2823	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐(𝐹 ∘ 𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
92	41, 44	opelxpd 5694	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
93	85, 92	fvco3d 7009	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
94		df-ov 7446	. . . . 5 ⊢ (𝑐(𝐹 ∘ 𝐷)𝑏) = ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑏⟩)
95		df-ov 7446	. . . . . 6 ⊢ (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
96	95	fveq2i 6911	. . . . 5 ⊢ (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
97	93, 94, 96	3eqtr4g 2823	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
98	91, 97	oveq12d 7461	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑐(𝐹 ∘ 𝐷)𝑎) +e (𝑐(𝐹 ∘ 𝐷)𝑏)) = ((𝐹‘(𝑐𝐷𝑎)) +e (𝐹‘(𝑐𝐷𝑏))))
99	83, 84, 98	3brtr4d 5140	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) ≤ ((𝑐(𝐹 ∘ 𝐷)𝑎) +e (𝑐(𝐹 ∘ 𝐷)𝑏)))
100	2, 16, 36, 99	isxmetd 24675	1 ⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⟨cop 4838   class class class wbr 5106   × cxp 5653   ∘ ccom 5659   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  0cc0 11227  +∞cpnf 11367  ℝ^*cxr 11369   ≤ cle 11371   +_e cxad 13266  [,]cicc 13506  ∞Metcxmet 21693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-rp 13148  df-xneg 13268  df-xadd 13269  df-xmul 13270  df-icc 13510  df-xmet 21701

This theorem is used by:  stdbdxmet  24864