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Theorem comet 23650

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 6802	. 2 ⊢ (𝜑 → 𝑋 ∈ V)
3		comet.2	. . 3 ⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ^*)
4		xmetf 23463	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
6	5	ffnd 6597	. . . 4 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
7		xmetcl 23465	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎𝐷𝑏) ∈ ℝ^*)
8		xmetge0 23478	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → 0 ≤ (𝑎𝐷𝑏))
9		elxrge0 13171	. . . . . . . 8 ⊢ ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ^* ∧ 0 ≤ (𝑎𝐷𝑏)))
10	7, 8, 9	sylanbrc 582	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
11	10	3expb 1118	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
12	1, 11	sylan 579	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
13	12	ralrimivva 3116	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
14		ffnov 7392	. . . 4 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
15	6, 13, 14	sylanbrc 582	. . 3 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
16	3, 15	fcod 6622	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐷):(𝑋 × 𝑋)⟶ℝ^*)
17		opelxpi 5625	. . . . . 6 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
18		fvco3 6861	. . . . . 6 ⊢ ((𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
19	5, 17, 18	syl2an 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
20		df-ov 7271	. . . . 5 ⊢ (𝑎(𝐹 ∘ 𝐷)𝑏) = ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩)
21		df-ov 7271	. . . . . 6 ⊢ (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
22	21	fveq2i 6771	. . . . 5 ⊢ (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
23	19, 20, 22	3eqtr4g 2804	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
24	23	eqeq1d 2741	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎(𝐹 ∘ 𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
25		fveq2 6768	. . . . . 6 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝐹‘𝑥) = (𝐹‘(𝑎𝐷𝑏)))
26	25	eqeq1d 2741	. . . . 5 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
27		eqeq1 2743	. . . . 5 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝑥 = 0 ↔ (𝑎𝐷𝑏) = 0))
28	26, 27	bibi12d 345	. . . 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 3109	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
31	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
32	28, 31, 12	rspcdva 3562	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
33		xmeteq0 23472	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
34	33	3expb 1118	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
35	1, 34	sylan 579	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
36	24, 32, 35	3bitrd 304	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎(𝐹 ∘ 𝐷)𝑏) = 0 ↔ 𝑎 = 𝑏))
37	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐹:(0[,]+∞)⟶ℝ^*)
38	12	3adantr3 1169	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
39	37, 38	ffvelrnd 6956	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ^*)
40	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
41		simpr3 1194	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑐 ∈ 𝑋)
42		simpr1 1192	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
43	40, 41, 42	fovrnd 7435	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
44		simpr2 1193	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
45	40, 41, 44	fovrnd 7435	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
46		ge0xaddcl 13176	. . . . . 6 ⊢ (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
47	43, 45, 46	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
48	37, 47	ffvelrnd 6956	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))) ∈ ℝ^*)
49	37, 43	ffvelrnd 6956	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ^*)
50	37, 45	ffvelrnd 6956	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ^*)
51	49, 50	xaddcld 13017	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ^*)
52		3anrot 1098	. . . . . . 7 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋))
53		xmettri2 23474	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
54	52, 53	sylan2br 594	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
55	1, 54	sylan 579	. . . . 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 3116	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
58	57	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
59		breq1 5081	. . . . . . . 8 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝑥 ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
60	25	breq1d 5088	. . . . . . . 8 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦)))
61	59, 60	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦))))
62		breq2 5082	. . . . . . . 8 ⊢ (𝑦 = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
63		fveq2 6768	. . . . . . . . 9 ⊢ (𝑦 = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) → (𝐹‘𝑦) = (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
64	63	breq2d 5090	. . . . . . . 8 ⊢ (𝑦 = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) → ((𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))))
65	62, 64	imbi12d 344	. . . . . . 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 3571	. . . . . 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 834	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))))
68	55, 67	mpd 15	. . . 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 3116	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +_𝑒 𝑦)) ≤ ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)))
71	70	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +_𝑒 𝑦)) ≤ ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)))
72		fvoveq1 7291	. . . . . . 7 ⊢ (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +_𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +_𝑒 𝑦)))
73		fveq2 6768	. . . . . . . 8 ⊢ (𝑥 = (𝑐𝐷𝑎) → (𝐹‘𝑥) = (𝐹‘(𝑐𝐷𝑎)))
74	73	oveq1d 7283	. . . . . . 7 ⊢ (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘𝑦)))
75	72, 74	breq12d 5091	. . . . . 6 ⊢ (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +_𝑒 𝑦)) ≤ ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +_𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘𝑦))))
76		oveq2 7276	. . . . . . . 8 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +_𝑒 𝑦) = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
77	76	fveq2d 6772	. . . . . . 7 ⊢ (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +_𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
78		fveq2 6768	. . . . . . . 8 ⊢ (𝑦 = (𝑐𝐷𝑏) → (𝐹‘𝑦) = (𝐹‘(𝑐𝐷𝑏)))
79	78	oveq2d 7284	. . . . . . 7 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))))
80	77, 79	breq12d 5091	. . . . . 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 3571	. . . . 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 834	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))))
83	39, 48, 51, 68, 82	xrletrd 12878	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))))
84	23	3adantr3 1169	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
85	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
86	41, 42	opelxpd 5626	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
87	85, 86	fvco3d 6862	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
88		df-ov 7271	. . . . 5 ⊢ (𝑐(𝐹 ∘ 𝐷)𝑎) = ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑎⟩)
89		df-ov 7271	. . . . . 6 ⊢ (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
90	89	fveq2i 6771	. . . . 5 ⊢ (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
91	87, 88, 90	3eqtr4g 2804	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐(𝐹 ∘ 𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
92	41, 44	opelxpd 5626	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
93	85, 92	fvco3d 6862	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
94		df-ov 7271	. . . . 5 ⊢ (𝑐(𝐹 ∘ 𝐷)𝑏) = ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑏⟩)
95		df-ov 7271	. . . . . 6 ⊢ (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
96	95	fveq2i 6771	. . . . 5 ⊢ (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
97	93, 94, 96	3eqtr4g 2804	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
98	91, 97	oveq12d 7286	. . 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 5110	. 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 23460	1 ⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∀wral 3065 Vcvv 3430 ⟨cop 4572 class class class wbr 5078 × cxp 5586 ∘ ccom 5592 Fn wfn 6425 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 0cc0 10855 +∞cpnf 10990 ℝ^*cxr 10992 ≤ cle 10994 +_𝑒 cxad 12828 [,]cicc 13064 ∞Metcxmet 20563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-2 12019 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-icc 13068 df-xmet 20571

This theorem is referenced by: stdbdxmet 23652

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