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Theorem comet 24013

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 6927	. 2 ⊢ (𝜑 → 𝑋 ∈ V)
3		comet.2	. . 3 ⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ^*)
4		xmetf 23826	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
6	5	ffnd 6715	. . . 4 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
7		xmetcl 23828	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎𝐷𝑏) ∈ ℝ^*)
8		xmetge0 23841	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → 0 ≤ (𝑎𝐷𝑏))
9		elxrge0 13430	. . . . . . . 8 ⊢ ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ^* ∧ 0 ≤ (𝑎𝐷𝑏)))
10	7, 8, 9	sylanbrc 583	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
11	10	3expb 1120	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
12	1, 11	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
13	12	ralrimivva 3200	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
14		ffnov 7531	. . . 4 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
15	6, 13, 14	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
16	3, 15	fcod 6740	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐷):(𝑋 × 𝑋)⟶ℝ^*)
17		opelxpi 5712	. . . . . 6 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
18		fvco3 6987	. . . . . 6 ⊢ ((𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
19	5, 17, 18	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
20		df-ov 7408	. . . . 5 ⊢ (𝑎(𝐹 ∘ 𝐷)𝑏) = ((𝐹 ∘ 𝐷)‘⟨𝑎, 𝑏⟩)
21		df-ov 7408	. . . . . 6 ⊢ (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
22	21	fveq2i 6891	. . . . 5 ⊢ (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
23	19, 20, 22	3eqtr4g 2797	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
24	23	eqeq1d 2734	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎(𝐹 ∘ 𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
25		fveq2 6888	. . . . . 6 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝐹‘𝑥) = (𝐹‘(𝑎𝐷𝑏)))
26	25	eqeq1d 2734	. . . . 5 ⊢ (𝑥 = (𝑎𝐷𝑏) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
27		eqeq1 2736	. . . . 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 3146	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
31	30	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))
32	28, 31, 12	rspcdva 3613	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
33		xmeteq0 23835	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
34	33	3expb 1120	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
35	1, 34	sylan 580	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐹:(0[,]+∞)⟶ℝ^*)
38	12	3adantr3 1171	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
39	37, 38	ffvelcdmd 7084	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ^*)
40	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
41		simpr3 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑐 ∈ 𝑋)
42		simpr1 1194	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
43	40, 41, 42	fovcdmd 7575	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
44		simpr2 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
45	40, 41, 44	fovcdmd 7575	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
46		ge0xaddcl 13435	. . . . . 6 ⊢ (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
47	43, 45, 46	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
48	37, 47	ffvelcdmd 7084	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))) ∈ ℝ^*)
49	37, 43	ffvelcdmd 7084	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ^*)
50	37, 45	ffvelcdmd 7084	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ^*)
51	49, 50	xaddcld 13276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ^*)
52		3anrot 1100	. . . . . . 7 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋))
53		xmettri2 23837	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
54	52, 53	sylan2br 595	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
55	1, 54	sylan 580	. . . . 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 3200	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
58	57	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
59		breq1 5150	. . . . . . . 8 ⊢ (𝑥 = (𝑎𝐷𝑏) → (𝑥 ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
60	25	breq1d 5157	. . . . . . . 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 5151	. . . . . . . 8 ⊢ (𝑦 = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
63		fveq2 6888	. . . . . . . . 9 ⊢ (𝑦 = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)) → (𝐹‘𝑦) = (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
64	63	breq2d 5159	. . . . . . . 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 3622	. . . . . 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 836	. . . . 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 3200	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +_𝑒 𝑦)) ≤ ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)))
71	70	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +_𝑒 𝑦)) ≤ ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)))
72		fvoveq1 7428	. . . . . . 7 ⊢ (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +_𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +_𝑒 𝑦)))
73		fveq2 6888	. . . . . . . 8 ⊢ (𝑥 = (𝑐𝐷𝑎) → (𝐹‘𝑥) = (𝐹‘(𝑐𝐷𝑎)))
74	73	oveq1d 7420	. . . . . . 7 ⊢ (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘𝑦)))
75	72, 74	breq12d 5160	. . . . . 6 ⊢ (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +_𝑒 𝑦)) ≤ ((𝐹‘𝑥) +_𝑒 (𝐹‘𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +_𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘𝑦))))
76		oveq2 7413	. . . . . . . 8 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +_𝑒 𝑦) = ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
77	76	fveq2d 6892	. . . . . . 7 ⊢ (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +_𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
78		fveq2 6888	. . . . . . . 8 ⊢ (𝑦 = (𝑐𝐷𝑏) → (𝐹‘𝑦) = (𝐹‘(𝑐𝐷𝑏)))
79	78	oveq2d 7421	. . . . . . 7 ⊢ (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))))
80	77, 79	breq12d 5160	. . . . . 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 3622	. . . . 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 836	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))))
83	39, 48, 51, 68, 82	xrletrd 13137	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +_𝑒 (𝐹‘(𝑐𝐷𝑏))))
84	23	3adantr3 1171	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑎(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
85	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
86	41, 42	opelxpd 5713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
87	85, 86	fvco3d 6988	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
88		df-ov 7408	. . . . 5 ⊢ (𝑐(𝐹 ∘ 𝐷)𝑎) = ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑎⟩)
89		df-ov 7408	. . . . . 6 ⊢ (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
90	89	fveq2i 6891	. . . . 5 ⊢ (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
91	87, 88, 90	3eqtr4g 2797	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐(𝐹 ∘ 𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
92	41, 44	opelxpd 5713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
93	85, 92	fvco3d 6988	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
94		df-ov 7408	. . . . 5 ⊢ (𝑐(𝐹 ∘ 𝐷)𝑏) = ((𝐹 ∘ 𝐷)‘⟨𝑐, 𝑏⟩)
95		df-ov 7408	. . . . . 6 ⊢ (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
96	95	fveq2i 6891	. . . . 5 ⊢ (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
97	93, 94, 96	3eqtr4g 2797	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐(𝐹 ∘ 𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
98	91, 97	oveq12d 7423	. . 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 5179	. 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 23823	1 ⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⟨cop 4633 class class class wbr 5147 × cxp 5673 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 0cc0 11106 +∞cpnf 11241 ℝ^*cxr 11243 ≤ cle 11245 +_𝑒 cxad 13086 [,]cicc 13323 ∞Metcxmet 20921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-xmet 20929

This theorem is referenced by: stdbdxmet 24015

W3C validator