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Theorem comet 13666
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 (𝜑𝐷 ∈ (∞Met‘𝑋))
comet.2 (𝜑𝐹:(0[,]+∞)⟶ℝ*)
comet.3 ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))
comet.4 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
comet.5 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
Assertion
Ref Expression
comet (𝜑 → (𝐹𝐷) ∈ (∞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.
StepHypRef Expression
1 xmetrel 13510 . . . 4 Rel ∞Met
2 comet.1 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
3 relelfvdm 5543 . . . 4 ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met)
41, 2, 3sylancr 414 . . 3 (𝜑𝑋 ∈ dom ∞Met)
54elexd 2750 . 2 (𝜑𝑋 ∈ V)
6 comet.2 . . 3 (𝜑𝐹:(0[,]+∞)⟶ℝ*)
7 xmetf 13517 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
82, 7syl 14 . . . . 5 (𝜑𝐷:(𝑋 × 𝑋)⟶ℝ*)
98ffnd 5362 . . . 4 (𝜑𝐷 Fn (𝑋 × 𝑋))
10 xmetcl 13519 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ ℝ*)
11 xmetge0 13532 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → 0 ≤ (𝑎𝐷𝑏))
12 elxrge0 9965 . . . . . . . 8 ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ* ∧ 0 ≤ (𝑎𝐷𝑏)))
1310, 11, 12sylanbrc 417 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
14133expb 1204 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
152, 14sylan 283 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
1615ralrimivva 2559 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
17 ffnov 5973 . . . 4 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
189, 16, 17sylanbrc 417 . . 3 (𝜑𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
19 fco 5377 . . 3 ((𝐹:(0[,]+∞)⟶ℝ*𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
206, 18, 19syl2anc 411 . 2 (𝜑 → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
21 opelxpi 4655 . . . . . 6 ((𝑎𝑋𝑏𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
22 fvco3 5583 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
238, 21, 22syl2an 289 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
24 df-ov 5872 . . . . 5 (𝑎(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑎, 𝑏⟩)
25 df-ov 5872 . . . . . 6 (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
2625fveq2i 5514 . . . . 5 (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
2723, 24, 263eqtr4g 2235 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
2827eqeq1d 2186 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
29 fveq2 5511 . . . . . 6 (𝑥 = (𝑎𝐷𝑏) → (𝐹𝑥) = (𝐹‘(𝑎𝐷𝑏)))
3029eqeq1d 2186 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
31 eqeq1 2184 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → (𝑥 = 0 ↔ (𝑎𝐷𝑏) = 0))
3230, 31bibi12d 235 . . . 4 (𝑥 = (𝑎𝐷𝑏) → (((𝐹𝑥) = 0 ↔ 𝑥 = 0) ↔ ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0)))
33 comet.3 . . . . . 6 ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3433ralrimiva 2550 . . . . 5 (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3534adantr 276 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3632, 35, 15rspcdva 2846 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
37 xmeteq0 13526 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
38373expb 1204 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
392, 38sylan 283 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
4028, 36, 393bitrd 214 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ 𝑎 = 𝑏))
416adantr 276 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐹:(0[,]+∞)⟶ℝ*)
42153adantr3 1158 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
4341, 42ffvelcdmd 5648 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ*)
4418adantr 276 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
45 simpr3 1005 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑐𝑋)
46 simpr1 1003 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑎𝑋)
4744, 45, 46fovcdmd 6013 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
48 simpr2 1004 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑏𝑋)
4944, 45, 48fovcdmd 6013 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
50 ge0xaddcl 9970 . . . . . 6 (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
5147, 49, 50syl2anc 411 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
5241, 51ffvelcdmd 5648 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ∈ ℝ*)
5341, 47ffvelcdmd 5648 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ*)
5441, 49ffvelcdmd 5648 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ*)
5553, 54xaddcld 9871 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ*)
56 3anrot 983 . . . . . . 7 ((𝑐𝑋𝑎𝑋𝑏𝑋) ↔ (𝑎𝑋𝑏𝑋𝑐𝑋))
57 xmettri2 13528 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
5856, 57sylan2br 288 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
592, 58sylan 283 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
60 comet.4 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
6160ralrimivva 2559 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
6261adantr 276 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
63 breq1 4003 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → (𝑥𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
6429breq1d 4010 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)))
6563, 64imbi12d 234 . . . . . . 7 (𝑥 = (𝑎𝐷𝑏) → ((𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦))))
66 breq2 4004 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
67 fveq2 5511 . . . . . . . . 9 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹𝑦) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
6867breq2d 4012 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6966, 68imbi12d 234 . . . . . . 7 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
7065, 69rspc2va 2855 . . . . . 6 ((((𝑎𝐷𝑏) ∈ (0[,]+∞) ∧ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞)) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦))) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
7142, 51, 62, 70syl21anc 1237 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
7259, 71mpd 13 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
73 comet.5 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7473ralrimivva 2559 . . . . . 6 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7574adantr 276 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
76 fvoveq1 5892 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)))
77 fveq2 5511 . . . . . . . 8 (𝑥 = (𝑐𝐷𝑎) → (𝐹𝑥) = (𝐹‘(𝑐𝐷𝑎)))
7877oveq1d 5884 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → ((𝐹𝑥) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)))
7976, 78breq12d 4013 . . . . . 6 (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦))))
80 oveq2 5877 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +𝑒 𝑦) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
8180fveq2d 5515 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
82 fveq2 5511 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → (𝐹𝑦) = (𝐹‘(𝑐𝐷𝑏)))
8382oveq2d 5885 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8481, 83breq12d 4013 . . . . . 6 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏)))))
8579, 84rspc2va 2855 . . . . 5 ((((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦))) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8647, 49, 75, 85syl21anc 1237 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8743, 52, 55, 72, 86xrletrd 9799 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
88273adantr3 1158 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
898adantr 276 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9045, 46opelxpd 4656 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
91 fvco3 5583 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
9289, 90, 91syl2anc 411 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
93 df-ov 5872 . . . . 5 (𝑐(𝐹𝐷)𝑎) = ((𝐹𝐷)‘⟨𝑐, 𝑎⟩)
94 df-ov 5872 . . . . . 6 (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
9594fveq2i 5514 . . . . 5 (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
9692, 93, 953eqtr4g 2235 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
9745, 48opelxpd 4656 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
98 fvco3 5583 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
9989, 97, 98syl2anc 411 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
100 df-ov 5872 . . . . 5 (𝑐(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑐, 𝑏⟩)
101 df-ov 5872 . . . . . 6 (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
102101fveq2i 5514 . . . . 5 (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
10399, 100, 1023eqtr4g 2235 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
10496, 103oveq12d 5887 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
10587, 88, 1043brtr4d 4032 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) ≤ ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)))
1065, 20, 40, 105isxmetd 13514 1 (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wral 2455  cop 3594   class class class wbr 4000   × cxp 4621  dom cdm 4623  ccom 4627  Rel wrel 4628   Fn wfn 5207  wf 5208  cfv 5212  (class class class)co 5869  0cc0 7802  +∞cpnf 7979  *cxr 7981  cle 7983   +𝑒 cxad 9757  [,]cicc 9878  ∞Metcxmet 13147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-2 8967  df-xadd 9760  df-icc 9882  df-xmet 13155
This theorem is referenced by:  bdxmet  13668
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