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Theorem comet 12694
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 12538 . . . 4 Rel ∞Met
2 comet.1 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
3 relelfvdm 5456 . . . 4 ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met)
41, 2, 3sylancr 410 . . 3 (𝜑𝑋 ∈ dom ∞Met)
54elexd 2699 . 2 (𝜑𝑋 ∈ V)
6 comet.2 . . 3 (𝜑𝐹:(0[,]+∞)⟶ℝ*)
7 xmetf 12545 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
82, 7syl 14 . . . . 5 (𝜑𝐷:(𝑋 × 𝑋)⟶ℝ*)
98ffnd 5276 . . . 4 (𝜑𝐷 Fn (𝑋 × 𝑋))
10 xmetcl 12547 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ ℝ*)
11 xmetge0 12560 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → 0 ≤ (𝑎𝐷𝑏))
12 elxrge0 9784 . . . . . . . 8 ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ* ∧ 0 ≤ (𝑎𝐷𝑏)))
1310, 11, 12sylanbrc 413 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
14133expb 1182 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
152, 14sylan 281 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
1615ralrimivva 2514 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
17 ffnov 5878 . . . 4 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
189, 16, 17sylanbrc 413 . . 3 (𝜑𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
19 fco 5291 . . 3 ((𝐹:(0[,]+∞)⟶ℝ*𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
206, 18, 19syl2anc 408 . 2 (𝜑 → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
21 opelxpi 4574 . . . . . 6 ((𝑎𝑋𝑏𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
22 fvco3 5495 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
238, 21, 22syl2an 287 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
24 df-ov 5780 . . . . 5 (𝑎(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑎, 𝑏⟩)
25 df-ov 5780 . . . . . 6 (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
2625fveq2i 5427 . . . . 5 (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
2723, 24, 263eqtr4g 2197 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
2827eqeq1d 2148 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
29 fveq2 5424 . . . . . 6 (𝑥 = (𝑎𝐷𝑏) → (𝐹𝑥) = (𝐹‘(𝑎𝐷𝑏)))
3029eqeq1d 2148 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
31 eqeq1 2146 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → (𝑥 = 0 ↔ (𝑎𝐷𝑏) = 0))
3230, 31bibi12d 234 . . . 4 (𝑥 = (𝑎𝐷𝑏) → (((𝐹𝑥) = 0 ↔ 𝑥 = 0) ↔ ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0)))
33 comet.3 . . . . . 6 ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3433ralrimiva 2505 . . . . 5 (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3534adantr 274 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3632, 35, 15rspcdva 2794 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
37 xmeteq0 12554 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
38373expb 1182 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
392, 38sylan 281 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
4028, 36, 393bitrd 213 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ 𝑎 = 𝑏))
416adantr 274 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐹:(0[,]+∞)⟶ℝ*)
42153adantr3 1142 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
4341, 42ffvelrnd 5559 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ*)
4418adantr 274 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
45 simpr3 989 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑐𝑋)
46 simpr1 987 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑎𝑋)
4744, 45, 46fovrnd 5918 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
48 simpr2 988 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑏𝑋)
4944, 45, 48fovrnd 5918 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
50 ge0xaddcl 9789 . . . . . 6 (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
5147, 49, 50syl2anc 408 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
5241, 51ffvelrnd 5559 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ∈ ℝ*)
5341, 47ffvelrnd 5559 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ*)
5441, 49ffvelrnd 5559 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ*)
5553, 54xaddcld 9690 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ*)
56 3anrot 967 . . . . . . 7 ((𝑐𝑋𝑎𝑋𝑏𝑋) ↔ (𝑎𝑋𝑏𝑋𝑐𝑋))
57 xmettri2 12556 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
5856, 57sylan2br 286 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
592, 58sylan 281 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
60 comet.4 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
6160ralrimivva 2514 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
6261adantr 274 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
63 breq1 3935 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → (𝑥𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
6429breq1d 3942 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)))
6563, 64imbi12d 233 . . . . . . 7 (𝑥 = (𝑎𝐷𝑏) → ((𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦))))
66 breq2 3936 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
67 fveq2 5424 . . . . . . . . 9 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹𝑦) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
6867breq2d 3944 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6966, 68imbi12d 233 . . . . . . 7 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
7065, 69rspc2va 2803 . . . . . 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 1215 . . . . 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 2514 . . . . . 6 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7574adantr 274 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
76 fvoveq1 5800 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)))
77 fveq2 5424 . . . . . . . 8 (𝑥 = (𝑐𝐷𝑎) → (𝐹𝑥) = (𝐹‘(𝑐𝐷𝑎)))
7877oveq1d 5792 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → ((𝐹𝑥) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)))
7976, 78breq12d 3945 . . . . . 6 (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦))))
80 oveq2 5785 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +𝑒 𝑦) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
8180fveq2d 5428 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
82 fveq2 5424 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → (𝐹𝑦) = (𝐹‘(𝑐𝐷𝑏)))
8382oveq2d 5793 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8481, 83breq12d 3945 . . . . . 6 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏)))))
8579, 84rspc2va 2803 . . . . 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 1215 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8743, 52, 55, 72, 86xrletrd 9618 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
88273adantr3 1142 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
898adantr 274 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9045, 46opelxpd 4575 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
91 fvco3 5495 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
9289, 90, 91syl2anc 408 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
93 df-ov 5780 . . . . 5 (𝑐(𝐹𝐷)𝑎) = ((𝐹𝐷)‘⟨𝑐, 𝑎⟩)
94 df-ov 5780 . . . . . 6 (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
9594fveq2i 5427 . . . . 5 (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
9692, 93, 953eqtr4g 2197 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
9745, 48opelxpd 4575 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
98 fvco3 5495 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
9989, 97, 98syl2anc 408 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
100 df-ov 5780 . . . . 5 (𝑐(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑐, 𝑏⟩)
101 df-ov 5780 . . . . . 6 (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
102101fveq2i 5427 . . . . 5 (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
10399, 100, 1023eqtr4g 2197 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
10496, 103oveq12d 5795 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
10587, 88, 1043brtr4d 3963 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) ≤ ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)))
1065, 20, 40, 105isxmetd 12542 1 (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wral 2416  cop 3530   class class class wbr 3932   × cxp 4540  dom cdm 4542  ccom 4546  Rel wrel 4547   Fn wfn 5121  wf 5122  cfv 5126  (class class class)co 5777  0cc0 7639  +∞cpnf 7816  *cxr 7818  cle 7820   +𝑒 cxad 9580  [,]cicc 9697  ∞Metcxmet 12175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-mulrcl 7738  ax-addcom 7739  ax-mulcom 7740  ax-addass 7741  ax-mulass 7742  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-1rid 7746  ax-0id 7747  ax-rnegex 7748  ax-precex 7749  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755  ax-pre-mulgt0 7756
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-po 4221  df-iso 4222  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-map 6547  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-2 8798  df-xadd 9583  df-icc 9701  df-xmet 12183
This theorem is referenced by:  bdxmet  12696
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