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Theorem comet 23869
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 comet.1 . . 3 (𝜑𝐷 ∈ (∞Met‘𝑋))
21elfvexd 6881 . 2 (𝜑𝑋 ∈ V)
3 comet.2 . . 3 (𝜑𝐹:(0[,]+∞)⟶ℝ*)
4 xmetf 23682 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
51, 4syl 17 . . . . 5 (𝜑𝐷:(𝑋 × 𝑋)⟶ℝ*)
65ffnd 6669 . . . 4 (𝜑𝐷 Fn (𝑋 × 𝑋))
7 xmetcl 23684 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ ℝ*)
8 xmetge0 23697 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → 0 ≤ (𝑎𝐷𝑏))
9 elxrge0 13374 . . . . . . . 8 ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ* ∧ 0 ≤ (𝑎𝐷𝑏)))
107, 8, 9sylanbrc 583 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
11103expb 1120 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
121, 11sylan 580 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
1312ralrimivva 3197 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
14 ffnov 7483 . . . 4 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
156, 13, 14sylanbrc 583 . . 3 (𝜑𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
163, 15fcod 6694 . 2 (𝜑 → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
17 opelxpi 5670 . . . . . 6 ((𝑎𝑋𝑏𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
18 fvco3 6940 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
195, 17, 18syl2an 596 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
20 df-ov 7360 . . . . 5 (𝑎(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑎, 𝑏⟩)
21 df-ov 7360 . . . . . 6 (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
2221fveq2i 6845 . . . . 5 (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
2319, 20, 223eqtr4g 2801 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
2423eqeq1d 2738 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
25 fveq2 6842 . . . . . 6 (𝑥 = (𝑎𝐷𝑏) → (𝐹𝑥) = (𝐹‘(𝑎𝐷𝑏)))
2625eqeq1d 2738 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
27 eqeq1 2740 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → (𝑥 = 0 ↔ (𝑎𝐷𝑏) = 0))
2826, 27bibi12d 345 . . . 4 (𝑥 = (𝑎𝐷𝑏) → (((𝐹𝑥) = 0 ↔ 𝑥 = 0) ↔ ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0)))
29 comet.3 . . . . . 6 ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3029ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3130adantr 481 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
3228, 31, 12rspcdva 3582 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
33 xmeteq0 23691 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
34333expb 1120 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
351, 34sylan 580 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
3624, 32, 353bitrd 304 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ 𝑎 = 𝑏))
373adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐹:(0[,]+∞)⟶ℝ*)
38123adantr3 1171 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
3937, 38ffvelcdmd 7036 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ*)
4015adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
41 simpr3 1196 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑐𝑋)
42 simpr1 1194 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑎𝑋)
4340, 41, 42fovcdmd 7526 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
44 simpr2 1195 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑏𝑋)
4540, 41, 44fovcdmd 7526 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
46 ge0xaddcl 13379 . . . . . 6 (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
4743, 45, 46syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
4837, 47ffvelcdmd 7036 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ∈ ℝ*)
4937, 43ffvelcdmd 7036 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ*)
5037, 45ffvelcdmd 7036 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ*)
5149, 50xaddcld 13220 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ*)
52 3anrot 1100 . . . . . . 7 ((𝑐𝑋𝑎𝑋𝑏𝑋) ↔ (𝑎𝑋𝑏𝑋𝑐𝑋))
53 xmettri2 23693 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
5452, 53sylan2br 595 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
551, 54sylan 580 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
56 comet.4 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
5756ralrimivva 3197 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
5857adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
59 breq1 5108 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → (𝑥𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
6025breq1d 5115 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)))
6159, 60imbi12d 344 . . . . . . 7 (𝑥 = (𝑎𝐷𝑏) → ((𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦))))
62 breq2 5109 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
63 fveq2 6842 . . . . . . . . 9 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹𝑦) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
6463breq2d 5117 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6562, 64imbi12d 344 . . . . . . 7 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
6661, 65rspc2va 3591 . . . . . 6 ((((𝑎𝐷𝑏) ∈ (0[,]+∞) ∧ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞)) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦))) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6738, 47, 58, 66syl21anc 836 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6855, 67mpd 15 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
69 comet.5 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7069ralrimivva 3197 . . . . . 6 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7170adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
72 fvoveq1 7380 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)))
73 fveq2 6842 . . . . . . . 8 (𝑥 = (𝑐𝐷𝑎) → (𝐹𝑥) = (𝐹‘(𝑐𝐷𝑎)))
7473oveq1d 7372 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → ((𝐹𝑥) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)))
7572, 74breq12d 5118 . . . . . 6 (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦))))
76 oveq2 7365 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +𝑒 𝑦) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
7776fveq2d 6846 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
78 fveq2 6842 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → (𝐹𝑦) = (𝐹‘(𝑐𝐷𝑏)))
7978oveq2d 7373 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8077, 79breq12d 5118 . . . . . 6 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏)))))
8175, 80rspc2va 3591 . . . . 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))))
8243, 45, 71, 81syl21anc 836 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8339, 48, 51, 68, 82xrletrd 13081 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
84233adantr3 1171 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
855adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
8641, 42opelxpd 5671 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
8785, 86fvco3d 6941 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
88 df-ov 7360 . . . . 5 (𝑐(𝐹𝐷)𝑎) = ((𝐹𝐷)‘⟨𝑐, 𝑎⟩)
89 df-ov 7360 . . . . . 6 (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
9089fveq2i 6845 . . . . 5 (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
9187, 88, 903eqtr4g 2801 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
9241, 44opelxpd 5671 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
9385, 92fvco3d 6941 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
94 df-ov 7360 . . . . 5 (𝑐(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑐, 𝑏⟩)
95 df-ov 7360 . . . . . 6 (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
9695fveq2i 6845 . . . . 5 (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
9793, 94, 963eqtr4g 2801 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
9891, 97oveq12d 7375 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
9983, 84, 983brtr4d 5137 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) ≤ ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)))
1002, 16, 36, 99isxmetd 23679 1 (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cop 4592   class class class wbr 5105   × cxp 5631  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  +∞cpnf 11186  *cxr 11188  cle 11190   +𝑒 cxad 13031  [,]cicc 13267  ∞Metcxmet 20781
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-2 12216  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-xmet 20789
This theorem is referenced by:  stdbdxmet  23871
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