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Theorem comet 22517
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 6381 . 2 (𝜑𝑋 ∈ V)
3 comet.2 . . 3 (𝜑𝐹:(0[,]+∞)⟶ℝ*)
4 xmetf 22333 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
51, 4syl 17 . . . . 5 (𝜑𝐷:(𝑋 × 𝑋)⟶ℝ*)
6 ffn 6204 . . . . 5 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
75, 6syl 17 . . . 4 (𝜑𝐷 Fn (𝑋 × 𝑋))
8 xmetcl 22335 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ ℝ*)
9 xmetge0 22348 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → 0 ≤ (𝑎𝐷𝑏))
10 elxrge0 12472 . . . . . . . 8 ((𝑎𝐷𝑏) ∈ (0[,]+∞) ↔ ((𝑎𝐷𝑏) ∈ ℝ* ∧ 0 ≤ (𝑎𝐷𝑏)))
118, 9, 10sylanbrc 701 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
12113expb 1114 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
131, 12sylan 489 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
1413ralrimivva 3107 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞))
15 ffnov 6927 . . . 4 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝐷𝑏) ∈ (0[,]+∞)))
167, 14, 15sylanbrc 701 . . 3 (𝜑𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
17 fco 6217 . . 3 ((𝐹:(0[,]+∞)⟶ℝ*𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
183, 16, 17syl2anc 696 . 2 (𝜑 → (𝐹𝐷):(𝑋 × 𝑋)⟶ℝ*)
19 opelxpi 5303 . . . . . 6 ((𝑎𝑋𝑏𝑋) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋))
20 fvco3 6435 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
215, 19, 20syl2an 495 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝐷)‘⟨𝑎, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩)))
22 df-ov 6814 . . . . 5 (𝑎(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑎, 𝑏⟩)
23 df-ov 6814 . . . . . 6 (𝑎𝐷𝑏) = (𝐷‘⟨𝑎, 𝑏⟩)
2423fveq2i 6353 . . . . 5 (𝐹‘(𝑎𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑎, 𝑏⟩))
2521, 22, 243eqtr4g 2817 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
2625eqeq1d 2760 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
27 comet.3 . . . . . 6 ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))
2827ralrimiva 3102 . . . . 5 (𝜑 → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
2928adantr 472 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0))
30 fveq2 6350 . . . . . . 7 (𝑥 = (𝑎𝐷𝑏) → (𝐹𝑥) = (𝐹‘(𝑎𝐷𝑏)))
3130eqeq1d 2760 . . . . . 6 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑎𝐷𝑏)) = 0))
32 eqeq1 2762 . . . . . 6 (𝑥 = (𝑎𝐷𝑏) → (𝑥 = 0 ↔ (𝑎𝐷𝑏) = 0))
3331, 32bibi12d 334 . . . . 5 (𝑥 = (𝑎𝐷𝑏) → (((𝐹𝑥) = 0 ↔ 𝑥 = 0) ↔ ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0)))
3433rspcv 3443 . . . 4 ((𝑎𝐷𝑏) ∈ (0[,]+∞) → (∀𝑥 ∈ (0[,]+∞)((𝐹𝑥) = 0 ↔ 𝑥 = 0) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0)))
3513, 29, 34sylc 65 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹‘(𝑎𝐷𝑏)) = 0 ↔ (𝑎𝐷𝑏) = 0))
36 xmeteq0 22342 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
37363expb 1114 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
381, 37sylan 489 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎𝐷𝑏) = 0 ↔ 𝑎 = 𝑏))
3926, 35, 383bitrd 294 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋)) → ((𝑎(𝐹𝐷)𝑏) = 0 ↔ 𝑎 = 𝑏))
403adantr 472 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐹:(0[,]+∞)⟶ℝ*)
41133adantr3 1177 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ∈ (0[,]+∞))
4240, 41ffvelrnd 6521 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ∈ ℝ*)
4316adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
44 simpr3 1238 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑐𝑋)
45 simpr1 1234 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑎𝑋)
4643, 44, 45fovrnd 6969 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑎) ∈ (0[,]+∞))
47 simpr2 1236 . . . . . . 7 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝑏𝑋)
4843, 44, 47fovrnd 6969 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐𝐷𝑏) ∈ (0[,]+∞))
49 ge0xaddcl 12477 . . . . . 6 (((𝑐𝐷𝑎) ∈ (0[,]+∞) ∧ (𝑐𝐷𝑏) ∈ (0[,]+∞)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
5046, 48, 49syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞))
5140, 50ffvelrnd 6521 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ∈ ℝ*)
5240, 46ffvelrnd 6521 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑎)) ∈ ℝ*)
5340, 48ffvelrnd 6521 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑐𝐷𝑏)) ∈ ℝ*)
5452, 53xaddcld 12322 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))) ∈ ℝ*)
55 3anrot 1087 . . . . . . 7 ((𝑐𝑋𝑎𝑋𝑏𝑋) ↔ (𝑎𝑋𝑏𝑋𝑐𝑋))
56 xmettri2 22344 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
5755, 56sylan2br 494 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
581, 57sylan 489 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
59 comet.4 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
6059ralrimivva 3107 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
6160adantr 472 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))
62 breq1 4805 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → (𝑥𝑦 ↔ (𝑎𝐷𝑏) ≤ 𝑦))
6330breq1d 4812 . . . . . . . 8 (𝑥 = (𝑎𝐷𝑏) → ((𝐹𝑥) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)))
6462, 63imbi12d 333 . . . . . . 7 (𝑥 = (𝑎𝐷𝑏) → ((𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦))))
65 breq2 4806 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝑎𝐷𝑏) ≤ 𝑦 ↔ (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
66 fveq2 6350 . . . . . . . . 9 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹𝑦) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
6766breq2d 4814 . . . . . . . 8 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → ((𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦) ↔ (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6865, 67imbi12d 333 . . . . . . 7 (𝑦 = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (((𝑎𝐷𝑏) ≤ 𝑦 → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹𝑦)) ↔ ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
6964, 68rspc2va 3460 . . . . . 6 ((((𝑎𝐷𝑏) ∈ (0[,]+∞) ∧ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ∈ (0[,]+∞)) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦))) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
7041, 50, 61, 69syl21anc 1476 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
7158, 70mpd 15 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
72 comet.5 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7372ralrimivva 3107 . . . . . 6 (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
7473adantr 472 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))
75 oveq1 6818 . . . . . . . 8 (𝑥 = (𝑐𝐷𝑎) → (𝑥 +𝑒 𝑦) = ((𝑐𝐷𝑎) +𝑒 𝑦))
7675fveq2d 6354 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → (𝐹‘(𝑥 +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)))
77 fveq2 6350 . . . . . . . 8 (𝑥 = (𝑐𝐷𝑎) → (𝐹𝑥) = (𝐹‘(𝑐𝐷𝑎)))
7877oveq1d 6826 . . . . . . 7 (𝑥 = (𝑐𝐷𝑎) → ((𝐹𝑥) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)))
7976, 78breq12d 4815 . . . . . 6 (𝑥 = (𝑐𝐷𝑎) → ((𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦))))
80 oveq2 6819 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → ((𝑐𝐷𝑎) +𝑒 𝑦) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
8180fveq2d 6354 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → (𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) = (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
82 fveq2 6350 . . . . . . . 8 (𝑦 = (𝑐𝐷𝑏) → (𝐹𝑦) = (𝐹‘(𝑐𝐷𝑏)))
8382oveq2d 6827 . . . . . . 7 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8481, 83breq12d 4815 . . . . . 6 (𝑦 = (𝑐𝐷𝑏) → ((𝐹‘((𝑐𝐷𝑎) +𝑒 𝑦)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹𝑦)) ↔ (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏)))))
8579, 84rspc2va 3460 . . . . 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))))
8646, 48, 74, 85syl21anc 1476 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
8742, 51, 54, 71, 86xrletrd 12184 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝐹‘(𝑎𝐷𝑏)) ≤ ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
88253adantr3 1177 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) = (𝐹‘(𝑎𝐷𝑏)))
895adantr 472 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
90 opelxpi 5303 . . . . . . 7 ((𝑐𝑋𝑎𝑋) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
9144, 45, 90syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋))
92 fvco3 6435 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑐, 𝑎⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
9389, 91, 92syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑎⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩)))
94 df-ov 6814 . . . . 5 (𝑐(𝐹𝐷)𝑎) = ((𝐹𝐷)‘⟨𝑐, 𝑎⟩)
95 df-ov 6814 . . . . . 6 (𝑐𝐷𝑎) = (𝐷‘⟨𝑐, 𝑎⟩)
9695fveq2i 6353 . . . . 5 (𝐹‘(𝑐𝐷𝑎)) = (𝐹‘(𝐷‘⟨𝑐, 𝑎⟩))
9793, 94, 963eqtr4g 2817 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑎) = (𝐹‘(𝑐𝐷𝑎)))
98 opelxpi 5303 . . . . . . 7 ((𝑐𝑋𝑏𝑋) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
9944, 47, 98syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋))
100 fvco3 6435 . . . . . 6 ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ⟨𝑐, 𝑏⟩ ∈ (𝑋 × 𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
10189, 99, 100syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝐹𝐷)‘⟨𝑐, 𝑏⟩) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩)))
102 df-ov 6814 . . . . 5 (𝑐(𝐹𝐷)𝑏) = ((𝐹𝐷)‘⟨𝑐, 𝑏⟩)
103 df-ov 6814 . . . . . 6 (𝑐𝐷𝑏) = (𝐷‘⟨𝑐, 𝑏⟩)
104103fveq2i 6353 . . . . 5 (𝐹‘(𝑐𝐷𝑏)) = (𝐹‘(𝐷‘⟨𝑐, 𝑏⟩))
105101, 102, 1043eqtr4g 2817 . . . 4 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑐(𝐹𝐷)𝑏) = (𝐹‘(𝑐𝐷𝑏)))
10697, 105oveq12d 6829 . . 3 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)) = ((𝐹‘(𝑐𝐷𝑎)) +𝑒 (𝐹‘(𝑐𝐷𝑏))))
10787, 88, 1063brtr4d 4834 . 2 ((𝜑 ∧ (𝑎𝑋𝑏𝑋𝑐𝑋)) → (𝑎(𝐹𝐷)𝑏) ≤ ((𝑐(𝐹𝐷)𝑎) +𝑒 (𝑐(𝐹𝐷)𝑏)))
1082, 18, 39, 107isxmetd 22330 1 (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  cop 4325   class class class wbr 4802   × cxp 5262  ccom 5268   Fn wfn 6042  wf 6043  cfv 6047  (class class class)co 6811  0cc0 10126  +∞cpnf 10261  *cxr 10263  cle 10265   +𝑒 cxad 12135  [,]cicc 12369  ∞Metcxmt 19931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-po 5185  df-so 5186  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-2 11269  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-icc 12373  df-xmet 19939
This theorem is referenced by:  stdbdxmet  22519
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