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Theorem dscmet 22789
Description: The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
Hypothesis
Ref Expression
dscmet.1 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))
Assertion
Ref Expression
dscmet (𝑋𝑉𝐷 ∈ (Met‘𝑋))
Distinct variable group:   𝑥,𝑦,𝑋
Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem dscmet
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 10380 . . . . . 6 0 ∈ ℝ
2 1re 10378 . . . . . 6 1 ∈ ℝ
31, 2ifcli 4353 . . . . 5 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
43rgen2w 3107 . . . 4 𝑥𝑋𝑦𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
5 dscmet.1 . . . . 5 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))
65fmpt2 7519 . . . 4 (∀𝑥𝑋𝑦𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
74, 6mpbi 222 . . 3 𝐷:(𝑋 × 𝑋)⟶ℝ
8 equequ1 2072 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
98ifbid 4329 . . . . . . . 8 (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10 equequ2 2073 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑤 = 𝑦𝑤 = 𝑣))
1110ifbid 4329 . . . . . . . 8 (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12 0nn0 11663 . . . . . . . . . 10 0 ∈ ℕ0
13 1nn0 11664 . . . . . . . . . 10 1 ∈ ℕ0
1412, 13ifcli 4353 . . . . . . . . 9 if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
1514elexi 3415 . . . . . . . 8 if(𝑤 = 𝑣, 0, 1) ∈ V
169, 11, 5, 15ovmpt2 7075 . . . . . . 7 ((𝑤𝑋𝑣𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
1716eqeq1d 2780 . . . . . 6 ((𝑤𝑋𝑣𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18 iffalse 4316 . . . . . . . . . 10 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19 ax-1ne0 10343 . . . . . . . . . . 11 1 ≠ 0
2019a1i 11 . . . . . . . . . 10 𝑤 = 𝑣 → 1 ≠ 0)
2118, 20eqnetrd 3036 . . . . . . . . 9 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
2221neneqd 2974 . . . . . . . 8 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
2322con4i 114 . . . . . . 7 (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24 iftrue 4313 . . . . . . 7 (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
2523, 24impbii 201 . . . . . 6 (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
2617, 25syl6bb 279 . . . . 5 ((𝑤𝑋𝑣𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
2712, 13ifcli 4353 . . . . . . . . . . 11 if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
2812, 13ifcli 4353 . . . . . . . . . . 11 if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
2927, 28nn0addcli 11685 . . . . . . . . . 10 (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30 elnn0 11648 . . . . . . . . . 10 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0 ↔ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ ∨ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0))
3129, 30mpbi 222 . . . . . . . . 9 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ ∨ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0)
32 breq1 4891 . . . . . . . . . . . 12 (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33 breq1 4891 . . . . . . . . . . . 12 (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34 0le1 10900 . . . . . . . . . . . 12 0 ≤ 1
352leidi 10911 . . . . . . . . . . . 12 1 ≤ 1
3632, 33, 34, 35keephyp 4376 . . . . . . . . . . 11 if(𝑤 = 𝑣, 0, 1) ≤ 1
37 nnge1 11408 . . . . . . . . . . 11 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ → 1 ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
3814nn0rei 11658 . . . . . . . . . . . 12 if(𝑤 = 𝑣, 0, 1) ∈ ℝ
3929nn0rei 11658 . . . . . . . . . . . 12 (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
4038, 2, 39letri 10507 . . . . . . . . . . 11 ((if(𝑤 = 𝑣, 0, 1) ≤ 1 ∧ 1 ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1))) → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
4136, 37, 40sylancr 581 . . . . . . . . . 10 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
4227nn0ge0i 11675 . . . . . . . . . . . . 13 0 ≤ if(𝑢 = 𝑤, 0, 1)
4328nn0ge0i 11675 . . . . . . . . . . . . 13 0 ≤ if(𝑢 = 𝑣, 0, 1)
4427nn0rei 11658 . . . . . . . . . . . . . 14 if(𝑢 = 𝑤, 0, 1) ∈ ℝ
4528nn0rei 11658 . . . . . . . . . . . . . 14 if(𝑢 = 𝑣, 0, 1) ∈ ℝ
4644, 45add20i 10920 . . . . . . . . . . . . 13 ((0 ≤ if(𝑢 = 𝑤, 0, 1) ∧ 0 ≤ if(𝑢 = 𝑣, 0, 1)) → ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0)))
4742, 43, 46mp2an 682 . . . . . . . . . . . 12 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0))
48 equequ2 2073 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → (𝑢 = 𝑣𝑢 = 𝑤))
4948ifbid 4329 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
5049eqeq1d 2780 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
5150, 48bibi12d 337 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52 equequ1 2072 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (𝑤 = 𝑣𝑢 = 𝑣))
5352ifbid 4329 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
5453eqeq1d 2780 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
5554, 52bibi12d 337 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
5655, 25chvarv 2361 . . . . . . . . . . . . . . . 16 (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
5751, 56chvarv 2361 . . . . . . . . . . . . . . 15 (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58 eqtr2 2800 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑤𝑢 = 𝑣) → 𝑤 = 𝑣)
5957, 56, 58syl2anb 591 . . . . . . . . . . . . . 14 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
6059iftrued 4315 . . . . . . . . . . . . 13 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
611leidi 10911 . . . . . . . . . . . . 13 0 ≤ 0
6260, 61syl6eqbr 4927 . . . . . . . . . . . 12 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
6347, 62sylbi 209 . . . . . . . . . . 11 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → if(𝑤 = 𝑣, 0, 1) ≤ 0)
64 id 22 . . . . . . . . . . 11 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0)
6563, 64breqtrrd 4916 . . . . . . . . . 10 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
6641, 65jaoi 846 . . . . . . . . 9 (((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ ∨ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
6731, 66mp1i 13 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
6816adantl 475 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69 eqeq12 2791 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑤) → (𝑥 = 𝑦𝑢 = 𝑤))
7069ifbid 4329 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
7127elexi 3415 . . . . . . . . . . 11 if(𝑢 = 𝑤, 0, 1) ∈ V
7270, 5, 71ovmpt2a 7070 . . . . . . . . . 10 ((𝑢𝑋𝑤𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
7372adantrr 707 . . . . . . . . 9 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74 eqeq12 2791 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑥 = 𝑦𝑢 = 𝑣))
7574ifbid 4329 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
7628elexi 3415 . . . . . . . . . . 11 if(𝑢 = 𝑣, 0, 1) ∈ V
7775, 5, 76ovmpt2a 7070 . . . . . . . . . 10 ((𝑢𝑋𝑣𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
7877adantrl 706 . . . . . . . . 9 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
7973, 78oveq12d 6942 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
8067, 68, 793brtr4d 4920 . . . . . . 7 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
8180expcom 404 . . . . . 6 ((𝑤𝑋𝑣𝑋) → (𝑢𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
8281ralrimiv 3147 . . . . 5 ((𝑤𝑋𝑣𝑋) → ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
8326, 82jca 507 . . . 4 ((𝑤𝑋𝑣𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
8483rgen2a 3159 . . 3 𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
857, 84pm3.2i 464 . 2 (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86 ismet 22540 . 2 (𝑋𝑉 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))))
8785, 86mpbiri 250 1 (𝑋𝑉𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836   = wceq 1601  wcel 2107  wne 2969  wral 3090  ifcif 4307   class class class wbr 4888   × cxp 5355  wf 6133  cfv 6137  (class class class)co 6924  cmpt2 6926  cr 10273  0cc0 10274  1c1 10275   + caddc 10277  cle 10414  cn 11378  0cn0 11646  Metcmet 20132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-er 8028  df-map 8144  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-n0 11647  df-met 20140
This theorem is referenced by:  dscopn  22790
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