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Theorem dscmet 22187
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 9919 . . . . . 6 0 ∈ ℝ
2 1re 9918 . . . . . 6 1 ∈ ℝ
31, 2keepel 4105 . . . . 5 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
43rgen2w 2909 . . . 4 𝑥𝑋𝑦𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
5 dscmet.1 . . . . 5 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))
65fmpt2 7126 . . . 4 (∀𝑥𝑋𝑦𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
74, 6mpbi 219 . . 3 𝐷:(𝑋 × 𝑋)⟶ℝ
8 equequ1 1939 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
98ifbid 4058 . . . . . . . 8 (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10 equequ2 1940 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑤 = 𝑦𝑤 = 𝑣))
1110ifbid 4058 . . . . . . . 8 (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12 0nn0 11184 . . . . . . . . . 10 0 ∈ ℕ0
13 1nn0 11185 . . . . . . . . . 10 1 ∈ ℕ0
1412, 13keepel 4105 . . . . . . . . 9 if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
1514elexi 3186 . . . . . . . 8 if(𝑤 = 𝑣, 0, 1) ∈ V
169, 11, 5, 15ovmpt2 6694 . . . . . . 7 ((𝑤𝑋𝑣𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
1716eqeq1d 2612 . . . . . 6 ((𝑤𝑋𝑣𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18 iffalse 4045 . . . . . . . . . 10 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19 ax-1ne0 9884 . . . . . . . . . . 11 1 ≠ 0
2019a1i 11 . . . . . . . . . 10 𝑤 = 𝑣 → 1 ≠ 0)
2118, 20eqnetrd 2849 . . . . . . . . 9 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
2221neneqd 2787 . . . . . . . 8 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
2322con4i 112 . . . . . . 7 (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24 iftrue 4042 . . . . . . 7 (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
2523, 24impbii 198 . . . . . 6 (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
2617, 25syl6bb 275 . . . . 5 ((𝑤𝑋𝑣𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
2712, 13keepel 4105 . . . . . . . . . . 11 if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
2812, 13keepel 4105 . . . . . . . . . . 11 if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
2927, 28nn0addcli 11207 . . . . . . . . . 10 (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30 elnn0 11171 . . . . . . . . . 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 219 . . . . . . . . 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 4586 . . . . . . . . . . . 12 (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33 breq1 4586 . . . . . . . . . . . 12 (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34 0le1 10430 . . . . . . . . . . . 12 0 ≤ 1
352leidi 10441 . . . . . . . . . . . 12 1 ≤ 1
3632, 33, 34, 35keephyp 4102 . . . . . . . . . . 11 if(𝑤 = 𝑣, 0, 1) ≤ 1
37 nnge1 10923 . . . . . . . . . . 11 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ → 1 ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
3814nn0rei 11180 . . . . . . . . . . . 12 if(𝑤 = 𝑣, 0, 1) ∈ ℝ
3929nn0rei 11180 . . . . . . . . . . . 12 (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
4038, 2, 39letri 10045 . . . . . . . . . . 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 694 . . . . . . . . . 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 11197 . . . . . . . . . . . . 13 0 ≤ if(𝑢 = 𝑤, 0, 1)
4328nn0ge0i 11197 . . . . . . . . . . . . 13 0 ≤ if(𝑢 = 𝑣, 0, 1)
4427nn0rei 11180 . . . . . . . . . . . . . 14 if(𝑢 = 𝑤, 0, 1) ∈ ℝ
4528nn0rei 11180 . . . . . . . . . . . . . 14 if(𝑢 = 𝑣, 0, 1) ∈ ℝ
4644, 45add20i 10450 . . . . . . . . . . . . 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 704 . . . . . . . . . . . 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 1940 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → (𝑢 = 𝑣𝑢 = 𝑤))
4948ifbid 4058 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
5049eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
5150, 48bibi12d 334 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52 equequ1 1939 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (𝑤 = 𝑣𝑢 = 𝑣))
5352ifbid 4058 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
5453eqeq1d 2612 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
5554, 52bibi12d 334 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
5655, 25chvarv 2251 . . . . . . . . . . . . . . . 16 (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
5751, 56chvarv 2251 . . . . . . . . . . . . . . 15 (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58 eqtr2 2630 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑤𝑢 = 𝑣) → 𝑤 = 𝑣)
5957, 56, 58syl2anb 495 . . . . . . . . . . . . . 14 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
6059iftrued 4044 . . . . . . . . . . . . 13 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
611leidi 10441 . . . . . . . . . . . . 13 0 ≤ 0
6260, 61syl6eqbr 4622 . . . . . . . . . . . 12 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
6347, 62sylbi 206 . . . . . . . . . . 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 4611 . . . . . . . . . 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 393 . . . . . . . . 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 481 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69 eqeq12 2623 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑤) → (𝑥 = 𝑦𝑢 = 𝑤))
7069ifbid 4058 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
7127elexi 3186 . . . . . . . . . . 11 if(𝑢 = 𝑤, 0, 1) ∈ V
7270, 5, 71ovmpt2a 6689 . . . . . . . . . 10 ((𝑢𝑋𝑤𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
7372adantrr 749 . . . . . . . . 9 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74 eqeq12 2623 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑥 = 𝑦𝑢 = 𝑣))
7574ifbid 4058 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
7628elexi 3186 . . . . . . . . . . 11 if(𝑢 = 𝑣, 0, 1) ∈ V
7775, 5, 76ovmpt2a 6689 . . . . . . . . . 10 ((𝑢𝑋𝑣𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
7877adantrl 748 . . . . . . . . 9 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
7973, 78oveq12d 6567 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
8067, 68, 793brtr4d 4615 . . . . . . 7 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
8180expcom 450 . . . . . 6 ((𝑤𝑋𝑣𝑋) → (𝑢𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
8281ralrimiv 2948 . . . . 5 ((𝑤𝑋𝑣𝑋) → ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
8326, 82jca 553 . . . 4 ((𝑤𝑋𝑣𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
8483rgen2a 2960 . . 3 𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
857, 84pm3.2i 470 . 2 (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86 ismet 21938 . 2 (𝑋𝑉 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))))
8785, 86mpbiri 247 1 (𝑋𝑉𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  ifcif 4036   class class class wbr 4583   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  cr 9814  0cc0 9815  1c1 9816   + caddc 9818  cle 9954  cn 10897  0cn0 11169  Metcme 19553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-met 19561
This theorem is referenced by:  dscopn  22188
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