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Theorem dscmet 23928
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 11157 . . . . . 6 0 ∈ ℝ
2 1re 11155 . . . . . 6 1 ∈ ℝ
31, 2ifcli 4533 . . . . 5 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
43rgen2w 3069 . . . 4 𝑥𝑋𝑦𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
5 dscmet.1 . . . . 5 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))
65fmpo 8000 . . . 4 (∀𝑥𝑋𝑦𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
74, 6mpbi 229 . . 3 𝐷:(𝑋 × 𝑋)⟶ℝ
8 equequ1 2028 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
98ifbid 4509 . . . . . . . 8 (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10 equequ2 2029 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑤 = 𝑦𝑤 = 𝑣))
1110ifbid 4509 . . . . . . . 8 (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12 0nn0 12428 . . . . . . . . . 10 0 ∈ ℕ0
13 1nn0 12429 . . . . . . . . . 10 1 ∈ ℕ0
1412, 13ifcli 4533 . . . . . . . . 9 if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
1514elexi 3464 . . . . . . . 8 if(𝑤 = 𝑣, 0, 1) ∈ V
169, 11, 5, 15ovmpo 7515 . . . . . . 7 ((𝑤𝑋𝑣𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
1716eqeq1d 2738 . . . . . 6 ((𝑤𝑋𝑣𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18 iffalse 4495 . . . . . . . . . 10 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19 ax-1ne0 11120 . . . . . . . . . . 11 1 ≠ 0
2019a1i 11 . . . . . . . . . 10 𝑤 = 𝑣 → 1 ≠ 0)
2118, 20eqnetrd 3011 . . . . . . . . 9 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
2221neneqd 2948 . . . . . . . 8 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
2322con4i 114 . . . . . . 7 (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24 iftrue 4492 . . . . . . 7 (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
2523, 24impbii 208 . . . . . 6 (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
2617, 25bitrdi 286 . . . . 5 ((𝑤𝑋𝑣𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
2712, 13ifcli 4533 . . . . . . . . . . 11 if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
2812, 13ifcli 4533 . . . . . . . . . . 11 if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
2927, 28nn0addcli 12450 . . . . . . . . . 10 (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30 elnn0 12415 . . . . . . . . . 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 229 . . . . . . . . 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 5108 . . . . . . . . . . . 12 (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33 breq1 5108 . . . . . . . . . . . 12 (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34 0le1 11678 . . . . . . . . . . . 12 0 ≤ 1
352leidi 11689 . . . . . . . . . . . 12 1 ≤ 1
3632, 33, 34, 35keephyp 4557 . . . . . . . . . . 11 if(𝑤 = 𝑣, 0, 1) ≤ 1
37 nnge1 12181 . . . . . . . . . . 11 ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ → 1 ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
3814nn0rei 12424 . . . . . . . . . . . 12 if(𝑤 = 𝑣, 0, 1) ∈ ℝ
3929nn0rei 12424 . . . . . . . . . . . 12 (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
4038, 2, 39letri 11284 . . . . . . . . . . 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 587 . . . . . . . . . 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 12440 . . . . . . . . . . . . 13 0 ≤ if(𝑢 = 𝑤, 0, 1)
4328nn0ge0i 12440 . . . . . . . . . . . . 13 0 ≤ if(𝑢 = 𝑣, 0, 1)
4427nn0rei 12424 . . . . . . . . . . . . . 14 if(𝑢 = 𝑤, 0, 1) ∈ ℝ
4528nn0rei 12424 . . . . . . . . . . . . . 14 if(𝑢 = 𝑣, 0, 1) ∈ ℝ
4644, 45add20i 11698 . . . . . . . . . . . . 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 690 . . . . . . . . . . . 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 2029 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → (𝑢 = 𝑣𝑢 = 𝑤))
4948ifbid 4509 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
5049eqeq1d 2738 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
5150, 48bibi12d 345 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52 equequ1 2028 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (𝑤 = 𝑣𝑢 = 𝑣))
5352ifbid 4509 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
5453eqeq1d 2738 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
5554, 52bibi12d 345 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
5655, 25chvarvv 2002 . . . . . . . . . . . . . . . 16 (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
5751, 56chvarvv 2002 . . . . . . . . . . . . . . 15 (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58 eqtr2 2760 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑤𝑢 = 𝑣) → 𝑤 = 𝑣)
5957, 56, 58syl2anb 598 . . . . . . . . . . . . . 14 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
6059iftrued 4494 . . . . . . . . . . . . 13 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
611leidi 11689 . . . . . . . . . . . . 13 0 ≤ 0
6260, 61eqbrtrdi 5144 . . . . . . . . . . . 12 ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
6347, 62sylbi 216 . . . . . . . . . . 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 5133 . . . . . . . . . 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 855 . . . . . . . . 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 482 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69 eqeq12 2753 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑤) → (𝑥 = 𝑦𝑢 = 𝑤))
7069ifbid 4509 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
7127elexi 3464 . . . . . . . . . . 11 if(𝑢 = 𝑤, 0, 1) ∈ V
7270, 5, 71ovmpoa 7510 . . . . . . . . . 10 ((𝑢𝑋𝑤𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
7372adantrr 715 . . . . . . . . 9 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74 eqeq12 2753 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑥 = 𝑦𝑢 = 𝑣))
7574ifbid 4509 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
7628elexi 3464 . . . . . . . . . . 11 if(𝑢 = 𝑣, 0, 1) ∈ V
7775, 5, 76ovmpoa 7510 . . . . . . . . . 10 ((𝑢𝑋𝑣𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
7877adantrl 714 . . . . . . . . 9 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
7973, 78oveq12d 7375 . . . . . . . 8 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
8067, 68, 793brtr4d 5137 . . . . . . 7 ((𝑢𝑋 ∧ (𝑤𝑋𝑣𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
8180expcom 414 . . . . . 6 ((𝑤𝑋𝑣𝑋) → (𝑢𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
8281ralrimiv 3142 . . . . 5 ((𝑤𝑋𝑣𝑋) → ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
8326, 82jca 512 . . . 4 ((𝑤𝑋𝑣𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
8483rgen2 3194 . . 3 𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
857, 84pm3.2i 471 . 2 (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86 ismet 23676 . 2 (𝑋𝑉 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤𝑋𝑣𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))))
8785, 86mpbiri 257 1 (𝑋𝑉𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  ifcif 4486   class class class wbr 5105   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  cle 11190  cn 12153  0cn0 12413  Metcmet 20782
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-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-pss 3929  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-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  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-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  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-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  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-nn 12154  df-n0 12414  df-met 20790
This theorem is referenced by:  dscopn  23929
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