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Theorem dscmet 23951
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 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(π‘₯ = 𝑦, 0, 1))
Assertion
Ref Expression
dscmet (𝑋 ∈ 𝑉 β†’ 𝐷 ∈ (Metβ€˜π‘‹))
Distinct variable group:   π‘₯,𝑦,𝑋
Allowed substitution hints:   𝐷(π‘₯,𝑦)   𝑉(π‘₯,𝑦)

Proof of Theorem dscmet
Dummy variables 𝑣 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 11165 . . . . . 6 0 ∈ ℝ
2 1re 11163 . . . . . 6 1 ∈ ℝ
31, 2ifcli 4537 . . . . 5 if(π‘₯ = 𝑦, 0, 1) ∈ ℝ
43rgen2w 3066 . . . 4 βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 if(π‘₯ = 𝑦, 0, 1) ∈ ℝ
5 dscmet.1 . . . . 5 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(π‘₯ = 𝑦, 0, 1))
65fmpo 8004 . . . 4 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 if(π‘₯ = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„)
74, 6mpbi 229 . . 3 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„
8 equequ1 2029 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ (π‘₯ = 𝑦 ↔ 𝑀 = 𝑦))
98ifbid 4513 . . . . . . . 8 (π‘₯ = 𝑀 β†’ if(π‘₯ = 𝑦, 0, 1) = if(𝑀 = 𝑦, 0, 1))
10 equequ2 2030 . . . . . . . . 9 (𝑦 = 𝑣 β†’ (𝑀 = 𝑦 ↔ 𝑀 = 𝑣))
1110ifbid 4513 . . . . . . . 8 (𝑦 = 𝑣 β†’ if(𝑀 = 𝑦, 0, 1) = if(𝑀 = 𝑣, 0, 1))
12 0nn0 12436 . . . . . . . . . 10 0 ∈ β„•0
13 1nn0 12437 . . . . . . . . . 10 1 ∈ β„•0
1412, 13ifcli 4537 . . . . . . . . 9 if(𝑀 = 𝑣, 0, 1) ∈ β„•0
1514elexi 3466 . . . . . . . 8 if(𝑀 = 𝑣, 0, 1) ∈ V
169, 11, 5, 15ovmpo 7519 . . . . . . 7 ((𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ (𝑀𝐷𝑣) = if(𝑀 = 𝑣, 0, 1))
1716eqeq1d 2735 . . . . . 6 ((𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ ((𝑀𝐷𝑣) = 0 ↔ if(𝑀 = 𝑣, 0, 1) = 0))
18 iffalse 4499 . . . . . . . . . 10 (Β¬ 𝑀 = 𝑣 β†’ if(𝑀 = 𝑣, 0, 1) = 1)
19 ax-1ne0 11128 . . . . . . . . . . 11 1 β‰  0
2019a1i 11 . . . . . . . . . 10 (Β¬ 𝑀 = 𝑣 β†’ 1 β‰  0)
2118, 20eqnetrd 3008 . . . . . . . . 9 (Β¬ 𝑀 = 𝑣 β†’ if(𝑀 = 𝑣, 0, 1) β‰  0)
2221neneqd 2945 . . . . . . . 8 (Β¬ 𝑀 = 𝑣 β†’ Β¬ if(𝑀 = 𝑣, 0, 1) = 0)
2322con4i 114 . . . . . . 7 (if(𝑀 = 𝑣, 0, 1) = 0 β†’ 𝑀 = 𝑣)
24 iftrue 4496 . . . . . . 7 (𝑀 = 𝑣 β†’ if(𝑀 = 𝑣, 0, 1) = 0)
2523, 24impbii 208 . . . . . 6 (if(𝑀 = 𝑣, 0, 1) = 0 ↔ 𝑀 = 𝑣)
2617, 25bitrdi 287 . . . . 5 ((𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ ((𝑀𝐷𝑣) = 0 ↔ 𝑀 = 𝑣))
2712, 13ifcli 4537 . . . . . . . . . . 11 if(𝑒 = 𝑀, 0, 1) ∈ β„•0
2812, 13ifcli 4537 . . . . . . . . . . 11 if(𝑒 = 𝑣, 0, 1) ∈ β„•0
2927, 28nn0addcli 12458 . . . . . . . . . 10 (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„•0
30 elnn0 12423 . . . . . . . . . 10 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„•0 ↔ ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„• ∨ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0))
3129, 30mpbi 229 . . . . . . . . 9 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„• ∨ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0)
32 breq1 5112 . . . . . . . . . . . 12 (0 = if(𝑀 = 𝑣, 0, 1) β†’ (0 ≀ 1 ↔ if(𝑀 = 𝑣, 0, 1) ≀ 1))
33 breq1 5112 . . . . . . . . . . . 12 (1 = if(𝑀 = 𝑣, 0, 1) β†’ (1 ≀ 1 ↔ if(𝑀 = 𝑣, 0, 1) ≀ 1))
34 0le1 11686 . . . . . . . . . . . 12 0 ≀ 1
352leidi 11697 . . . . . . . . . . . 12 1 ≀ 1
3632, 33, 34, 35keephyp 4561 . . . . . . . . . . 11 if(𝑀 = 𝑣, 0, 1) ≀ 1
37 nnge1 12189 . . . . . . . . . . 11 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„• β†’ 1 ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
3814nn0rei 12432 . . . . . . . . . . . 12 if(𝑀 = 𝑣, 0, 1) ∈ ℝ
3929nn0rei 12432 . . . . . . . . . . . 12 (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ ℝ
4038, 2, 39letri 11292 . . . . . . . . . . 11 ((if(𝑀 = 𝑣, 0, 1) ≀ 1 ∧ 1 ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1))) β†’ if(𝑀 = 𝑣, 0, 1) ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
4136, 37, 40sylancr 588 . . . . . . . . . 10 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„• β†’ if(𝑀 = 𝑣, 0, 1) ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
4227nn0ge0i 12448 . . . . . . . . . . . . 13 0 ≀ if(𝑒 = 𝑀, 0, 1)
4328nn0ge0i 12448 . . . . . . . . . . . . 13 0 ≀ if(𝑒 = 𝑣, 0, 1)
4427nn0rei 12432 . . . . . . . . . . . . . 14 if(𝑒 = 𝑀, 0, 1) ∈ ℝ
4528nn0rei 12432 . . . . . . . . . . . . . 14 if(𝑒 = 𝑣, 0, 1) ∈ ℝ
4644, 45add20i 11706 . . . . . . . . . . . . 13 ((0 ≀ if(𝑒 = 𝑀, 0, 1) ∧ 0 ≀ if(𝑒 = 𝑣, 0, 1)) β†’ ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0 ↔ (if(𝑒 = 𝑀, 0, 1) = 0 ∧ if(𝑒 = 𝑣, 0, 1) = 0)))
4742, 43, 46mp2an 691 . . . . . . . . . . . 12 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0 ↔ (if(𝑒 = 𝑀, 0, 1) = 0 ∧ if(𝑒 = 𝑣, 0, 1) = 0))
48 equequ2 2030 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑀 β†’ (𝑒 = 𝑣 ↔ 𝑒 = 𝑀))
4948ifbid 4513 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑀 β†’ if(𝑒 = 𝑣, 0, 1) = if(𝑒 = 𝑀, 0, 1))
5049eqeq1d 2735 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑀 β†’ (if(𝑒 = 𝑣, 0, 1) = 0 ↔ if(𝑒 = 𝑀, 0, 1) = 0))
5150, 48bibi12d 346 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑀 β†’ ((if(𝑒 = 𝑣, 0, 1) = 0 ↔ 𝑒 = 𝑣) ↔ (if(𝑒 = 𝑀, 0, 1) = 0 ↔ 𝑒 = 𝑀)))
52 equequ1 2029 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝑒 β†’ (𝑀 = 𝑣 ↔ 𝑒 = 𝑣))
5352ifbid 4513 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝑒 β†’ if(𝑀 = 𝑣, 0, 1) = if(𝑒 = 𝑣, 0, 1))
5453eqeq1d 2735 . . . . . . . . . . . . . . . . . 18 (𝑀 = 𝑒 β†’ (if(𝑀 = 𝑣, 0, 1) = 0 ↔ if(𝑒 = 𝑣, 0, 1) = 0))
5554, 52bibi12d 346 . . . . . . . . . . . . . . . . 17 (𝑀 = 𝑒 β†’ ((if(𝑀 = 𝑣, 0, 1) = 0 ↔ 𝑀 = 𝑣) ↔ (if(𝑒 = 𝑣, 0, 1) = 0 ↔ 𝑒 = 𝑣)))
5655, 25chvarvv 2003 . . . . . . . . . . . . . . . 16 (if(𝑒 = 𝑣, 0, 1) = 0 ↔ 𝑒 = 𝑣)
5751, 56chvarvv 2003 . . . . . . . . . . . . . . 15 (if(𝑒 = 𝑀, 0, 1) = 0 ↔ 𝑒 = 𝑀)
58 eqtr2 2757 . . . . . . . . . . . . . . 15 ((𝑒 = 𝑀 ∧ 𝑒 = 𝑣) β†’ 𝑀 = 𝑣)
5957, 56, 58syl2anb 599 . . . . . . . . . . . . . 14 ((if(𝑒 = 𝑀, 0, 1) = 0 ∧ if(𝑒 = 𝑣, 0, 1) = 0) β†’ 𝑀 = 𝑣)
6059iftrued 4498 . . . . . . . . . . . . 13 ((if(𝑒 = 𝑀, 0, 1) = 0 ∧ if(𝑒 = 𝑣, 0, 1) = 0) β†’ if(𝑀 = 𝑣, 0, 1) = 0)
611leidi 11697 . . . . . . . . . . . . 13 0 ≀ 0
6260, 61eqbrtrdi 5148 . . . . . . . . . . . 12 ((if(𝑒 = 𝑀, 0, 1) = 0 ∧ if(𝑒 = 𝑣, 0, 1) = 0) β†’ if(𝑀 = 𝑣, 0, 1) ≀ 0)
6347, 62sylbi 216 . . . . . . . . . . 11 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0 β†’ if(𝑀 = 𝑣, 0, 1) ≀ 0)
64 id 22 . . . . . . . . . . 11 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0 β†’ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0)
6563, 64breqtrrd 5137 . . . . . . . . . 10 ((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0 β†’ if(𝑀 = 𝑣, 0, 1) ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
6641, 65jaoi 856 . . . . . . . . 9 (((if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) ∈ β„• ∨ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)) = 0) β†’ if(𝑀 = 𝑣, 0, 1) ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
6731, 66mp1i 13 . . . . . . . 8 ((𝑒 ∈ 𝑋 ∧ (𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ if(𝑀 = 𝑣, 0, 1) ≀ (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
6816adantl 483 . . . . . . . 8 ((𝑒 ∈ 𝑋 ∧ (𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ (𝑀𝐷𝑣) = if(𝑀 = 𝑣, 0, 1))
69 eqeq12 2750 . . . . . . . . . . . 12 ((π‘₯ = 𝑒 ∧ 𝑦 = 𝑀) β†’ (π‘₯ = 𝑦 ↔ 𝑒 = 𝑀))
7069ifbid 4513 . . . . . . . . . . 11 ((π‘₯ = 𝑒 ∧ 𝑦 = 𝑀) β†’ if(π‘₯ = 𝑦, 0, 1) = if(𝑒 = 𝑀, 0, 1))
7127elexi 3466 . . . . . . . . . . 11 if(𝑒 = 𝑀, 0, 1) ∈ V
7270, 5, 71ovmpoa 7514 . . . . . . . . . 10 ((𝑒 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋) β†’ (𝑒𝐷𝑀) = if(𝑒 = 𝑀, 0, 1))
7372adantrr 716 . . . . . . . . 9 ((𝑒 ∈ 𝑋 ∧ (𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ (𝑒𝐷𝑀) = if(𝑒 = 𝑀, 0, 1))
74 eqeq12 2750 . . . . . . . . . . . 12 ((π‘₯ = 𝑒 ∧ 𝑦 = 𝑣) β†’ (π‘₯ = 𝑦 ↔ 𝑒 = 𝑣))
7574ifbid 4513 . . . . . . . . . . 11 ((π‘₯ = 𝑒 ∧ 𝑦 = 𝑣) β†’ if(π‘₯ = 𝑦, 0, 1) = if(𝑒 = 𝑣, 0, 1))
7628elexi 3466 . . . . . . . . . . 11 if(𝑒 = 𝑣, 0, 1) ∈ V
7775, 5, 76ovmpoa 7514 . . . . . . . . . 10 ((𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ (𝑒𝐷𝑣) = if(𝑒 = 𝑣, 0, 1))
7877adantrl 715 . . . . . . . . 9 ((𝑒 ∈ 𝑋 ∧ (𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ (𝑒𝐷𝑣) = if(𝑒 = 𝑣, 0, 1))
7973, 78oveq12d 7379 . . . . . . . 8 ((𝑒 ∈ 𝑋 ∧ (𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣)) = (if(𝑒 = 𝑀, 0, 1) + if(𝑒 = 𝑣, 0, 1)))
8067, 68, 793brtr4d 5141 . . . . . . 7 ((𝑒 ∈ 𝑋 ∧ (𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣)))
8180expcom 415 . . . . . 6 ((𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ (𝑒 ∈ 𝑋 β†’ (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣))))
8281ralrimiv 3139 . . . . 5 ((𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ βˆ€π‘’ ∈ 𝑋 (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣)))
8326, 82jca 513 . . . 4 ((𝑀 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ (((𝑀𝐷𝑣) = 0 ↔ 𝑀 = 𝑣) ∧ βˆ€π‘’ ∈ 𝑋 (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣))))
8483rgen2 3191 . . 3 βˆ€π‘€ ∈ 𝑋 βˆ€π‘£ ∈ 𝑋 (((𝑀𝐷𝑣) = 0 ↔ 𝑀 = 𝑣) ∧ βˆ€π‘’ ∈ 𝑋 (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣)))
857, 84pm3.2i 472 . 2 (𝐷:(𝑋 Γ— 𝑋)βŸΆβ„ ∧ βˆ€π‘€ ∈ 𝑋 βˆ€π‘£ ∈ 𝑋 (((𝑀𝐷𝑣) = 0 ↔ 𝑀 = 𝑣) ∧ βˆ€π‘’ ∈ 𝑋 (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣))))
86 ismet 23699 . 2 (𝑋 ∈ 𝑉 β†’ (𝐷 ∈ (Metβ€˜π‘‹) ↔ (𝐷:(𝑋 Γ— 𝑋)βŸΆβ„ ∧ βˆ€π‘€ ∈ 𝑋 βˆ€π‘£ ∈ 𝑋 (((𝑀𝐷𝑣) = 0 ↔ 𝑀 = 𝑣) ∧ βˆ€π‘’ ∈ 𝑋 (𝑀𝐷𝑣) ≀ ((𝑒𝐷𝑀) + (𝑒𝐷𝑣))))))
8785, 86mpbiri 258 1 (𝑋 ∈ 𝑉 β†’ 𝐷 ∈ (Metβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  ifcif 4490   class class class wbr 5109   Γ— cxp 5635  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  β„cr 11058  0cc0 11059  1c1 11060   + caddc 11062   ≀ cle 11198  β„•cn 12161  β„•0cn0 12421  Metcmet 20805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-met 20813
This theorem is referenced by:  dscopn  23952
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