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Mirrors > Home > MPE Home > Th. List > gcd0val | Structured version Visualization version GIF version |
Description: The value, by convention, of the gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcd0val | ⊢ (0 gcd 0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11722 | . . 3 ⊢ 0 ∈ ℤ | |
2 | gcdval 15598 | . . 3 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ) → (0 gcd 0) = if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < ))) | |
3 | 1, 1, 2 | mp2an 683 | . 2 ⊢ (0 gcd 0) = if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) |
4 | eqid 2825 | . . 3 ⊢ 0 = 0 | |
5 | iftrue 4314 | . . 3 ⊢ ((0 = 0 ∧ 0 = 0) → if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) = 0) | |
6 | 4, 4, 5 | mp2an 683 | . 2 ⊢ if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) = 0 |
7 | 3, 6 | eqtri 2849 | 1 ⊢ (0 gcd 0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1656 ∈ wcel 2164 {crab 3121 ifcif 4308 class class class wbr 4875 (class class class)co 6910 supcsup 8621 ℝcr 10258 0cc0 10259 < clt 10398 ℤcz 11711 ∥ cdvds 15364 gcd cgcd 15596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-i2m1 10327 ax-rnegex 10330 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-neg 10595 df-z 11712 df-gcd 15597 |
This theorem is referenced by: gcddvds 15605 gcdcl 15608 gcdeq0 15618 gcd0id 15620 bezout 15640 mulgcd 15645 nn0gcdsq 15838 |
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