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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 11716 | . . 3 ⊢ 0 ∈ ℤ | |
2 | gcdval 15592 | . . 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 685 | . 2 ⊢ (0 gcd 0) = if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) |
4 | eqid 2826 | . . 3 ⊢ 0 = 0 | |
5 | iftrue 4313 | . . 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 685 | . 2 ⊢ if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) = 0 |
7 | 3, 6 | eqtri 2850 | 1 ⊢ (0 gcd 0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3122 ifcif 4307 class class class wbr 4874 (class class class)co 6906 supcsup 8616 ℝcr 10252 0cc0 10253 < clt 10392 ℤcz 11705 ∥ cdvds 15358 gcd cgcd 15590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-i2m1 10321 ax-rnegex 10324 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-pnf 10394 df-mnf 10395 df-ltxr 10397 df-neg 10589 df-z 11706 df-gcd 15591 |
This theorem is referenced by: gcddvds 15599 gcdcl 15602 gcdeq0 15612 gcd0id 15614 bezout 15634 mulgcd 15639 nn0gcdsq 15832 |
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