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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 12622 | . . 3 ⊢ 0 ∈ ℤ | |
2 | gcdval 16530 | . . 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 692 | . 2 ⊢ (0 gcd 0) = if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) |
4 | eqid 2735 | . . 3 ⊢ 0 = 0 | |
5 | iftrue 4537 | . . 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 692 | . 2 ⊢ if((0 = 0 ∧ 0 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 0 ∧ 𝑛 ∥ 0)}, ℝ, < )) = 0 |
7 | 3, 6 | eqtri 2763 | 1 ⊢ (0 gcd 0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ifcif 4531 class class class wbr 5148 (class class class)co 7431 supcsup 9478 ℝcr 11152 0cc0 11153 < clt 11293 ℤcz 12611 ∥ cdvds 16287 gcd cgcd 16528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-i2m1 11221 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-neg 11493 df-z 12612 df-gcd 16529 |
This theorem is referenced by: gcddvds 16537 gcdcl 16540 gcdeq0 16551 gcd0id 16553 bezout 16577 mulgcd 16582 nn0rppwr 16595 nn0expgcd 16598 nn0gcdsq 16786 |
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