Description: The gcd of 420 and 8 is 4. (Contributed by metakunt, 25-Apr-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | 420gcd8e4 | |- ( ; ; 4 2 0 gcd 8 ) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn | |- 8 e. NN |
|
2 | 4nn | |- 4 e. NN |
|
3 | 5nn0 | |- 5 e. NN0 |
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4 | 2nn | |- 2 e. NN |
|
5 | 3 4 | decnncl | |- ; 5 2 e. NN |
6 | 5 | nnzi | |- ; 5 2 e. ZZ |
7 | 1 2 6 | gcdaddmzz2nncomi | |- ( 8 gcd 4 ) = ( 8 gcd ( ( ; 5 2 x. 8 ) + 4 ) ) |
8 | 4nn0 | |- 4 e. NN0 |
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9 | 1nn0 | |- 1 e. NN0 |
|
10 | 8 9 | deccl | |- ; 4 1 e. NN0 |
11 | 6nn0 | |- 6 e. NN0 |
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12 | 0nn0 | |- 0 e. NN0 |
|
13 | eqid | |- ; ; 4 1 6 = ; ; 4 1 6 |
|
14 | 8 | dec0h | |- 4 = ; 0 4 |
15 | 1p1e2 | |- ( 1 + 1 ) = 2 |
|
16 | 10 | nn0cni | |- ; 4 1 e. CC |
17 | 16 | addid1i | |- ( ; 4 1 + 0 ) = ; 4 1 |
18 | 8 9 15 17 | decsuc | |- ( ( ; 4 1 + 0 ) + 1 ) = ; 4 2 |
19 | 6p4e10 | |- ( 6 + 4 ) = ; 1 0 |
|
20 | 10 11 12 8 13 14 18 19 | decaddc2 | |- ( ; ; 4 1 6 + 4 ) = ; ; 4 2 0 |
21 | 8nn0 | |- 8 e. NN0 |
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22 | 2nn0 | |- 2 e. NN0 |
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23 | eqid | |- ; 5 2 = ; 5 2 |
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24 | 0p1e1 | |- ( 0 + 1 ) = 1 |
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25 | 8t5e40 | |- ( 8 x. 5 ) = ; 4 0 |
|
26 | 8 12 24 25 | decsuc | |- ( ( 8 x. 5 ) + 1 ) = ; 4 1 |
27 | 8t2e16 | |- ( 8 x. 2 ) = ; 1 6 |
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28 | 21 3 22 23 11 9 26 27 | decmul2c | |- ( 8 x. ; 5 2 ) = ; ; 4 1 6 |
29 | 1 5 | mulcomnni | |- ( 8 x. ; 5 2 ) = ( ; 5 2 x. 8 ) |
30 | 28 29 | eqtr3i | |- ; ; 4 1 6 = ( ; 5 2 x. 8 ) |
31 | 30 | oveq1i | |- ( ; ; 4 1 6 + 4 ) = ( ( ; 5 2 x. 8 ) + 4 ) |
32 | 20 31 | eqtr3i | |- ; ; 4 2 0 = ( ( ; 5 2 x. 8 ) + 4 ) |
33 | 32 | oveq2i | |- ( 8 gcd ; ; 4 2 0 ) = ( 8 gcd ( ( ; 5 2 x. 8 ) + 4 ) ) |
34 | 7 33 | eqtr4i | |- ( 8 gcd 4 ) = ( 8 gcd ; ; 4 2 0 ) |
35 | 1 2 | gcdcomnni | |- ( 8 gcd 4 ) = ( 4 gcd 8 ) |
36 | 4t2e8 | |- ( 4 x. 2 ) = 8 |
|
37 | 36 | oveq2i | |- ( 4 gcd ( 4 x. 2 ) ) = ( 4 gcd 8 ) |
38 | 2 4 | gcdmultiplei | |- ( 4 gcd ( 4 x. 2 ) ) = 4 |
39 | 37 38 | eqtr3i | |- ( 4 gcd 8 ) = 4 |
40 | 35 39 | eqtri | |- ( 8 gcd 4 ) = 4 |
41 | 8 4 | decnncl | |- ; 4 2 e. NN |
42 | 41 | decnncl2 | |- ; ; 4 2 0 e. NN |
43 | 1 42 | gcdcomnni | |- ( 8 gcd ; ; 4 2 0 ) = ( ; ; 4 2 0 gcd 8 ) |
44 | 34 40 43 | 3eqtr3ri | |- ( ; ; 4 2 0 gcd 8 ) = 4 |