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Mirrors > Home > MPE Home > Th. List > Mathboxes > 60gcd6e6 | Structured version Visualization version GIF version |
Description: The gcd of 60 and 6 is 6. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
60gcd6e6 | ⊢ (;60 gcd 6) = 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 12112 | . . 3 ⊢ 6 ∈ ℕ | |
2 | 1 | decnncl2 12511 | . . 3 ⊢ ;60 ∈ ℕ |
3 | 1, 2 | gcdcomnni 40197 | . 2 ⊢ (6 gcd ;60) = (;60 gcd 6) |
4 | 1 | nnnn0i 12291 | . . . . . 6 ⊢ 6 ∈ ℕ0 |
5 | 1nn0 12299 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 0nn0 12298 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | eqid 2736 | . . . . . 6 ⊢ ;10 = ;10 | |
8 | 6cn 12114 | . . . . . . 7 ⊢ 6 ∈ ℂ | |
9 | 8 | mulid2i 11030 | . . . . . 6 ⊢ (1 · 6) = 6 |
10 | 8 | mul02i 11214 | . . . . . 6 ⊢ (0 · 6) = 0 |
11 | 4, 5, 6, 7, 9, 10 | decmul1 12551 | . . . . 5 ⊢ (;10 · 6) = ;60 |
12 | 10nn 12503 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
13 | 12, 1 | mulcomnni 40196 | . . . . 5 ⊢ (;10 · 6) = (6 · ;10) |
14 | 11, 13 | eqtr3i 2766 | . . . 4 ⊢ ;60 = (6 · ;10) |
15 | 14 | oveq2i 7318 | . . 3 ⊢ (6 gcd ;60) = (6 gcd (6 · ;10)) |
16 | 1, 12 | gcdmultiplei 40202 | . . 3 ⊢ (6 gcd (6 · ;10)) = 6 |
17 | 15, 16 | eqtri 2764 | . 2 ⊢ (6 gcd ;60) = 6 |
18 | 3, 17 | eqtr3i 2766 | 1 ⊢ (;60 gcd 6) = 6 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 (class class class)co 7307 0cc0 10921 1c1 10922 · cmul 10926 6c6 12082 ;cdc 12487 gcd cgcd 16250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-sup 9249 df-inf 9250 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-5 12089 df-6 12090 df-7 12091 df-8 12092 df-9 12093 df-n0 12284 df-z 12370 df-dec 12488 df-uz 12633 df-rp 12781 df-seq 13772 df-exp 13833 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-abs 14996 df-dvds 16013 df-gcd 16251 |
This theorem is referenced by: 60lcm6e60 40217 |
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