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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 12278 | . . 3 ⊢ 6 ∈ ℕ | |
2 | 1 | decnncl2 12678 | . . 3 ⊢ ;60 ∈ ℕ |
3 | 1, 2 | gcdcomnni 40594 | . 2 ⊢ (6 gcd ;60) = (;60 gcd 6) |
4 | 1 | nnnn0i 12457 | . . . . . 6 ⊢ 6 ∈ ℕ0 |
5 | 1nn0 12465 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 0nn0 12464 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | eqid 2731 | . . . . . 6 ⊢ ;10 = ;10 | |
8 | 6cn 12280 | . . . . . . 7 ⊢ 6 ∈ ℂ | |
9 | 8 | mullidi 11196 | . . . . . 6 ⊢ (1 · 6) = 6 |
10 | 8 | mul02i 11380 | . . . . . 6 ⊢ (0 · 6) = 0 |
11 | 4, 5, 6, 7, 9, 10 | decmul1 12718 | . . . . 5 ⊢ (;10 · 6) = ;60 |
12 | 10nn 12670 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
13 | 12, 1 | mulcomnni 40593 | . . . . 5 ⊢ (;10 · 6) = (6 · ;10) |
14 | 11, 13 | eqtr3i 2761 | . . . 4 ⊢ ;60 = (6 · ;10) |
15 | 14 | oveq2i 7399 | . . 3 ⊢ (6 gcd ;60) = (6 gcd (6 · ;10)) |
16 | 1, 12 | gcdmultiplei 40599 | . . 3 ⊢ (6 gcd (6 · ;10)) = 6 |
17 | 15, 16 | eqtri 2759 | . 2 ⊢ (6 gcd ;60) = 6 |
18 | 3, 17 | eqtr3i 2761 | 1 ⊢ (;60 gcd 6) = 6 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7388 0cc0 11087 1c1 11088 · cmul 11092 6c6 12248 ;cdc 12654 gcd cgcd 16412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 ax-cnex 11143 ax-resscn 11144 ax-1cn 11145 ax-icn 11146 ax-addcl 11147 ax-addrcl 11148 ax-mulcl 11149 ax-mulrcl 11150 ax-mulcom 11151 ax-addass 11152 ax-mulass 11153 ax-distr 11154 ax-i2m1 11155 ax-1ne0 11156 ax-1rid 11157 ax-rnegex 11158 ax-rrecex 11159 ax-cnre 11160 ax-pre-lttri 11161 ax-pre-lttrn 11162 ax-pre-ltadd 11163 ax-pre-mulgt0 11164 ax-pre-sup 11165 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3374 df-reu 3375 df-rab 3429 df-v 3471 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-op 4624 df-uni 4897 df-iun 4987 df-br 5137 df-opab 5199 df-mpt 5220 df-tr 5254 df-id 5562 df-eprel 5568 df-po 5576 df-so 5577 df-fr 5619 df-we 5621 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6284 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7344 df-ov 7391 df-oprab 7392 df-mpo 7393 df-om 7834 df-2nd 7953 df-frecs 8243 df-wrecs 8274 df-recs 8348 df-rdg 8387 df-er 8681 df-en 8918 df-dom 8919 df-sdom 8920 df-sup 9414 df-inf 9415 df-pnf 11227 df-mnf 11228 df-xr 11229 df-ltxr 11230 df-le 11231 df-sub 11423 df-neg 11424 df-div 11849 df-nn 12190 df-2 12252 df-3 12253 df-4 12254 df-5 12255 df-6 12256 df-7 12257 df-8 12258 df-9 12259 df-n0 12450 df-z 12536 df-dec 12655 df-uz 12800 df-rp 12952 df-seq 13944 df-exp 14005 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-dvds 16175 df-gcd 16413 |
This theorem is referenced by: 60lcm6e60 40614 |
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