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Mirrors > Home > ILE Home > Th. List > gcdzeq | Unicode version |
Description: A positive integer ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
gcdzeq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 8867 |
. . . . 5
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2 | gcddvds 11398 |
. . . . 5
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3 | 1, 2 | sylan 278 |
. . . 4
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4 | 3 | simprd 113 |
. . 3
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5 | breq1 3870 |
. . 3
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6 | 4, 5 | syl5ibcom 154 |
. 2
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7 | 1 | adantr 271 |
. . . . . 6
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8 | iddvds 11252 |
. . . . . 6
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9 | 7, 8 | syl 14 |
. . . . 5
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10 | simpr 109 |
. . . . . 6
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11 | nnne0 8548 |
. . . . . . . 8
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12 | simpl 108 |
. . . . . . . . 9
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13 | 12 | necon3ai 2311 |
. . . . . . . 8
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14 | 11, 13 | syl 14 |
. . . . . . 7
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15 | 14 | adantr 271 |
. . . . . 6
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16 | dvdslegcd 11399 |
. . . . . 6
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17 | 7, 7, 10, 15, 16 | syl31anc 1184 |
. . . . 5
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18 | 9, 17 | mpand 421 |
. . . 4
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19 | 3 | simpld 111 |
. . . . 5
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20 | gcdcl 11401 |
. . . . . . . 8
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21 | 1, 20 | sylan 278 |
. . . . . . 7
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22 | 21 | nn0zd 8965 |
. . . . . 6
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23 | simpl 108 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | dvdsle 11288 |
. . . . . 6
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25 | 22, 23, 24 | syl2anc 404 |
. . . . 5
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26 | 19, 25 | mpd 13 |
. . . 4
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27 | 18, 26 | jctild 310 |
. . 3
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28 | 21 | nn0red 8825 |
. . . 4
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29 | nnre 8527 |
. . . . 5
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30 | 29 | adantr 271 |
. . . 4
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31 | 28, 30 | letri3d 7697 |
. . 3
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32 | 27, 31 | sylibrd 168 |
. 2
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33 | 6, 32 | impbid 128 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 ax-caucvg 7562 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-sup 6759 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-n0 8772 df-z 8849 df-uz 9119 df-q 9204 df-rp 9234 df-fz 9574 df-fzo 9703 df-fl 9826 df-mod 9879 df-iseq 10002 df-seq3 10003 df-exp 10086 df-cj 10407 df-re 10408 df-im 10409 df-rsqrt 10562 df-abs 10563 df-dvds 11240 df-gcd 11382 |
This theorem is referenced by: gcdeq 11455 isevengcd2 11580 |
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