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Mirrors > Home > ILE Home > Th. List > prmdvdsexpr | Unicode version |
Description: If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
prmdvdsexpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8673 |
. . 3
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2 | prmdvdsexpb 11402 |
. . . . . 6
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3 | 2 | biimpd 142 |
. . . . 5
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4 | 3 | 3expia 1145 |
. . . 4
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5 | prmnn 11366 |
. . . . . . . . . 10
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6 | 5 | adantl 271 |
. . . . . . . . 9
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7 | 6 | nncnd 8434 |
. . . . . . . 8
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8 | 7 | exp0d 10076 |
. . . . . . 7
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9 | 8 | breq2d 3857 |
. . . . . 6
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10 | nprmdvds1 11395 |
. . . . . . . 8
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11 | 10 | pm2.21d 584 |
. . . . . . 7
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12 | 11 | adantr 270 |
. . . . . 6
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13 | 9, 12 | sylbid 148 |
. . . . 5
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14 | oveq2 5660 |
. . . . . . 7
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15 | 14 | breq2d 3857 |
. . . . . 6
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16 | 15 | imbi1d 229 |
. . . . 5
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17 | 13, 16 | syl5ibrcom 155 |
. . . 4
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18 | 4, 17 | jaod 672 |
. . 3
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19 | 1, 18 | syl5bi 150 |
. 2
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20 | 19 | 3impia 1140 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 ax-arch 7462 ax-caucvg 7463 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3394 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-frec 6156 df-1o 6181 df-2o 6182 df-er 6290 df-en 6456 df-sup 6677 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-2 8479 df-3 8480 df-4 8481 df-n0 8672 df-z 8749 df-uz 9018 df-q 9103 df-rp 9133 df-fz 9423 df-fzo 9550 df-fl 9673 df-mod 9726 df-iseq 9849 df-seq3 9850 df-exp 9951 df-cj 10272 df-re 10273 df-im 10274 df-rsqrt 10427 df-abs 10428 df-dvds 11071 df-gcd 11213 df-prm 11364 |
This theorem is referenced by: (None) |
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