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Mirrors > Home > MPE Home > Th. List > pcfaclem | Structured version Visualization version Unicode version |
Description: Lemma for pcfac 14923. (Contributed by Mario Carneiro, 20-May-2014.) |
Ref | Expression |
---|---|
pcfaclem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ge0 10919 |
. . . 4
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2 | 1 | 3ad2ant1 1051 |
. . 3
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3 | nn0re 10902 |
. . . . 5
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4 | 3 | 3ad2ant1 1051 |
. . . 4
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5 | prmnn 14704 |
. . . . . . 7
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6 | 5 | 3ad2ant3 1053 |
. . . . . 6
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7 | eluznn0 11251 |
. . . . . . 7
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8 | 7 | 3adant3 1050 |
. . . . . 6
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9 | 6, 8 | nnexpcld 12475 |
. . . . 5
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10 | 9 | nnred 10646 |
. . . 4
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11 | 9 | nngt0d 10675 |
. . . 4
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12 | ge0div 10494 |
. . . 4
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13 | 4, 10, 11, 12 | syl3anc 1292 |
. . 3
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14 | 2, 13 | mpbid 215 |
. 2
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15 | 8 | nn0red 10950 |
. . . . . 6
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16 | eluzle 11195 |
. . . . . . 7
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17 | 16 | 3ad2ant2 1052 |
. . . . . 6
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18 | prmuz2 14721 |
. . . . . . . 8
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19 | 18 | 3ad2ant3 1053 |
. . . . . . 7
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20 | bernneq3 12438 |
. . . . . . 7
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21 | 19, 8, 20 | syl2anc 673 |
. . . . . 6
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22 | 4, 15, 10, 17, 21 | lelttrd 9810 |
. . . . 5
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23 | 9 | nncnd 10647 |
. . . . . 6
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24 | 23 | mulid1d 9678 |
. . . . 5
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25 | 22, 24 | breqtrrd 4422 |
. . . 4
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26 | 1red 9676 |
. . . . 5
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27 | ltdivmul 10502 |
. . . . 5
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28 | 4, 26, 10, 11, 27 | syl112anc 1296 |
. . . 4
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29 | 25, 28 | mpbird 240 |
. . 3
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30 | 0p1e1 10743 |
. . 3
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31 | 29, 30 | syl6breqr 4436 |
. 2
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32 | 4, 9 | nndivred 10680 |
. . 3
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33 | 0z 10972 |
. . 3
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34 | flbi 12084 |
. . 3
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35 | 32, 33, 34 | sylancl 675 |
. 2
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36 | 14, 31, 35 | mpbir2and 936 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 ax-pre-sup 9635 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-1o 7200 df-2o 7201 df-oadd 7204 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-sup 7974 df-inf 7975 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-div 10292 df-nn 10632 df-2 10690 df-n0 10894 df-z 10962 df-uz 11183 df-fl 12061 df-seq 12252 df-exp 12311 df-dvds 14383 df-prm 14702 |
This theorem is referenced by: pcfac 14923 |
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