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Mirrors > Home > MPE Home > Th. List > 3prm | Structured version Visualization version Unicode version |
Description: 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
3prm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3z 10970 |
. . 3
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2 | 1lt3 10778 |
. . 3
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3 | eluz2b1 11230 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | mpbir2an 931 |
. 2
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5 | elfz1eq 11810 |
. . . . 5
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6 | 2z 10969 |
. . . . . . . 8
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7 | iddvds 14316 |
. . . . . . . 8
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8 | 2nn 10767 |
. . . . . . . . 9
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9 | 1lt2 10776 |
. . . . . . . . 9
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10 | ndvdsp1 14390 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 6, 8, 9, 10 | mp3an 1364 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 6, 7, 11 | mp2b 10 |
. . . . . . 7
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13 | df-3 10669 |
. . . . . . . 8
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14 | 13 | breq2i 4410 |
. . . . . . 7
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15 | 12, 14 | mtbir 301 |
. . . . . 6
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16 | breq1 4405 |
. . . . . 6
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17 | 15, 16 | mtbiri 305 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 5, 17 | syl 17 |
. . . 4
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19 | 3m1e2 10726 |
. . . . 5
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20 | 19 | oveq2i 6301 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 18, 20 | eleq2s 2547 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | rgen 2747 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | isprm3 14633 |
. 2
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24 | 4, 22, 23 | mpbir2an 931 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-2o 7183 df-oadd 7186 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-sup 7956 df-inf 7957 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-3 10669 df-n0 10870 df-z 10938 df-uz 11160 df-rp 11303 df-fz 11785 df-seq 12214 df-exp 12273 df-cj 13162 df-re 13163 df-im 13164 df-sqrt 13298 df-abs 13299 df-dvds 14306 df-prm 14623 |
This theorem is referenced by: 3lcm2e6 14681 prmo3 14999 4001lem4 15115 lt6abl 17529 ppi3 24098 cht3 24100 bpos1 24211 6gbe 38872 7gbo 38873 8gbe 38874 9gboa 38875 11gboa 38876 bgoldbwt 38878 bgoldbst 38879 nnsum3primesle9 38889 nnsum4primeseven 38895 nnsum4primesevenALTV 38896 zlmodzxznm 40343 |
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