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Mirrors > Home > MPE Home > Th. List > prmirredOLD | Structured version Unicode version |
Description: The irreducible elements
of ![]() |
Ref | Expression |
---|---|
prmirredOLD.1 |
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prmirredOLD.2 |
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Ref | Expression |
---|---|
prmirredOLD |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmirredOLD.2 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | zsubrg 17961 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() | |
3 | prmirredOLD.1 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | subrgbas 16966 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 2, 4 | ax-mp 5 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 1, 5 | irredcl 16888 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | elnn0 10668 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | ax-1 6 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 3 | subrgrng 16960 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 2, 9 | ax-mp 5 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() |
11 | subrgsubg 16963 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 2, 11 | ax-mp 5 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() |
13 | cnfld0 17935 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 3, 13 | subg0 15775 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 12, 14 | ax-mp 5 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 1, 15 | irredn0 16887 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 10, 16 | mpan 670 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | necon2bi 2682 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 18 | pm2.21d 106 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 8, 19 | jaoi 379 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 7, 20 | sylbi 195 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | prmnn 13854 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 22 | a1i 11 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 3, 1 | prmirredlemOLD 18015 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | a1i 11 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 21, 23, 25 | pm5.21ndd 354 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | nn0re 10675 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | nn0ge0 10692 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 27, 28 | absidd 12997 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | eleq1d 2518 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 26, 30 | bitr4d 256 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31 | adantl 466 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 3, 1 | prmirredlemOLD 18015 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33 | adantl 466 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | eqid 2450 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 1, 35, 5 | irrednegb 16895 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 10, 36 | mpan 670 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | eqid 2450 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 3, 38, 35 | subginv 15776 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 12, 39 | mpan 670 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | zcn 10738 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
42 | cnfldneg 17937 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 41, 42 | syl 16 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 40, 43 | eqtr3d 2492 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 44 | eleq1d 2518 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 37, 45 | bitrd 253 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 46 | adantr 465 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | zre 10737 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
49 | 48 | adantr 465 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | nnnn0 10673 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
51 | 50 | nn0ge0d 10726 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 51 | adantl 466 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 49 | le0neg1d 9998 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 52, 53 | mpbird 232 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 49, 54 | absnidd 12988 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | 55 | eleq1d 2518 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57 | 34, 47, 56 | 3bitr4d 285 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 57 | adantrl 715 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | elznn0nn 10747 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
60 | 59 | biimpi 194 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
61 | 32, 58, 60 | mpjaodan 784 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 6, 61 | biadan2 642 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1592 ax-4 1603 ax-5 1671 ax-6 1709 ax-7 1729 ax-8 1759 ax-9 1761 ax-10 1776 ax-11 1781 ax-12 1793 ax-13 1944 ax-ext 2429 ax-rep 4487 ax-sep 4497 ax-nul 4505 ax-pow 4554 ax-pr 4615 ax-un 6458 ax-cnex 9425 ax-resscn 9426 ax-1cn 9427 ax-icn 9428 ax-addcl 9429 ax-addrcl 9430 ax-mulcl 9431 ax-mulrcl 9432 ax-mulcom 9433 ax-addass 9434 ax-mulass 9435 ax-distr 9436 ax-i2m1 9437 ax-1ne0 9438 ax-1rid 9439 ax-rnegex 9440 ax-rrecex 9441 ax-cnre 9442 ax-pre-lttri 9443 ax-pre-lttrn 9444 ax-pre-ltadd 9445 ax-pre-mulgt0 9446 ax-pre-sup 9447 ax-addf 9448 ax-mulf 9449 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1702 df-eu 2263 df-mo 2264 df-clab 2436 df-cleq 2442 df-clel 2445 df-nfc 2598 df-ne 2643 df-nel 2644 df-ral 2797 df-rex 2798 df-reu 2799 df-rmo 2800 df-rab 2801 df-v 3056 df-sbc 3271 df-csb 3373 df-dif 3415 df-un 3417 df-in 3419 df-ss 3426 df-pss 3428 df-nul 3722 df-if 3876 df-pw 3946 df-sn 3962 df-pr 3964 df-tp 3966 df-op 3968 df-uni 4176 df-int 4213 df-iun 4257 df-br 4377 df-opab 4435 df-mpt 4436 df-tr 4470 df-eprel 4716 df-id 4720 df-po 4725 df-so 4726 df-fr 4763 df-we 4765 df-ord 4806 df-on 4807 df-lim 4808 df-suc 4809 df-xp 4930 df-rel 4931 df-cnv 4932 df-co 4933 df-dm 4934 df-rn 4935 df-res 4936 df-ima 4937 df-iota 5465 df-fun 5504 df-fn 5505 df-f 5506 df-f1 5507 df-fo 5508 df-f1o 5509 df-fv 5510 df-riota 6137 df-ov 6179 df-oprab 6180 df-mpt2 6181 df-om 6563 df-1st 6663 df-2nd 6664 df-tpos 6831 df-recs 6918 df-rdg 6952 df-1o 7006 df-2o 7007 df-oadd 7010 df-er 7187 df-en 7397 df-dom 7398 df-sdom 7399 df-fin 7400 df-sup 7778 df-pnf 9507 df-mnf 9508 df-xr 9509 df-ltxr 9510 df-le 9511 df-sub 9684 df-neg 9685 df-div 10081 df-nn 10410 df-2 10467 df-3 10468 df-4 10469 df-5 10470 df-6 10471 df-7 10472 df-8 10473 df-9 10474 df-10 10475 df-n0 10667 df-z 10734 df-dec 10843 df-uz 10949 df-rp 11079 df-fz 11525 df-seq 11894 df-exp 11953 df-cj 12676 df-re 12677 df-im 12678 df-sqr 12812 df-abs 12813 df-dvds 13624 df-prm 13852 df-gz 14079 df-struct 14264 df-ndx 14265 df-slot 14266 df-base 14267 df-sets 14268 df-ress 14269 df-plusg 14339 df-mulr 14340 df-starv 14341 df-tset 14345 df-ple 14346 df-ds 14348 df-unif 14349 df-0g 14468 df-mnd 15503 df-grp 15633 df-minusg 15634 df-subg 15766 df-cmn 16369 df-mgp 16683 df-ur 16695 df-rng 16739 df-cring 16740 df-oppr 16807 df-dvdsr 16825 df-unit 16826 df-irred 16827 df-invr 16856 df-dvr 16867 df-drng 16926 df-subrg 16955 df-cnfld 17914 |
This theorem is referenced by: (None) |
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