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Mirrors > Home > MPE Home > Th. List > Mathboxes > 41prothprmlem1 | Structured version Visualization version Unicode version |
Description: Lemma 1 for 41prothprm 41536. (Contributed by AV, 4-Jul-2020.) |
Ref | Expression |
---|---|
41prothprm.p | ; |
Ref | Expression |
---|---|
41prothprmlem1 | ; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 41prothprm.p | . . . . . 6 ; | |
2 | dfdec10 11497 | . . . . . 6 ; ; | |
3 | 1, 2 | eqtri 2644 | . . . . 5 ; |
4 | 3 | oveq1i 6660 | . . . 4 ; |
5 | 10nn 11514 | . . . . . . 7 ; | |
6 | 5 | nncni 11030 | . . . . . 6 ; |
7 | 4cn 11098 | . . . . . 6 | |
8 | 6, 7 | mulcli 10045 | . . . . 5 ; |
9 | pncan1 10454 | . . . . 5 ; ; ; | |
10 | 8, 9 | ax-mp 5 | . . . 4 ; ; |
11 | 4, 10 | eqtri 2644 | . . 3 ; |
12 | 11 | oveq1i 6660 | . 2 ; |
13 | 2cn 11091 | . . . 4 | |
14 | 2ne0 11113 | . . . 4 | |
15 | 6, 7, 13, 14 | divassi 10781 | . . 3 ; ; |
16 | 4d2e2 11184 | . . . . 5 | |
17 | 16 | oveq2i 6661 | . . . 4 ; ; |
18 | 2nn0 11309 | . . . . 5 | |
19 | 18 | dec0u 11520 | . . . 4 ; ; |
20 | 17, 19 | eqtri 2644 | . . 3 ; ; |
21 | 15, 20 | eqtri 2644 | . 2 ; ; |
22 | 12, 21 | eqtri 2644 | 1 ; |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 caddc 9939 cmul 9941 cmin 10266 cdiv 10684 c2 11070 c4 11072 ;cdc 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 |
This theorem is referenced by: 41prothprmlem2 41535 |
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