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Mirrors > Home > ILE Home > Th. List > zesq | Unicode version |
Description: An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
zesq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 8356 | . . . . . . 7 | |
2 | sqval 9534 | . . . . . . 7 | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | 3 | oveq1d 5547 | . . . . 5 |
5 | 2cnd 8112 | . . . . . 6 | |
6 | 2ap0 8132 | . . . . . . 7 # | |
7 | 6 | a1i 9 | . . . . . 6 # |
8 | 1, 1, 5, 7 | divassapd 7912 | . . . . 5 |
9 | 4, 8 | eqtrd 2113 | . . . 4 |
10 | 9 | adantr 270 | . . 3 |
11 | zmulcl 8404 | . . 3 | |
12 | 10, 11 | eqeltrd 2155 | . 2 |
13 | 1 | adantr 270 | . . . . . . . . . . 11 |
14 | sqcl 9537 | . . . . . . . . . . 11 | |
15 | 13, 14 | syl 14 | . . . . . . . . . 10 |
16 | peano2cn 7243 | . . . . . . . . . 10 | |
17 | 15, 16 | syl 14 | . . . . . . . . 9 |
18 | 17 | halfcld 8275 | . . . . . . . 8 |
19 | 18, 13 | pncand 7420 | . . . . . . 7 |
20 | binom21 9586 | . . . . . . . . . . . . 13 | |
21 | 13, 20 | syl 14 | . . . . . . . . . . . 12 |
22 | peano2cn 7243 | . . . . . . . . . . . . . 14 | |
23 | 13, 22 | syl 14 | . . . . . . . . . . . . 13 |
24 | sqval 9534 | . . . . . . . . . . . . 13 | |
25 | 23, 24 | syl 14 | . . . . . . . . . . . 12 |
26 | 2cn 8110 | . . . . . . . . . . . . . 14 | |
27 | mulcl 7100 | . . . . . . . . . . . . . 14 | |
28 | 26, 13, 27 | sylancr 405 | . . . . . . . . . . . . 13 |
29 | 1cnd 7135 | . . . . . . . . . . . . 13 | |
30 | 15, 28, 29 | add32d 7276 | . . . . . . . . . . . 12 |
31 | 21, 25, 30 | 3eqtr3d 2121 | . . . . . . . . . . 11 |
32 | 31 | oveq1d 5547 | . . . . . . . . . 10 |
33 | 2cnd 8112 | . . . . . . . . . . 11 | |
34 | 6 | a1i 9 | . . . . . . . . . . 11 # |
35 | 23, 23, 33, 34 | divassapd 7912 | . . . . . . . . . 10 |
36 | 17, 28, 33, 34 | divdirapd 7915 | . . . . . . . . . . 11 |
37 | 13, 33, 34 | divcanap3d 7882 | . . . . . . . . . . . 12 |
38 | 37 | oveq2d 5548 | . . . . . . . . . . 11 |
39 | 36, 38 | eqtrd 2113 | . . . . . . . . . 10 |
40 | 32, 35, 39 | 3eqtr3d 2121 | . . . . . . . . 9 |
41 | peano2z 8387 | . . . . . . . . . 10 | |
42 | zmulcl 8404 | . . . . . . . . . 10 | |
43 | 41, 42 | sylan 277 | . . . . . . . . 9 |
44 | 40, 43 | eqeltrrd 2156 | . . . . . . . 8 |
45 | simpl 107 | . . . . . . . 8 | |
46 | 44, 45 | zsubcld 8474 | . . . . . . 7 |
47 | 19, 46 | eqeltrrd 2156 | . . . . . 6 |
48 | 47 | ex 113 | . . . . 5 |
49 | 48 | con3d 593 | . . . 4 |
50 | zsqcl 9546 | . . . . 5 | |
51 | zeo2 8453 | . . . . 5 | |
52 | 50, 51 | syl 14 | . . . 4 |
53 | zeo2 8453 | . . . 4 | |
54 | 49, 52, 53 | 3imtr4d 201 | . . 3 |
55 | 54 | imp 122 | . 2 |
56 | 12, 55 | impbida 560 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cmul 6986 cmin 7279 # cap 7681 cdiv 7760 c2 8089 cz 8351 cexp 9475 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: nnesq 9592 sqrt2irrlem 10540 |
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