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Mirrors > Home > ILE Home > Th. List > msqznn | Unicode version |
Description: The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.) |
Ref | Expression |
---|---|
msqznn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zmulcl 9265 | . . . 4 | |
2 | 1 | anidms 395 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | 0z 9223 | . . . . 5 | |
5 | zapne 9286 | . . . . 5 # | |
6 | 4, 5 | mpan2 423 | . . . 4 # |
7 | 6 | pm5.32i 451 | . . 3 # |
8 | zre 9216 | . . . 4 | |
9 | apsqgt0 8520 | . . . 4 # | |
10 | 8, 9 | sylan 281 | . . 3 # |
11 | 7, 10 | sylbir 134 | . 2 |
12 | elnnz 9222 | . 2 | |
13 | 3, 11, 12 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2141 wne 2340 class class class wbr 3989 (class class class)co 5853 cr 7773 cc0 7774 cmul 7779 clt 7954 # cap 8500 cn 8878 cz 9212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-inn 8879 df-n0 9136 df-z 9213 |
This theorem is referenced by: qreccl 9601 |
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