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Mirrors > Home > ILE Home > Th. List > sq1 | GIF version |
Description: The square of 1 is 1. (Contributed by NM, 22-Aug-1999.) |
Ref | Expression |
---|---|
sq1 | ⊢ (1↑2) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9277 | . 2 ⊢ 2 ∈ ℤ | |
2 | 1exp 10544 | . 2 ⊢ (2 ∈ ℤ → (1↑2) = 1) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (1↑2) = 1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5872 1c1 7809 2c2 8966 ℤcz 9249 ↑cexp 10514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-n0 9173 df-z 9250 df-uz 9525 df-seqfrec 10441 df-exp 10515 |
This theorem is referenced by: neg1sqe1 10609 binom21 10627 binom2sub1 10629 sqrt1 11048 sinbnd 11753 cosbnd 11754 cos1bnd 11760 cos2bnd 11761 cos01gt0 11763 sqnprm 12128 numdensq 12194 nn0sqrtelqelz 12198 sinhalfpilem 14083 cos2pi 14096 tangtx 14130 coskpi 14140 lgslem1 14272 lgsne0 14310 lgssq 14312 lgssq2 14313 1lgs 14315 lgs1 14316 lgsdinn0 14320 2sqlem9 14331 2sqlem10 14332 |
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