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Mirrors > Home > ILE Home > Th. List > faclbnd2 | Unicode version |
Description: A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
Ref | Expression |
---|---|
faclbnd2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq2 10709 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 2t2e4 9139 |
. . . . . 6
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3 | 1, 2 | eqtr4i 2217 |
. . . . 5
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4 | 3 | oveq2i 5930 |
. . . 4
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5 | 2cn 9055 |
. . . . . 6
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6 | expp1 10620 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | mpan 424 |
. . . . 5
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8 | 7 | oveq1d 5934 |
. . . 4
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9 | 4, 8 | eqtrid 2238 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | expcl 10631 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 5, 10 | mpan 424 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 5 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 2ap0 9077 |
. . . . 5
![]() ![]() ![]() | |
14 | 13 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 11, 12, 12, 12, 14, 14 | divmuldivapd 8853 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 2div2e1 9117 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | oveq2i 5930 |
. . . 4
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18 | 11 | halfcld 9230 |
. . . . 5
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19 | 18 | mulridd 8038 |
. . . 4
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20 | 17, 19 | eqtrid 2238 |
. . 3
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21 | 9, 15, 20 | 3eqtr2rd 2233 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 2nn0 9260 |
. . . 4
![]() ![]() ![]() ![]() | |
23 | faclbnd 10815 |
. . . 4
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24 | 22, 23 | mpan 424 |
. . 3
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25 | 2re 9054 |
. . . . 5
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26 | peano2nn0 9283 |
. . . . 5
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27 | reexpcl 10630 |
. . . . 5
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28 | 25, 26, 27 | sylancr 414 |
. . . 4
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29 | faccl 10809 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 29 | nnred 8997 |
. . . 4
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31 | 4re 9061 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
32 | 1, 31 | eqeltri 2266 |
. . . . . 6
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33 | 4pos 9081 |
. . . . . . 7
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34 | 33, 1 | breqtrri 4057 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 32, 34 | pm3.2i 272 |
. . . . 5
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36 | ledivmul 8898 |
. . . . 5
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37 | 35, 36 | mp3an3 1337 |
. . . 4
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38 | 28, 30, 37 | syl2anc 411 |
. . 3
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39 | 24, 38 | mpbird 167 |
. 2
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40 | 21, 39 | eqbrtrd 4052 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-rp 9723 df-seqfrec 10522 df-exp 10613 df-fac 10800 |
This theorem is referenced by: ege2le3 11817 |
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