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Mirrors > Home > ILE Home > Th. List > mulgt1 | Unicode version |
Description: The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
Ref | Expression |
---|---|
mulgt1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 |
. . . . 5
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2 | 1 | a1i 9 |
. . . 4
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3 | 0lt1 8087 |
. . . . . . . . 9
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4 | 0re 7960 |
. . . . . . . . . 10
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5 | 1re 7959 |
. . . . . . . . . 10
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6 | lttr 8034 |
. . . . . . . . . 10
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7 | 4, 5, 6 | mp3an12 1327 |
. . . . . . . . 9
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8 | 3, 7 | mpani 430 |
. . . . . . . 8
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9 | 8 | adantr 276 |
. . . . . . 7
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10 | ltmul2 8816 |
. . . . . . . . . . 11
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11 | 10 | biimpd 144 |
. . . . . . . . . 10
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12 | 5, 11 | mp3an1 1324 |
. . . . . . . . 9
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13 | 12 | exp32 365 |
. . . . . . . 8
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14 | 13 | impcom 125 |
. . . . . . 7
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15 | 9, 14 | syld 45 |
. . . . . 6
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16 | 15 | impd 254 |
. . . . 5
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17 | ax-1rid 7921 |
. . . . . . 7
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18 | 17 | adantr 276 |
. . . . . 6
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19 | 18 | breq1d 4015 |
. . . . 5
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20 | 16, 19 | sylibd 149 |
. . . 4
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21 | 2, 20 | jcad 307 |
. . 3
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22 | remulcl 7942 |
. . . 4
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23 | lttr 8034 |
. . . . 5
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24 | 5, 23 | mp3an1 1324 |
. . . 4
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25 | 22, 24 | syldan 282 |
. . 3
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26 | 21, 25 | syld 45 |
. 2
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27 | 26 | imp 124 |
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 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-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-lttrn 7928 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 |
This theorem depends on definitions: df-bi 117 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-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-sub 8133 df-neg 8134 |
This theorem is referenced by: mulgt1d 8896 addltmul 9158 uz2mulcl 9611 |
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