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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 108 |
. . . . 5
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2 | 1 | a1i 9 |
. . . 4
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3 | 0lt1 7812 |
. . . . . . . . 9
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4 | 0re 7690 |
. . . . . . . . . 10
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5 | 1re 7689 |
. . . . . . . . . 10
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6 | lttr 7761 |
. . . . . . . . . 10
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7 | 4, 5, 6 | mp3an12 1288 |
. . . . . . . . 9
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8 | 3, 7 | mpani 424 |
. . . . . . . 8
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9 | 8 | adantr 272 |
. . . . . . 7
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10 | ltmul2 8524 |
. . . . . . . . . . 11
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11 | 10 | biimpd 143 |
. . . . . . . . . 10
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12 | 5, 11 | mp3an1 1285 |
. . . . . . . . 9
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13 | 12 | exp32 360 |
. . . . . . . 8
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14 | 13 | impcom 124 |
. . . . . . 7
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15 | 9, 14 | syld 45 |
. . . . . 6
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16 | 15 | impd 252 |
. . . . 5
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17 | ax-1rid 7652 |
. . . . . . 7
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18 | 17 | adantr 272 |
. . . . . 6
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19 | 18 | breq1d 3905 |
. . . . 5
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20 | 16, 19 | sylibd 148 |
. . . 4
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21 | 2, 20 | jcad 303 |
. . 3
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22 | remulcl 7672 |
. . . 4
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23 | lttr 7761 |
. . . . 5
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24 | 5, 23 | mp3an1 1285 |
. . . 4
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25 | 22, 24 | syldan 278 |
. . 3
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26 | 21, 25 | syld 45 |
. 2
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27 | 26 | imp 123 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-precex 7655 ax-cnre 7656 ax-pre-lttrn 7659 ax-pre-ltadd 7661 ax-pre-mulgt0 7662 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-ltxr 7729 df-sub 7858 df-neg 7859 |
This theorem is referenced by: mulgt1d 8604 addltmul 8860 uz2mulcl 9304 |
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