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Mirrors > Home > ILE Home > Th. List > ltaddnq | Unicode version |
Description: The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
ltaddnq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2nq 7407 |
. . . . . . 7
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2 | 1nq 7367 |
. . . . . . . 8
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3 | addclnq 7376 |
. . . . . . . . 9
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4 | 2, 2, 3 | mp2an 426 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | ltmnqg 7402 |
. . . . . . . 8
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6 | 2, 4, 5 | mp3an12 1327 |
. . . . . . 7
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7 | 1, 6 | mpbii 148 |
. . . . . 6
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8 | mulidnq 7390 |
. . . . . 6
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9 | distrnqg 7388 |
. . . . . . . 8
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10 | 2, 2, 9 | mp3an23 1329 |
. . . . . . 7
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11 | 8, 8 | oveq12d 5895 |
. . . . . . 7
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12 | 10, 11 | eqtrd 2210 |
. . . . . 6
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13 | 7, 8, 12 | 3brtr3d 4036 |
. . . . 5
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14 | 13 | adantl 277 |
. . . 4
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15 | simpr 110 |
. . . . 5
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16 | addclnq 7376 |
. . . . . . 7
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17 | 16 | anidms 397 |
. . . . . 6
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18 | 17 | adantl 277 |
. . . . 5
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19 | simpl 109 |
. . . . 5
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20 | ltanqg 7401 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 15, 18, 19, 20 | syl3anc 1238 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 14, 21 | mpbid 147 |
. . 3
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23 | addcomnqg 7382 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | addcomnqg 7382 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | adantl 277 |
. . . 4
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26 | addassnqg 7383 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 26 | adantl 277 |
. . . 4
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28 | 19, 15, 15, 25, 27 | caov12d 6058 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 22, 23, 28 | 3brtr3d 4036 |
. 2
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30 | addclnq 7376 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | ltanqg 7401 |
. . 3
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32 | 19, 30, 15, 31 | syl3anc 1238 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 29, 32 | mpbird 167 |
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-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-1o 6419 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-mpq 7346 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-mqqs 7351 df-1nqqs 7352 df-ltnqqs 7354 |
This theorem is referenced by: ltexnqq 7409 nsmallnqq 7413 subhalfnqq 7415 ltbtwnnqq 7416 prarloclemarch2 7420 ltexprlemm 7601 ltexprlemopl 7602 addcanprleml 7615 addcanprlemu 7616 recexprlemm 7625 cauappcvgprlemm 7646 cauappcvgprlemopl 7647 cauappcvgprlem2 7661 caucvgprlemnkj 7667 caucvgprlemnbj 7668 caucvgprlemm 7669 caucvgprlemopl 7670 caucvgprprlemnjltk 7692 caucvgprprlemopl 7698 suplocexprlemmu 7719 |
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