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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 7019 |
. . . . . . 7
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2 | 1nq 6979 |
. . . . . . . 8
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3 | addclnq 6988 |
. . . . . . . . 9
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4 | 2, 2, 3 | mp2an 418 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | ltmnqg 7014 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 2, 4, 5 | mp3an12 1264 |
. . . . . . 7
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7 | 1, 6 | mpbii 147 |
. . . . . 6
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8 | mulidnq 7002 |
. . . . . 6
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9 | distrnqg 7000 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 2, 2, 9 | mp3an23 1266 |
. . . . . . 7
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11 | 8, 8 | oveq12d 5684 |
. . . . . . 7
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12 | 10, 11 | eqtrd 2121 |
. . . . . 6
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13 | 7, 8, 12 | 3brtr3d 3880 |
. . . . 5
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14 | 13 | adantl 272 |
. . . 4
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15 | simpr 109 |
. . . . 5
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16 | addclnq 6988 |
. . . . . . 7
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17 | 16 | anidms 390 |
. . . . . 6
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18 | 17 | adantl 272 |
. . . . 5
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19 | simpl 108 |
. . . . 5
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20 | ltanqg 7013 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 15, 18, 19, 20 | syl3anc 1175 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 14, 21 | mpbid 146 |
. . 3
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23 | addcomnqg 6994 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | addcomnqg 6994 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | adantl 272 |
. . . 4
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26 | addassnqg 6995 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 26 | adantl 272 |
. . . 4
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28 | 19, 15, 15, 25, 27 | caov12d 5840 |
. . 3
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29 | 22, 23, 28 | 3brtr3d 3880 |
. 2
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30 | addclnq 6988 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | ltanqg 7013 |
. . 3
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32 | 19, 30, 15, 31 | syl3anc 1175 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 29, 32 | mpbird 166 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6917 df-pli 6918 df-mi 6919 df-lti 6920 df-plpq 6957 df-mpq 6958 df-enq 6960 df-nqqs 6961 df-plqqs 6962 df-mqqs 6963 df-1nqqs 6964 df-ltnqqs 6966 |
This theorem is referenced by: ltexnqq 7021 nsmallnqq 7025 subhalfnqq 7027 ltbtwnnqq 7028 prarloclemarch2 7032 ltexprlemm 7213 ltexprlemopl 7214 addcanprleml 7227 addcanprlemu 7228 recexprlemm 7237 cauappcvgprlemm 7258 cauappcvgprlemopl 7259 cauappcvgprlem2 7273 caucvgprlemnkj 7279 caucvgprlemnbj 7280 caucvgprlemm 7281 caucvgprlemopl 7282 caucvgprprlemnjltk 7304 caucvgprprlemopl 7310 |
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