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Mirrors > Home > MPE Home > Th. List > posdif | Structured version Unicode version |
Description: Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
posdif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubcl 9787 |
. . . 4
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2 | 1 | ancoms 453 |
. . 3
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3 | simpl 457 |
. . 3
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4 | ltaddpos 9943 |
. . 3
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5 | 2, 3, 4 | syl2anc 661 |
. 2
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6 | recn 9486 |
. . . 4
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7 | recn 9486 |
. . . 4
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8 | pncan3 9732 |
. . . 4
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9 | 6, 7, 8 | syl2an 477 |
. . 3
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10 | 9 | breq2d 4415 |
. 2
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11 | 5, 10 | bitr2d 254 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-resscn 9453 ax-1cn 9454 ax-icn 9455 ax-addcl 9456 ax-addrcl 9457 ax-mulcl 9458 ax-mulrcl 9459 ax-mulcom 9460 ax-addass 9461 ax-mulass 9462 ax-distr 9463 ax-i2m1 9464 ax-1ne0 9465 ax-1rid 9466 ax-rnegex 9467 ax-rrecex 9468 ax-cnre 9469 ax-pre-lttri 9470 ax-pre-lttrn 9471 ax-pre-ltadd 9472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-po 4752 df-so 4753 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-er 7214 df-en 7424 df-dom 7425 df-sdom 7426 df-pnf 9534 df-mnf 9535 df-ltxr 9537 df-sub 9711 df-neg 9712 |
This theorem is referenced by: posdifi 10004 posdifd 10040 nnsub 10474 nn0sub 10744 znnsub 10805 rpnnen1lem5 11097 difrp 11138 qbtwnre 11283 expnbnd 12113 expmulnbnd 12116 swrd0 12448 swrdccatin12lem3 12502 eflt 13522 cos01gt0 13596 ndvdsadd 13733 nn0seqcvgd 13866 cshwshashlem2 14244 dvcvx 21628 abelthlem7 22039 sinq12gt0 22105 cosq14gt0 22108 cosne0 22122 tanregt0 22131 logdivlti 22205 logcnlem4 22226 scvxcvx 22515 perfectlem2 22705 rplogsumlem2 22870 dchrisum0flblem1 22893 mblfinlem3 28598 mblfinlem4 28599 dvasin 28648 geomcau 28823 bfp 28891 eluzgtdifelfzo 30387 clwlkisclwwlklem2fv2 30613 numclwwlkovf2ex 30847 |
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