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Mirrors > Home > ILE Home > Th. List > gt0ap0d | GIF version |
Description: Positive implies apart from zero. Because of the way we define #, 𝐴 must be an element of ℝ, not just ℝ*. (Contributed by Jim Kingdon, 27-Feb-2020.) |
Ref | Expression |
---|---|
gt0ap0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
gt0ap0d.2 | ⊢ (𝜑 → 0 < 𝐴) |
Ref | Expression |
---|---|
gt0ap0d | ⊢ (𝜑 → 𝐴 # 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gt0ap0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | gt0ap0d.2 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | gt0ap0 8388 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 0cc0 7620 < clt 7800 # cap 8343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 |
This theorem is referenced by: prodgt0gt0 8609 prodgt0 8610 ltdiv1 8626 ltmuldiv 8632 ledivmul 8635 lt2mul2div 8637 lemuldiv 8639 ltrec 8641 lerec 8642 ltrec1 8646 lerec2 8647 ledivdiv 8648 lediv2 8649 ltdiv23 8650 lediv23 8651 lediv12a 8652 recp1lt1 8657 ledivp1 8661 nnap0 8749 rpap0 9458 modq0 10102 mulqmod0 10103 negqmod0 10104 modqlt 10106 modqdiffl 10108 modqid0 10123 modqcyc 10132 modqmuladdnn0 10141 q2txmodxeq0 10157 modqdi 10165 ltexp2a 10345 leexp2a 10346 expnbnd 10415 expcanlem 10462 expcan 10463 resqrexlemover 10782 resqrexlemcalc1 10786 resqrexlemcalc2 10787 ltabs 10859 divcnv 11266 expcnvre 11272 georeclim 11282 geoisumr 11287 cvgratnnlembern 11292 cvgratnnlemfm 11298 cvgratz 11301 cnopnap 12763 tangtx 12919 |
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