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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 8412 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 0cc0 7644 < clt 7824 # cap 8367 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 |
This theorem is referenced by: prodgt0gt0 8633 prodgt0 8634 ltdiv1 8650 ltmuldiv 8656 ledivmul 8659 lt2mul2div 8661 lemuldiv 8663 ltrec 8665 lerec 8666 ltrec1 8670 lerec2 8671 ledivdiv 8672 lediv2 8673 ltdiv23 8674 lediv23 8675 lediv12a 8676 recp1lt1 8681 ledivp1 8685 nnap0 8773 rpap0 9487 modq0 10133 mulqmod0 10134 negqmod0 10135 modqlt 10137 modqdiffl 10139 modqid0 10154 modqcyc 10163 modqmuladdnn0 10172 q2txmodxeq0 10188 modqdi 10196 ltexp2a 10376 leexp2a 10377 expnbnd 10446 expcanlem 10493 expcan 10494 resqrexlemover 10814 resqrexlemcalc1 10818 resqrexlemcalc2 10819 ltabs 10891 divcnv 11298 expcnvre 11304 georeclim 11314 geoisumr 11319 cvgratnnlembern 11324 cvgratnnlemfm 11330 cvgratz 11333 cnopnap 12802 reeff1oleme 12901 tangtx 12967 trirec0 13412 |
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