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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 8596 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 class class class wbr 4015 ℝcr 7823 0cc0 7824 < clt 8005 # cap 8551 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 |
This theorem is referenced by: prodgt0gt0 8821 prodgt0 8822 ltdiv1 8838 ltmuldiv 8844 ledivmul 8847 lt2mul2div 8849 lemuldiv 8851 ltrec 8853 lerec 8854 ltrec1 8858 lerec2 8859 ledivdiv 8860 lediv2 8861 ltdiv23 8862 lediv23 8863 lediv12a 8864 recp1lt1 8869 ledivp1 8873 nnap0 8961 rpap0 9683 modq0 10342 mulqmod0 10343 negqmod0 10344 modqlt 10346 modqdiffl 10348 modqid0 10363 modqcyc 10372 modqmuladdnn0 10381 q2txmodxeq0 10397 modqdi 10405 ltexp2a 10585 leexp2a 10586 expnbnd 10657 expcanlem 10708 expcan 10709 resqrexlemover 11032 resqrexlemcalc1 11036 resqrexlemcalc2 11037 ltabs 11109 divcnv 11518 expcnvre 11524 georeclim 11534 geoisumr 11539 cvgratnnlembern 11544 cvgratnnlemfm 11550 cvgratz 11553 cnopnap 14321 reeff1oleme 14420 tangtx 14486 trirec0 15020 ltlenmkv 15046 |
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