![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 8199 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0) | |
4 | 1, 2, 3 | syl2anc 404 | 1 ⊢ (𝜑 → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 class class class wbr 3867 ℝcr 7446 0cc0 7447 < clt 7619 # cap 8155 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-ltxr 7624 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 |
This theorem is referenced by: prodgt0gt0 8409 prodgt0 8410 ltdiv1 8426 ltmuldiv 8432 ledivmul 8435 lt2mul2div 8437 lemuldiv 8439 ltrec 8441 lerec 8442 ltrec1 8446 lerec2 8447 ledivdiv 8448 lediv2 8449 ltdiv23 8450 lediv23 8451 lediv12a 8452 recp1lt1 8457 ledivp1 8461 nnap0 8549 rpap0 9249 modq0 9885 mulqmod0 9886 negqmod0 9887 modqlt 9889 modqdiffl 9891 modqid0 9906 modqcyc 9915 modqmuladdnn0 9924 q2txmodxeq0 9940 modqdi 9948 ltexp2a 10138 leexp2a 10139 expnbnd 10208 expcanlem 10255 expcan 10256 resqrexlemover 10574 resqrexlemcalc1 10578 resqrexlemcalc2 10579 ltabs 10651 divcnv 11055 expcnvre 11061 georeclim 11071 geoisumr 11076 cvgratnnlembern 11081 cvgratnnlemfm 11087 cvgratz 11090 |
Copyright terms: Public domain | W3C validator |