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Mirrors > Home > MPE Home > Th. List > qnumgt0 | Structured version Visualization version GIF version |
Description: A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
qnumgt0 | ⊢ (𝐴 ∈ ℚ → (0 < 𝐴 ↔ 0 < (numer‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10442 | . . 3 ⊢ (𝐴 ∈ ℚ → 0 ∈ ℝ) | |
2 | qre 12166 | . . 3 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
3 | qdencl 15936 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | |
4 | 3 | nnred 11455 | . . 3 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℝ) |
5 | 3 | nngt0d 11488 | . . 3 ⊢ (𝐴 ∈ ℚ → 0 < (denom‘𝐴)) |
6 | ltmul1 11290 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((denom‘𝐴) ∈ ℝ ∧ 0 < (denom‘𝐴))) → (0 < 𝐴 ↔ (0 · (denom‘𝐴)) < (𝐴 · (denom‘𝐴)))) | |
7 | 1, 2, 4, 5, 6 | syl112anc 1355 | . 2 ⊢ (𝐴 ∈ ℚ → (0 < 𝐴 ↔ (0 · (denom‘𝐴)) < (𝐴 · (denom‘𝐴)))) |
8 | 3 | nncnd 11456 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℂ) |
9 | 8 | mul02d 10637 | . . 3 ⊢ (𝐴 ∈ ℚ → (0 · (denom‘𝐴)) = 0) |
10 | qmuldeneqnum 15942 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴 · (denom‘𝐴)) = (numer‘𝐴)) | |
11 | 9, 10 | breq12d 4939 | . 2 ⊢ (𝐴 ∈ ℚ → ((0 · (denom‘𝐴)) < (𝐴 · (denom‘𝐴)) ↔ 0 < (numer‘𝐴))) |
12 | 7, 11 | bitrd 271 | 1 ⊢ (𝐴 ∈ ℚ → (0 < 𝐴 ↔ 0 < (numer‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2051 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 0cc0 10334 · cmul 10339 < clt 10473 ℚcq 12161 numercnumer 15928 denomcdenom 15929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-z 11793 df-uz 12058 df-q 12162 df-rp 12204 df-fl 12976 df-mod 13052 df-seq 13184 df-exp 13244 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-dvds 15467 df-gcd 15703 df-numer 15930 df-denom 15931 |
This theorem is referenced by: qgt0numnn 15946 |
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