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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > numdenneg | Structured version Visualization version GIF version |
Description: Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.) |
Ref | Expression |
---|---|
numdenneg | ⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qnegcl 12113 | . 2 ⊢ (𝑄 ∈ ℚ → -𝑄 ∈ ℚ) | |
2 | qnumcl 15852 | . . 3 ⊢ (𝑄 ∈ ℚ → (numer‘𝑄) ∈ ℤ) | |
3 | 2 | znegcld 11836 | . 2 ⊢ (𝑄 ∈ ℚ → -(numer‘𝑄) ∈ ℤ) |
4 | qdencl 15853 | . 2 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℕ) | |
5 | 4 | nnzd 11833 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℤ) |
6 | neggcd 15650 | . . . 4 ⊢ (((numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℤ) → (-(numer‘𝑄) gcd (denom‘𝑄)) = ((numer‘𝑄) gcd (denom‘𝑄))) | |
7 | 2, 5, 6 | syl2anc 579 | . . 3 ⊢ (𝑄 ∈ ℚ → (-(numer‘𝑄) gcd (denom‘𝑄)) = ((numer‘𝑄) gcd (denom‘𝑄))) |
8 | qnumdencoprm 15857 | . . 3 ⊢ (𝑄 ∈ ℚ → ((numer‘𝑄) gcd (denom‘𝑄)) = 1) | |
9 | 7, 8 | eqtrd 2813 | . 2 ⊢ (𝑄 ∈ ℚ → (-(numer‘𝑄) gcd (denom‘𝑄)) = 1) |
10 | qeqnumdivden 15858 | . . . 4 ⊢ (𝑄 ∈ ℚ → 𝑄 = ((numer‘𝑄) / (denom‘𝑄))) | |
11 | 10 | negeqd 10616 | . . 3 ⊢ (𝑄 ∈ ℚ → -𝑄 = -((numer‘𝑄) / (denom‘𝑄))) |
12 | 2 | zcnd 11835 | . . . 4 ⊢ (𝑄 ∈ ℚ → (numer‘𝑄) ∈ ℂ) |
13 | 4 | nncnd 11392 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℂ) |
14 | 4 | nnne0d 11425 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ≠ 0) |
15 | 12, 13, 14 | divnegd 11164 | . . 3 ⊢ (𝑄 ∈ ℚ → -((numer‘𝑄) / (denom‘𝑄)) = (-(numer‘𝑄) / (denom‘𝑄))) |
16 | 11, 15 | eqtrd 2813 | . 2 ⊢ (𝑄 ∈ ℚ → -𝑄 = (-(numer‘𝑄) / (denom‘𝑄))) |
17 | qnumdenbi 15856 | . . 3 ⊢ ((-𝑄 ∈ ℚ ∧ -(numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℕ) → (((-(numer‘𝑄) gcd (denom‘𝑄)) = 1 ∧ -𝑄 = (-(numer‘𝑄) / (denom‘𝑄))) ↔ ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))) | |
18 | 17 | biimpa 470 | . 2 ⊢ (((-𝑄 ∈ ℚ ∧ -(numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℕ) ∧ ((-(numer‘𝑄) gcd (denom‘𝑄)) = 1 ∧ -𝑄 = (-(numer‘𝑄) / (denom‘𝑄)))) → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
19 | 1, 3, 4, 9, 16, 18 | syl32anc 1446 | 1 ⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 1c1 10273 -cneg 10607 / cdiv 11032 ℕcn 11374 ℤcz 11728 ℚcq 12095 gcd cgcd 15622 numercnumer 15845 denomcdenom 15846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-gcd 15623 df-numer 15847 df-denom 15848 |
This theorem is referenced by: divnumden2 30128 |
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