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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 12845 | . 2 ⊢ (𝑄 ∈ ℚ → -𝑄 ∈ ℚ) | |
2 | qnumcl 16574 | . . 3 ⊢ (𝑄 ∈ ℚ → (numer‘𝑄) ∈ ℤ) | |
3 | 2 | znegcld 12567 | . 2 ⊢ (𝑄 ∈ ℚ → -(numer‘𝑄) ∈ ℤ) |
4 | qdencl 16575 | . 2 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℕ) | |
5 | 4 | nnzd 12484 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℤ) |
6 | neggcd 16362 | . . . 4 ⊢ (((numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℤ) → (-(numer‘𝑄) gcd (denom‘𝑄)) = ((numer‘𝑄) gcd (denom‘𝑄))) | |
7 | 2, 5, 6 | syl2anc 584 | . . 3 ⊢ (𝑄 ∈ ℚ → (-(numer‘𝑄) gcd (denom‘𝑄)) = ((numer‘𝑄) gcd (denom‘𝑄))) |
8 | qnumdencoprm 16579 | . . 3 ⊢ (𝑄 ∈ ℚ → ((numer‘𝑄) gcd (denom‘𝑄)) = 1) | |
9 | 7, 8 | eqtrd 2777 | . 2 ⊢ (𝑄 ∈ ℚ → (-(numer‘𝑄) gcd (denom‘𝑄)) = 1) |
10 | qeqnumdivden 16580 | . . . 4 ⊢ (𝑄 ∈ ℚ → 𝑄 = ((numer‘𝑄) / (denom‘𝑄))) | |
11 | 10 | negeqd 11353 | . . 3 ⊢ (𝑄 ∈ ℚ → -𝑄 = -((numer‘𝑄) / (denom‘𝑄))) |
12 | 2 | zcnd 12566 | . . . 4 ⊢ (𝑄 ∈ ℚ → (numer‘𝑄) ∈ ℂ) |
13 | 4 | nncnd 12127 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℂ) |
14 | 4 | nnne0d 12161 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ≠ 0) |
15 | 12, 13, 14 | divnegd 11902 | . . 3 ⊢ (𝑄 ∈ ℚ → -((numer‘𝑄) / (denom‘𝑄)) = (-(numer‘𝑄) / (denom‘𝑄))) |
16 | 11, 15 | eqtrd 2777 | . 2 ⊢ (𝑄 ∈ ℚ → -𝑄 = (-(numer‘𝑄) / (denom‘𝑄))) |
17 | qnumdenbi 16578 | . . 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 477 | . 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 1378 | 1 ⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 1c1 11010 -cneg 11344 / cdiv 11770 ℕcn 12111 ℤcz 12457 ℚcq 12827 gcd cgcd 16333 numercnumer 16567 denomcdenom 16568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-dvds 16096 df-gcd 16334 df-numer 16569 df-denom 16570 |
This theorem is referenced by: divnumden2 31538 |
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