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Mirrors > Home > MPE Home > Th. List > numdensq | Structured version Visualization version GIF version |
Description: Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
numdensq | ⊢ (𝐴 ∈ ℚ → ((numer‘(𝐴↑2)) = ((numer‘𝐴)↑2) ∧ (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qnumdencoprm 16073 | . . . 4 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) gcd (denom‘𝐴)) = 1) | |
2 | 1 | oveq1d 7160 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (1↑2)) |
3 | qnumcl 16068 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ) | |
4 | qdencl 16069 | . . . . 5 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | |
5 | 4 | nnzd 12074 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℤ) |
6 | zgcdsq 16081 | . . . 4 ⊢ (((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℤ) → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2))) | |
7 | 3, 5, 6 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2))) |
8 | sq1 13546 | . . . 4 ⊢ (1↑2) = 1 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℚ → (1↑2) = 1) |
10 | 2, 7, 9 | 3eqtr3d 2861 | . 2 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2)) = 1) |
11 | qeqnumdivden 16074 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴))) | |
12 | 11 | oveq1d 7160 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴) / (denom‘𝐴))↑2)) |
13 | 3 | zcnd 12076 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℂ) |
14 | 4 | nncnd 11642 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℂ) |
15 | 4 | nnne0d 11675 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ≠ 0) |
16 | 13, 14, 15 | sqdivd 13511 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) / (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
17 | 12, 16 | eqtrd 2853 | . 2 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
18 | qsqcl 13483 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ) | |
19 | zsqcl 13482 | . . . 4 ⊢ ((numer‘𝐴) ∈ ℤ → ((numer‘𝐴)↑2) ∈ ℤ) | |
20 | 3, 19 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴)↑2) ∈ ℤ) |
21 | 4 | nnsqcld 13593 | . . 3 ⊢ (𝐴 ∈ ℚ → ((denom‘𝐴)↑2) ∈ ℕ) |
22 | qnumdenbi 16072 | . . 3 ⊢ (((𝐴↑2) ∈ ℚ ∧ ((numer‘𝐴)↑2) ∈ ℤ ∧ ((denom‘𝐴)↑2) ∈ ℕ) → (((((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2)) = 1 ∧ (𝐴↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) ↔ ((numer‘(𝐴↑2)) = ((numer‘𝐴)↑2) ∧ (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2)))) | |
23 | 18, 20, 21, 22 | syl3anc 1363 | . 2 ⊢ (𝐴 ∈ ℚ → (((((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2)) = 1 ∧ (𝐴↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) ↔ ((numer‘(𝐴↑2)) = ((numer‘𝐴)↑2) ∧ (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2)))) |
24 | 10, 17, 23 | mpbi2and 708 | 1 ⊢ (𝐴 ∈ ℚ → ((numer‘(𝐴↑2)) = ((numer‘𝐴)↑2) ∧ (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 1c1 10526 / cdiv 11285 ℕcn 11626 2c2 11680 ℤcz 11969 ℚcq 12336 ↑cexp 13417 gcd cgcd 15831 numercnumer 16061 denomcdenom 16062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-numer 16063 df-denom 16064 |
This theorem is referenced by: numsq 16083 densq 16084 |
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