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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 16075 | . . . 4 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) gcd (denom‘𝐴)) = 1) | |
2 | 1 | oveq1d 7150 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (1↑2)) |
3 | qnumcl 16070 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ) | |
4 | qdencl 16071 | . . . . 5 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | |
5 | 4 | nnzd 12074 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℤ) |
6 | zgcdsq 16083 | . . . 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 587 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2))) |
8 | sq1 13554 | . . . 4 ⊢ (1↑2) = 1 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℚ → (1↑2) = 1) |
10 | 2, 7, 9 | 3eqtr3d 2841 | . 2 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2)) = 1) |
11 | qeqnumdivden 16076 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴))) | |
12 | 11 | oveq1d 7150 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴) / (denom‘𝐴))↑2)) |
13 | 3 | zcnd 12076 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℂ) |
14 | 4 | nncnd 11641 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℂ) |
15 | 4 | nnne0d 11675 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ≠ 0) |
16 | 13, 14, 15 | sqdivd 13519 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) / (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
17 | 12, 16 | eqtrd 2833 | . 2 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
18 | qsqcl 13491 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ) | |
19 | zsqcl 13490 | . . . 4 ⊢ ((numer‘𝐴) ∈ ℤ → ((numer‘𝐴)↑2) ∈ ℤ) | |
20 | 3, 19 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴)↑2) ∈ ℤ) |
21 | 4 | nnsqcld 13601 | . . 3 ⊢ (𝐴 ∈ ℚ → ((denom‘𝐴)↑2) ∈ ℕ) |
22 | qnumdenbi 16074 | . . 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 1368 | . 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 711 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 1c1 10527 / cdiv 11286 ℕcn 11625 2c2 11680 ℤcz 11969 ℚcq 12336 ↑cexp 13425 gcd cgcd 15833 numercnumer 16063 denomcdenom 16064 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-numer 16065 df-denom 16066 |
This theorem is referenced by: numsq 16085 densq 16086 |
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