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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 16087 | . . . 4 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) gcd (denom‘𝐴)) = 1) | |
2 | 1 | oveq1d 7173 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (1↑2)) |
3 | qnumcl 16082 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ) | |
4 | qdencl 16083 | . . . . 5 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | |
5 | 4 | nnzd 12089 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℤ) |
6 | zgcdsq 16095 | . . . 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 586 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2))) |
8 | sq1 13561 | . . . 4 ⊢ (1↑2) = 1 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℚ → (1↑2) = 1) |
10 | 2, 7, 9 | 3eqtr3d 2866 | . 2 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2)) = 1) |
11 | qeqnumdivden 16088 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴))) | |
12 | 11 | oveq1d 7173 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴) / (denom‘𝐴))↑2)) |
13 | 3 | zcnd 12091 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℂ) |
14 | 4 | nncnd 11656 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℂ) |
15 | 4 | nnne0d 11690 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ≠ 0) |
16 | 13, 14, 15 | sqdivd 13526 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) / (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
17 | 12, 16 | eqtrd 2858 | . 2 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
18 | qsqcl 13498 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ) | |
19 | zsqcl 13497 | . . . 4 ⊢ ((numer‘𝐴) ∈ ℤ → ((numer‘𝐴)↑2) ∈ ℤ) | |
20 | 3, 19 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴)↑2) ∈ ℤ) |
21 | 4 | nnsqcld 13608 | . . 3 ⊢ (𝐴 ∈ ℚ → ((denom‘𝐴)↑2) ∈ ℕ) |
22 | qnumdenbi 16086 | . . 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 1367 | . 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 710 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 1c1 10540 / cdiv 11299 ℕcn 11640 2c2 11695 ℤcz 11984 ℚcq 12351 ↑cexp 13432 gcd cgcd 15845 numercnumer 16075 denomcdenom 16076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-numer 16077 df-denom 16078 |
This theorem is referenced by: numsq 16097 densq 16098 |
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