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Mirrors > Home > ILE Home > Th. List > numdensq | 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 11663 | . . . 4 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) gcd (denom‘𝐴)) = 1) | |
2 | 1 | oveq1d 5721 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (1↑2)) |
3 | qnumcl 11658 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ) | |
4 | qdencl 11659 | . . . . 5 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ) | |
5 | 4 | nnzd 9024 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℤ) |
6 | zgcdsq 11671 | . . . 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 406 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) gcd (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2))) |
8 | sq1 10227 | . . . 4 ⊢ (1↑2) = 1 | |
9 | 8 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℚ → (1↑2) = 1) |
10 | 2, 7, 9 | 3eqtr3d 2140 | . 2 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴)↑2) gcd ((denom‘𝐴)↑2)) = 1) |
11 | qeqnumdivden 11664 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 = ((numer‘𝐴) / (denom‘𝐴))) | |
12 | 11 | oveq1d 5721 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴) / (denom‘𝐴))↑2)) |
13 | 3 | zcnd 9026 | . . . 4 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℂ) |
14 | 4 | nncnd 8592 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℂ) |
15 | 4 | nnap0d 8624 | . . . 4 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) # 0) |
16 | 13, 14, 15 | sqdivapd 10278 | . . 3 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) / (denom‘𝐴))↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
17 | 12, 16 | eqtrd 2132 | . 2 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) = (((numer‘𝐴)↑2) / ((denom‘𝐴)↑2))) |
18 | qsqcl 10205 | . . 3 ⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ) | |
19 | zsqcl 10204 | . . . 4 ⊢ ((numer‘𝐴) ∈ ℤ → ((numer‘𝐴)↑2) ∈ ℤ) | |
20 | 3, 19 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴)↑2) ∈ ℤ) |
21 | 4 | nnsqcld 10286 | . . 3 ⊢ (𝐴 ∈ ℚ → ((denom‘𝐴)↑2) ∈ ℕ) |
22 | qnumdenbi 11662 | . . 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 1184 | . 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 895 | 1 ⊢ (𝐴 ∈ ℚ → ((numer‘(𝐴↑2)) = ((numer‘𝐴)↑2) ∧ (denom‘(𝐴↑2)) = ((denom‘𝐴)↑2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1299 ∈ wcel 1448 ‘cfv 5059 (class class class)co 5706 1c1 7501 / cdiv 8293 ℕcn 8578 2c2 8629 ℤcz 8906 ℚcq 9261 ↑cexp 10133 gcd cgcd 11430 numercnumer 11651 denomcdenom 11652 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-sup 6786 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-fz 9632 df-fzo 9761 df-fl 9884 df-mod 9937 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-dvds 11289 df-gcd 11431 df-numer 11653 df-denom 11654 |
This theorem is referenced by: numsq 11673 densq 11674 |
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