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Theorem mul2sq 24858
Description: Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypothesis
Ref Expression
2sq.1 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
Assertion
Ref Expression
mul2sq ((𝐴𝑆𝐵𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

Proof of Theorem mul2sq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2sq.1 . . 3 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
212sqlem1 24856 . 2 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
312sqlem1 24856 . 2 (𝐵𝑆 ↔ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2))
4 reeanv 3082 . . 3 (∃𝑥 ∈ ℤ[i] ∃𝑦 ∈ ℤ[i] (𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) ↔ (∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2) ∧ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2)))
5 gzmulcl 15423 . . . . . . 7 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (𝑥 · 𝑦) ∈ ℤ[i])
6 gzcn 15417 . . . . . . . . . 10 (𝑥 ∈ ℤ[i] → 𝑥 ∈ ℂ)
7 gzcn 15417 . . . . . . . . . 10 (𝑦 ∈ ℤ[i] → 𝑦 ∈ ℂ)
8 absmul 13825 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
96, 7, 8syl2an 492 . . . . . . . . 9 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
109oveq1d 6539 . . . . . . . 8 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((abs‘(𝑥 · 𝑦))↑2) = (((abs‘𝑥) · (abs‘𝑦))↑2))
116abscld 13966 . . . . . . . . . 10 (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℝ)
1211recnd 9921 . . . . . . . . 9 (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℂ)
137abscld 13966 . . . . . . . . . 10 (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℝ)
1413recnd 9921 . . . . . . . . 9 (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℂ)
15 sqmul 12740 . . . . . . . . 9 (((abs‘𝑥) ∈ ℂ ∧ (abs‘𝑦) ∈ ℂ) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
1612, 14, 15syl2an 492 . . . . . . . 8 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
1710, 16eqtr2d 2641 . . . . . . 7 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2))
18 fveq2 6085 . . . . . . . . . 10 (𝑧 = (𝑥 · 𝑦) → (abs‘𝑧) = (abs‘(𝑥 · 𝑦)))
1918oveq1d 6539 . . . . . . . . 9 (𝑧 = (𝑥 · 𝑦) → ((abs‘𝑧)↑2) = ((abs‘(𝑥 · 𝑦))↑2))
2019eqeq2d 2616 . . . . . . . 8 (𝑧 = (𝑥 · 𝑦) → ((((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2) ↔ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2)))
2120rspcev 3278 . . . . . . 7 (((𝑥 · 𝑦) ∈ ℤ[i] ∧ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2)) → ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
225, 17, 21syl2anc 690 . . . . . 6 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
2312sqlem1 24856 . . . . . 6 ((((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆 ↔ ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
2422, 23sylibr 222 . . . . 5 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆)
25 oveq12 6533 . . . . . 6 ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
2625eleq1d 2668 . . . . 5 ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → ((𝐴 · 𝐵) ∈ 𝑆 ↔ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆))
2724, 26syl5ibrcom 235 . . . 4 ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆))
2827rexlimivv 3014 . . 3 (∃𝑥 ∈ ℤ[i] ∃𝑦 ∈ ℤ[i] (𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)
294, 28sylbir 223 . 2 ((∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2) ∧ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)
302, 3, 29syl2anb 494 1 ((𝐴𝑆𝐵𝑆) → (𝐴 · 𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wrex 2893  cmpt 4634  ran crn 5026  cfv 5787  (class class class)co 6524  cc 9787   · cmul 9794  2c2 10914  cexp 12674  abscabs 13765  ℤ[i]cgz 15414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-sup 8205  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-seq 12616  df-exp 12675  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-gz 15415
This theorem is referenced by: (None)
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