Theorem mul2sq 27365

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	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

Proof of Theorem mul2sq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sq.1	. . 3 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
2	1	2sqlem1 27363	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
3	1	2sqlem1 27363	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2))
4		reeanv 3223	. . 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 16907	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (𝑥 · 𝑦) ∈ ℤ[i])
6		gzcn 16901	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ[i] → 𝑥 ∈ ℂ)
7		gzcn 16901	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℤ[i] → 𝑦 ∈ ℂ)
8		absmul 15274	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
9	6, 7, 8	syl2an 595	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
10	9	oveq1d 7435	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((abs‘(𝑥 · 𝑦))↑2) = (((abs‘𝑥) · (abs‘𝑦))↑2))
11	6	abscld 15416	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℝ)
12	11	recnd 11273	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℂ)
13	7	abscld 15416	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℝ)
14	13	recnd 11273	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℂ)
15		sqmul 14116	. . . . . . . . 9 ⊢ (((abs‘𝑥) ∈ ℂ ∧ (abs‘𝑦) ∈ ℂ) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
16	12, 14, 15	syl2an 595	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
17	10, 16	eqtr2d 2769	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2))
18		fveq2 6897	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → (abs‘𝑧) = (abs‘(𝑥 · 𝑦)))
19	18	oveq1d 7435	. . . . . . . 8 ⊢ (𝑧 = (𝑥 · 𝑦) → ((abs‘𝑧)↑2) = ((abs‘(𝑥 · 𝑦))↑2))
20	19	rspceeqv 3631	. . . . . . 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))
21	5, 17, 20	syl2anc 583	. . . . . 6 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
22	1	2sqlem1 27363	. . . . . 6 ⊢ ((((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆 ↔ ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
23	21, 22	sylibr 233	. . . . 5 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆)
24		oveq12 7429	. . . . . 6 ⊢ ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
25	24	eleq1d 2814	. . . . 5 ⊢ ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → ((𝐴 · 𝐵) ∈ 𝑆 ↔ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆))
26	23, 25	syl5ibrcom 246	. . . 4 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆))
27	26	rexlimivv 3196	. . 3 ⊢ (∃𝑥 ∈ ℤ[i] ∃𝑦 ∈ ℤ[i] (𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)
28	4, 27	sylbir 234	. 2 ⊢ ((∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2) ∧ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)
29	2, 3, 28	syl2anb 597	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3067   ↦ cmpt 5231  ran crn 5679  ‘cfv 6548  (class class class)co 7420  ℂcc 11137   · cmul 11144  2c2 12298  ↑cexp 14059  abscabs 15214  ℤ[i]cgz 16898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9466  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-n0 12504  df-z 12590  df-uz 12854  df-rp 13008  df-seq 14000  df-exp 14060  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-gz 16899

This theorem is referenced by: (None)