Theorem mul2sq 27655

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 27653	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
3	1	2sqlem1 27653	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2))
4		reeanv 3234	. . 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 17030	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (𝑥 · 𝑦) ∈ ℤ[i])
6		gzcn 17024	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ[i] → 𝑥 ∈ ℂ)
7		gzcn 17024	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℤ[i] → 𝑦 ∈ ℂ)
8		absmul 15381	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
9	6, 7, 8	syl2an 608	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
10	9	oveq1d 7428	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((abs‘(𝑥 · 𝑦))↑2) = (((abs‘𝑥) · (abs‘𝑦))↑2))
11	6	abscld 15526	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℝ)
12	11	recnd 11261	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℂ)
13	7	abscld 15526	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℝ)
14	13	recnd 11261	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℂ)
15		sqmul 14183	. . . . . . . . 9 ⊢ (((abs‘𝑥) ∈ ℂ ∧ (abs‘𝑦) ∈ ℂ) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
16	12, 14, 15	syl2an 608	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
17	10, 16	eqtr2d 2796	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2))
18		fveq2 6878	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → (abs‘𝑧) = (abs‘(𝑥 · 𝑦)))
19	18	oveq1d 7428	. . . . . . . 8 ⊢ (𝑧 = (𝑥 · 𝑦) → ((abs‘𝑧)↑2) = ((abs‘(𝑥 · 𝑦))↑2))
20	19	rspceeqv 3599	. . . . . . 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 596	. . . . . 6 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
22	1	2sqlem1 27653	. . . . . 6 ⊢ ((((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆 ↔ ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2))
23	21, 22	sylibr 237	. . . . 5 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆)
24		oveq12 7422	. . . . . 6 ⊢ ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)))
25	24	eleq1d 2845	. . . . 5 ⊢ ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → ((𝐴 · 𝐵) ∈ 𝑆 ↔ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆))
26	23, 25	syl5ibrcom 250	. . . 4 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆))
27	26	rexlimivv 3204	. . 3 ⊢ (∃𝑥 ∈ ℤ[i] ∃𝑦 ∈ ℤ[i] (𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)
28	4, 27	sylbir 238	. 2 ⊢ ((∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2) ∧ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)
29	2, 3, 28	syl2anb 610	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ↦ cmpt 5186  ran crn 5656  ‘cfv 6533  (class class class)co 7413  ℂcc 11122   · cmul 11129  2c2 12319  ↑cexp 14125  abscabs 15321  ℤ[i]cgz 17021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-z 12616  df-uz 12888  df-rp 13043  df-seq 14066  df-exp 14126  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-gz 17022

This theorem is used by: (None)