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Mirrors > Home > MPE Home > Th. List > mul2sq | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
2sq.1 | ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) |
Ref | Expression |
---|---|
mul2sq | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sq.1 | . . 3 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) | |
2 | 1 | 2sqlem1 24942 | . 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
3 | 1 | 2sqlem1 24942 | . 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2)) |
4 | reeanv 3086 | . . 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 15480 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (𝑥 · 𝑦) ∈ ℤ[i]) | |
6 | gzcn 15474 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ[i] → 𝑥 ∈ ℂ) | |
7 | gzcn 15474 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℤ[i] → 𝑦 ∈ ℂ) | |
8 | absmul 13882 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) | |
9 | 6, 7, 8 | syl2an 493 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
10 | 9 | oveq1d 6564 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((abs‘(𝑥 · 𝑦))↑2) = (((abs‘𝑥) · (abs‘𝑦))↑2)) |
11 | 6 | abscld 14023 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℝ) |
12 | 11 | recnd 9947 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ[i] → (abs‘𝑥) ∈ ℂ) |
13 | 7 | abscld 14023 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℝ) |
14 | 13 | recnd 9947 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℤ[i] → (abs‘𝑦) ∈ ℂ) |
15 | sqmul 12788 | . . . . . . . . 9 ⊢ (((abs‘𝑥) ∈ ℂ ∧ (abs‘𝑦) ∈ ℂ) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2))) | |
16 | 12, 14, 15 | syl2an 493 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥) · (abs‘𝑦))↑2) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2))) |
17 | 10, 16 | eqtr2d 2645 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2)) |
18 | fveq2 6103 | . . . . . . . . . 10 ⊢ (𝑧 = (𝑥 · 𝑦) → (abs‘𝑧) = (abs‘(𝑥 · 𝑦))) | |
19 | 18 | oveq1d 6564 | . . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → ((abs‘𝑧)↑2) = ((abs‘(𝑥 · 𝑦))↑2)) |
20 | 19 | eqeq2d 2620 | . . . . . . . 8 ⊢ (𝑧 = (𝑥 · 𝑦) → ((((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2) ↔ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘(𝑥 · 𝑦))↑2))) |
21 | 20 | rspcev 3282 | . . . . . . 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)) |
22 | 5, 17, 21 | syl2anc 691 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2)) |
23 | 1 | 2sqlem1 24942 | . . . . . 6 ⊢ ((((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆 ↔ ∃𝑧 ∈ ℤ[i] (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) = ((abs‘𝑧)↑2)) |
24 | 22, 23 | sylibr 223 | . . . . 5 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆) |
25 | oveq12 6558 | . . . . . 6 ⊢ ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) = (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2))) | |
26 | 25 | eleq1d 2672 | . . . . 5 ⊢ ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → ((𝐴 · 𝐵) ∈ 𝑆 ↔ (((abs‘𝑥)↑2) · ((abs‘𝑦)↑2)) ∈ 𝑆)) |
27 | 24, 26 | syl5ibrcom 236 | . . . 4 ⊢ ((𝑥 ∈ ℤ[i] ∧ 𝑦 ∈ ℤ[i]) → ((𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆)) |
28 | 27 | rexlimivv 3018 | . . 3 ⊢ (∃𝑥 ∈ ℤ[i] ∃𝑦 ∈ ℤ[i] (𝐴 = ((abs‘𝑥)↑2) ∧ 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆) |
29 | 4, 28 | sylbir 224 | . 2 ⊢ ((∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2) ∧ ∃𝑦 ∈ ℤ[i] 𝐵 = ((abs‘𝑦)↑2)) → (𝐴 · 𝐵) ∈ 𝑆) |
30 | 2, 3, 29 | syl2anb 495 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 · cmul 9820 2c2 10947 ↑cexp 12722 abscabs 13822 ℤ[i]cgz 15471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-gz 15472 |
This theorem is referenced by: (None) |
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