2sqreuopb - Metamath Proof Explorer

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Theorem 2sqreuopb 27413

Description: There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Alternate ordered pair variant of 2sqreunnltb 27406. (Contributed by AV, 3-Jul-2023.)

**Assertion**
Ref	Expression
2sqreuopb	⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))))

Distinct variable group: 𝑃,𝑎,𝑏,𝑝

**Proof of Theorem 2sqreuopb**
Step	Hyp	Ref	Expression
1		2sqreuopnnltb 27412	. 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)((1^st ‘𝑝) < (2^nd ‘𝑝) ∧ (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)) = 𝑃)))
2		breq12 5107	. . . 4 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑎 < 𝑏 ↔ (1^st ‘𝑝) < (2^nd ‘𝑝)))
3		simpl 482	. . . . . . 7 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → 𝑎 = (1^st ‘𝑝))
4	3	oveq1d 7384	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑎↑2) = ((1^st ‘𝑝)↑2))
5		simpr 484	. . . . . . 7 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → 𝑏 = (2^nd ‘𝑝))
6	5	oveq1d 7384	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑏↑2) = ((2^nd ‘𝑝)↑2))
7	4, 6	oveq12d 7387	. . . . 5 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → ((𝑎↑2) + (𝑏↑2)) = (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)))
8	7	eqeq1d 2731	. . . 4 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)) = 𝑃))
9	2, 8	anbi12d 632	. . 3 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ((1^st ‘𝑝) < (2^nd ‘𝑝) ∧ (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)) = 𝑃)))
10	9	opreuopreu 7992	. 2 ⊢ (∃!𝑝 ∈ (ℕ × ℕ)((1^st ‘𝑝) < (2^nd ‘𝑝) ∧ (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)) = 𝑃) ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
11	1, 10	bitrdi 287	1 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∃!wreu 3349 ⟨cop 4591 class class class wbr 5102 × cxp 5629 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 1c1 11047 + caddc 11049 < clt 11186 ℕcn 12164 2c2 12219 4c4 12221 mod cmo 13809 ↑cexp 14004 ℙcprime 16618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 ax-pre-sup 11124 ax-addf 11125 ax-mulf 11126

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9832 df-card 9870 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-div 11814 df-nn 12165 df-2 12227 df-3 12228 df-4 12229 df-5 12230 df-6 12231 df-7 12232 df-8 12233 df-9 12234 df-n0 12421 df-xnn0 12494 df-z 12508 df-dec 12628 df-uz 12772 df-q 12886 df-rp 12930 df-fz 13447 df-fzo 13594 df-fl 13732 df-mod 13810 df-seq 13945 df-exp 14005 df-hash 14274 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16200 df-gcd 16442 df-prm 16619 df-phi 16713 df-pc 16785 df-gz 16878 df-struct 17094 df-sets 17111 df-slot 17129 df-ndx 17141 df-base 17157 df-ress 17178 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17381 df-gsum 17382 df-prds 17387 df-pws 17389 df-imas 17448 df-qus 17449 df-mre 17524 df-mrc 17525 df-acs 17527 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-mulg 18983 df-subg 19038 df-nsg 19039 df-eqg 19040 df-ghm 19128 df-cntz 19232 df-cmn 19697 df-abl 19698 df-mgp 20062 df-rng 20074 df-ur 20103 df-srg 20108 df-ring 20156 df-cring 20157 df-oppr 20258 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-rhm 20393 df-nzr 20434 df-subrng 20467 df-subrg 20491 df-rlreg 20615 df-domn 20616 df-idom 20617 df-drng 20652 df-field 20653 df-lmod 20801 df-lss 20871 df-lsp 20911 df-sra 21113 df-rgmod 21114 df-lidl 21151 df-rsp 21152 df-2idl 21193 df-cnfld 21298 df-zring 21390 df-zrh 21446 df-zn 21449 df-assa 21796 df-asp 21797 df-ascl 21798 df-psr 21852 df-mvr 21853 df-mpl 21854 df-opsr 21856 df-evls 22015 df-evl 22016 df-psr1 22098 df-vr1 22099 df-ply1 22100 df-coe1 22101 df-evl1 22237 df-mdeg 25994 df-deg1 25995 df-mon1 26070 df-uc1p 26071 df-q1p 26072 df-r1p 26073 df-lgs 27240

This theorem is referenced by: (None)

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