2sqreuopb - Metamath Proof Explorer

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Theorem 2sqreuopb 27448

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 27441. (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 27447	. 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 5128	. . . 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 7428	. . . . . 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 7428	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑏↑2) = ((2^nd ‘𝑝)↑2))
7	4, 6	oveq12d 7431	. . . . 5 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → ((𝑎↑2) + (𝑏↑2)) = (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)))
8	7	eqeq1d 2736	. . . 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 8041	. 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 1539 ∃wex 1778 ∈ wcel 2107 ∃!wreu 3361 ⟨cop 4612 class class class wbr 5123 × cxp 5663 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 2^nd c2nd 7995 1c1 11138 + caddc 11140 < clt 11277 ℕcn 12248 2c2 12303 4c4 12305 mod cmo 13891 ↑cexp 14084 ℙcprime 16690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215 ax-addf 11216 ax-mulf 11217

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7870 df-1st 7996 df-2nd 7997 df-supp 8168 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8727 df-ec 8729 df-qs 8733 df-map 8850 df-pm 8851 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-fsupp 9384 df-sup 9464 df-inf 9465 df-oi 9532 df-dju 9923 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-xnn0 12583 df-z 12597 df-dec 12717 df-uz 12861 df-q 12973 df-rp 13017 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-hash 14352 df-cj 15120 df-re 15121 df-im 15122 df-sqrt 15256 df-abs 15257 df-dvds 16273 df-gcd 16514 df-prm 16691 df-phi 16785 df-pc 16857 df-gz 16950 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17230 df-ress 17253 df-plusg 17286 df-mulr 17287 df-starv 17288 df-sca 17289 df-vsca 17290 df-ip 17291 df-tset 17292 df-ple 17293 df-ds 17295 df-unif 17296 df-hom 17297 df-cco 17298 df-0g 17457 df-gsum 17458 df-prds 17463 df-pws 17465 df-imas 17524 df-qus 17525 df-mre 17600 df-mrc 17601 df-acs 17603 df-mgm 18622 df-sgrp 18701 df-mnd 18717 df-mhm 18765 df-submnd 18766 df-grp 18923 df-minusg 18924 df-sbg 18925 df-mulg 19055 df-subg 19110 df-nsg 19111 df-eqg 19112 df-ghm 19200 df-cntz 19304 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-srg 20152 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-rhm 20440 df-nzr 20481 df-subrng 20514 df-subrg 20538 df-rlreg 20662 df-domn 20663 df-idom 20664 df-drng 20699 df-field 20700 df-lmod 20828 df-lss 20898 df-lsp 20938 df-sra 21140 df-rgmod 21141 df-lidl 21180 df-rsp 21181 df-2idl 21222 df-cnfld 21327 df-zring 21420 df-zrh 21476 df-zn 21479 df-assa 21827 df-asp 21828 df-ascl 21829 df-psr 21883 df-mvr 21884 df-mpl 21885 df-opsr 21887 df-evls 22046 df-evl 22047 df-psr1 22129 df-vr1 22130 df-ply1 22131 df-coe1 22132 df-evl1 22268 df-mdeg 26030 df-deg1 26031 df-mon1 26106 df-uc1p 26107 df-q1p 26108 df-r1p 26109 df-lgs 27275

This theorem is referenced by: (None)

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