2sqreuopb - Metamath Proof Explorer

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Theorem 2sqreuopb 27538

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 27531. (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 27537	. 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 5156	. . . 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 7453	. . . . . 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 7453	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑏↑2) = ((2^nd ‘𝑝)↑2))
7	4, 6	oveq12d 7456	. . . . 5 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → ((𝑎↑2) + (𝑏↑2)) = (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)))
8	7	eqeq1d 2739	. . . 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 8067	. 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 2108 ∃!wreu 3378 ⟨cop 4640 class class class wbr 5151 × cxp 5691 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020 2^nd c2nd 8021 1c1 11163 + caddc 11165 < clt 11302 ℕcn 12273 2c2 12328 4c4 12330 mod cmo 13915 ↑cexp 14108 ℙcprime 16714

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-pre-sup 11240 ax-addf 11241 ax-mulf 11242

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-se 5646 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-isom 6578 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-of 7704 df-ofr 7705 df-om 7895 df-1st 8022 df-2nd 8023 df-supp 8194 df-tpos 8259 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8753 df-ec 8755 df-qs 8759 df-map 8876 df-pm 8877 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-fsupp 9409 df-sup 9489 df-inf 9490 df-oi 9557 df-dju 9948 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-div 11928 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-xnn0 12607 df-z 12621 df-dec 12741 df-uz 12886 df-q 12998 df-rp 13042 df-fz 13554 df-fzo 13701 df-fl 13838 df-mod 13916 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-dvds 16297 df-gcd 16538 df-prm 16715 df-phi 16809 df-pc 16880 df-gz 16973 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-starv 17322 df-sca 17323 df-vsca 17324 df-ip 17325 df-tset 17326 df-ple 17327 df-ds 17329 df-unif 17330 df-hom 17331 df-cco 17332 df-0g 17497 df-gsum 17498 df-prds 17503 df-pws 17505 df-imas 17564 df-qus 17565 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-nzr 20539 df-subrng 20572 df-subrg 20596 df-rlreg 20720 df-domn 20721 df-idom 20722 df-drng 20757 df-field 20758 df-lmod 20886 df-lss 20957 df-lsp 20997 df-sra 21199 df-rgmod 21200 df-lidl 21245 df-rsp 21246 df-2idl 21287 df-cnfld 21392 df-zring 21485 df-zrh 21541 df-zn 21544 df-assa 21900 df-asp 21901 df-ascl 21902 df-psr 21956 df-mvr 21957 df-mpl 21958 df-opsr 21960 df-evls 22125 df-evl 22126 df-psr1 22206 df-vr1 22207 df-ply1 22208 df-coe1 22209 df-evl1 22345 df-mdeg 26120 df-deg1 26121 df-mon1 26196 df-uc1p 26197 df-q1p 26198 df-r1p 26199 df-lgs 27365

This theorem is referenced by: (None)

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