2sqreuopb - Metamath Proof Explorer

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Theorem 2sqreuopb 27385

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 27378. (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 27384	. 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 5114	. . . 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 7404	. . . . . 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 7404	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑏↑2) = ((2^nd ‘𝑝)↑2))
7	4, 6	oveq12d 7407	. . . . 5 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → ((𝑎↑2) + (𝑏↑2)) = (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)))
8	7	eqeq1d 2732	. . . 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 8015	. 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 3354 ⟨cop 4597 class class class wbr 5109 × cxp 5638 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 1c1 11075 + caddc 11077 < clt 11214 ℕcn 12187 2c2 12242 4c4 12244 mod cmo 13837 ↑cexp 14032 ℙcprime 16647

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 ax-mulf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-ofr 7656 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-er 8673 df-ec 8675 df-qs 8679 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-sup 9399 df-inf 9400 df-oi 9469 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-seq 13973 df-exp 14033 df-hash 14302 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-dvds 16229 df-gcd 16471 df-prm 16648 df-phi 16742 df-pc 16814 df-gz 16907 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17410 df-gsum 17411 df-prds 17416 df-pws 17418 df-imas 17477 df-qus 17478 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-nsg 19062 df-eqg 19063 df-ghm 19151 df-cntz 19255 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-srg 20102 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-rhm 20387 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-lmod 20774 df-lss 20844 df-lsp 20884 df-sra 21086 df-rgmod 21087 df-lidl 21124 df-rsp 21125 df-2idl 21166 df-cnfld 21271 df-zring 21363 df-zrh 21419 df-zn 21422 df-assa 21768 df-asp 21769 df-ascl 21770 df-psr 21824 df-mvr 21825 df-mpl 21826 df-opsr 21828 df-evls 21987 df-evl 21988 df-psr1 22070 df-vr1 22071 df-ply1 22072 df-coe1 22073 df-evl1 22209 df-mdeg 25966 df-deg1 25967 df-mon1 26042 df-uc1p 26043 df-q1p 26044 df-r1p 26045 df-lgs 27212

This theorem is referenced by: (None)

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