2sqreuopb - Metamath Proof Explorer

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Theorem 2sqreuopb 27492

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 27485. (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 27491	. 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 5149	. . . 4 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑎 < 𝑏 ↔ (1^st ‘𝑝) < (2^nd ‘𝑝)))
3		simpl 481	. . . . . . 7 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → 𝑎 = (1^st ‘𝑝))
4	3	oveq1d 7429	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑎↑2) = ((1^st ‘𝑝)↑2))
5		simpr 483	. . . . . . 7 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → 𝑏 = (2^nd ‘𝑝))
6	5	oveq1d 7429	. . . . . 6 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → (𝑏↑2) = ((2^nd ‘𝑝)↑2))
7	4, 6	oveq12d 7432	. . . . 5 ⊢ ((𝑎 = (1^st ‘𝑝) ∧ 𝑏 = (2^nd ‘𝑝)) → ((𝑎↑2) + (𝑏↑2)) = (((1^st ‘𝑝)↑2) + ((2^nd ‘𝑝)↑2)))
8	7	eqeq1d 2728	. . . 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 630	. . 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 8038	. 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 286	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 205 ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∃!wreu 3363 ⟨cop 4630 class class class wbr 5144 × cxp 5671 ‘cfv 6544 (class class class)co 7414 1^st c1st 7991 2^nd c2nd 7992 1c1 11148 + caddc 11150 < clt 11287 ℕcn 12256 2c2 12311 4c4 12313 mod cmo 13881 ↑cexp 14073 ℙcprime 16665

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226 ax-mulf 11227

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-ofr 7681 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-sup 9476 df-inf 9477 df-oi 9544 df-dju 9935 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-xnn0 12589 df-z 12603 df-dec 12722 df-uz 12867 df-q 12977 df-rp 13021 df-fz 13531 df-fzo 13674 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-hash 14341 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-dvds 16250 df-gcd 16488 df-prm 16666 df-phi 16761 df-pc 16832 df-gz 16925 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-starv 17274 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-hom 17283 df-cco 17284 df-0g 17449 df-gsum 17450 df-prds 17455 df-pws 17457 df-imas 17516 df-qus 17517 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201 df-cntz 19305 df-cmn 19774 df-abl 19775 df-mgp 20112 df-rng 20130 df-ur 20159 df-srg 20164 df-ring 20212 df-cring 20213 df-oppr 20310 df-dvdsr 20333 df-unit 20334 df-invr 20364 df-dvr 20377 df-rhm 20448 df-nzr 20489 df-subrng 20522 df-subrg 20547 df-rlreg 20666 df-domn 20667 df-idom 20668 df-drng 20703 df-field 20704 df-lmod 20832 df-lss 20903 df-lsp 20943 df-sra 21145 df-rgmod 21146 df-lidl 21191 df-rsp 21192 df-2idl 21233 df-cnfld 21338 df-zring 21431 df-zrh 21487 df-zn 21490 df-assa 21845 df-asp 21846 df-ascl 21847 df-psr 21900 df-mvr 21901 df-mpl 21902 df-opsr 21904 df-evls 22081 df-evl 22082 df-psr1 22163 df-vr1 22164 df-ply1 22165 df-coe1 22166 df-evl1 22302 df-mdeg 26074 df-deg1 26075 df-mon1 26153 df-uc1p 26154 df-q1p 26155 df-r1p 26156 df-lgs 27319

This theorem is referenced by: (None)

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