2sqreultb - Metamath Proof Explorer

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Theorem 2sqreultb 27410

Description: There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers iff 𝑃≡1 (mod 4). (Contributed by AV, 10-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 25-Jun-2023.)

**Hypothesis**
Ref	Expression
2sqreult.1	⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

**Assertion**
Ref	Expression
2sqreultb	⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ (∃!𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝜑 ∧ ∃!𝑏 ∈ ℕ₀ ∃𝑎 ∈ ℕ₀ 𝜑)))

Distinct variable group: 𝑃,𝑎,𝑏 Allowed substitution hints: 𝜑(𝑎,𝑏)

**Proof of Theorem 2sqreultb**
Step	Hyp	Ref	Expression
1		2sqreultblem 27399	. 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
2		2sqreult.1	. . . . . 6 ⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
3	2	bicomi 224	. . . . 5 ⊢ ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ 𝜑)
4	3	reubii 3349	. . . 4 ⊢ (∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ₀ 𝜑)
5	4	reubii 3349	. . 3 ⊢ (∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ 𝜑)
6	5	a1i 11	. 2 ⊢ (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ 𝜑))
7	2	2sqreulem4 27405	. . 3 ⊢ ∀𝑎 ∈ ℕ₀ ∃*𝑏 ∈ ℕ₀ 𝜑
8		2reu1 3831	. . 3 ⊢ (∀𝑎 ∈ ℕ₀ ∃*𝑏 ∈ ℕ₀ 𝜑 → (∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ 𝜑 ↔ (∃!𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝜑 ∧ ∃!𝑏 ∈ ℕ₀ ∃𝑎 ∈ ℕ₀ 𝜑)))
9	7, 8	mp1i 13	. 2 ⊢ (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ 𝜑 ↔ (∃!𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝜑 ∧ ∃!𝑏 ∈ ℕ₀ ∃𝑎 ∈ ℕ₀ 𝜑)))
10	1, 6, 9	3bitrd 305	1 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ (∃!𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝜑 ∧ ∃!𝑏 ∈ ℕ₀ ∃𝑎 ∈ ℕ₀ 𝜑)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3049 ∃wrex 3059 ∃!wreu 3338 ∃*wrmo 3339 class class class wbr 5074 (class class class)co 7356 1c1 11028 + caddc 11030 < clt 11168 2c2 12225 4c4 12227 ℕ₀cn0 12426 mod cmo 13817 ↑cexp 14012 ℙcprime 16629

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-er 8632 df-ec 8634 df-qs 8638 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-hash 14282 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16211 df-gcd 16453 df-prm 16630 df-phi 16725 df-pc 16797 df-gz 16890 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-imas 17461 df-qus 17462 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-nsg 19089 df-eqg 19090 df-ghm 19177 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-srg 20157 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-rhm 20441 df-nzr 20479 df-subrng 20512 df-subrg 20536 df-rlreg 20660 df-domn 20661 df-idom 20662 df-drng 20697 df-field 20698 df-lmod 20846 df-lss 20916 df-lsp 20956 df-sra 21157 df-rgmod 21158 df-lidl 21195 df-rsp 21196 df-2idl 21237 df-cnfld 21342 df-zring 21416 df-zrh 21472 df-zn 21475 df-assa 21822 df-asp 21823 df-ascl 21824 df-psr 21878 df-mvr 21879 df-mpl 21880 df-opsr 21882 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evl1 22269 df-mdeg 26008 df-deg1 26009 df-mon1 26084 df-uc1p 26085 df-q1p 26086 df-r1p 26087 df-lgs 27246

This theorem is referenced by: 2sqreuopltb 27416

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