2sqreultb - Metamath Proof Explorer

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Theorem 2sqreultb 27347

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 27336	. 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 223	. . . . 5 ⊢ ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ 𝜑)
4	3	reubii 3379	. . . 4 ⊢ (∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ₀ 𝜑)
5	4	reubii 3379	. . 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 27342	. . 3 ⊢ ∀𝑎 ∈ ℕ₀ ∃*𝑏 ∈ ℕ₀ 𝜑
8		2reu1 3886	. . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ∃!wreu 3368 ∃*wrmo 3369 class class class wbr 5141 (class class class)co 7405 1c1 11113 + caddc 11115 < clt 11252 2c2 12271 4c4 12273 ℕ₀cn0 12476 mod cmo 13840 ↑cexp 14032 ℙcprime 16615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-prm 16616 df-phi 16708 df-pc 16779 df-gz 16872 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-imas 17463 df-qus 17464 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-nzr 20415 df-subrng 20446 df-subrg 20471 df-drng 20589 df-field 20590 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-rlreg 21193 df-domn 21194 df-idom 21195 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 df-assa 21748 df-asp 21749 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-evls 21977 df-evl 21978 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-coe1 22057 df-evl1 22190 df-mdeg 25943 df-deg1 25944 df-mon1 26021 df-uc1p 26022 df-q1p 26023 df-r1p 26024 df-lgs 27183

This theorem is referenced by: 2sqreuopltb 27353

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