2sqreultb - Metamath Proof Explorer

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Theorem 2sqreultb 27379

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 27368	. 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 3380	. . . 4 ⊢ (∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ₀ 𝜑)
5	4	reubii 3380	. . 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 27374	. . 3 ⊢ ∀𝑎 ∈ ℕ₀ ∃*𝑏 ∈ ℕ₀ 𝜑
8		2reu1 3887	. . 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 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ∃!wreu 3369 ∃*wrmo 3370 class class class wbr 5142 (class class class)co 7414 1c1 11131 + caddc 11133 < clt 11270 2c2 12289 4c4 12291 ℕ₀cn0 12494 mod cmo 13858 ↑cexp 14050 ℙcprime 16633

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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-ec 8720 df-qs 8724 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-inf 9458 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-dvds 16223 df-gcd 16461 df-prm 16634 df-phi 16726 df-pc 16797 df-gz 16890 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-imas 17481 df-qus 17482 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-nsg 19070 df-eqg 19071 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-nzr 20441 df-subrng 20472 df-subrg 20497 df-drng 20615 df-field 20616 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rsp 21094 df-2idl 21133 df-rlreg 21219 df-domn 21220 df-idom 21221 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-zn 21419 df-assa 21774 df-asp 21775 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-evls 22005 df-evl 22006 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-evl1 22222 df-mdeg 25975 df-deg1 25976 df-mon1 26053 df-uc1p 26054 df-q1p 26055 df-r1p 26056 df-lgs 27215

This theorem is referenced by: 2sqreuopltb 27385

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