2sqreultb - Metamath Proof Explorer

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Theorem 2sqreultb 27420

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 27409	. 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 3383	. . . 4 ⊢ (∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ₀ 𝜑)
5	4	reubii 3383	. . 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 27415	. . 3 ⊢ ∀𝑎 ∈ ℕ₀ ∃*𝑏 ∈ ℕ₀ 𝜑
8		2reu1 3892	. . 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 304	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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 ∃!wreu 3372 ∃*wrmo 3373 class class class wbr 5152 (class class class)co 7426 1c1 11149 + caddc 11151 < clt 11288 2c2 12307 4c4 12309 ℕ₀cn0 12512 mod cmo 13876 ↑cexp 14068 ℙcprime 16651

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-er 8733 df-ec 8735 df-qs 8739 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-inf 9476 df-oi 9543 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-xnn0 12585 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-dvds 16241 df-gcd 16479 df-prm 16652 df-phi 16744 df-pc 16815 df-gz 16908 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-imas 17499 df-qus 17500 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-nsg 19093 df-eqg 19094 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-srg 20141 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-nzr 20466 df-subrng 20497 df-subrg 20522 df-drng 20640 df-field 20641 df-lmod 20759 df-lss 20830 df-lsp 20870 df-sra 21072 df-rgmod 21073 df-lidl 21118 df-rsp 21119 df-2idl 21158 df-rlreg 21244 df-domn 21245 df-idom 21246 df-cnfld 21294 df-zring 21387 df-zrh 21443 df-zn 21446 df-assa 21801 df-asp 21802 df-ascl 21803 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-evls 22035 df-evl 22036 df-psr1 22117 df-vr1 22118 df-ply1 22119 df-coe1 22120 df-evl1 22254 df-mdeg 26016 df-deg1 26017 df-mon1 26094 df-uc1p 26095 df-q1p 26096 df-r1p 26097 df-lgs 27256

This theorem is referenced by: 2sqreuopltb 27426

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