Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2sqreunn | Structured version Visualization version GIF version |
Description: There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two positive integers. See 2sqnn 26178 for the existence of such a decomposition. (Contributed by AV, 11-Jun-2023.) (Revised by AV, 25-Jun-2023.) |
Ref | Expression |
---|---|
2sqreu.1 | ⊢ (𝜑 ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
Ref | Expression |
---|---|
2sqreunn | ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreunnlem1 26188 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
2 | 2sqreu.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
3 | 2 | bicomi 227 | . . . . 5 ⊢ ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ 𝜑) |
4 | 3 | reubii 3295 | . . . 4 ⊢ (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ 𝜑) |
5 | 4 | reubii 3295 | . . 3 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ 𝜑) |
6 | 2 | 2sqreunnlem2 26194 | . . . 4 ⊢ ∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑 |
7 | 2reu1 3789 | . . . 4 ⊢ (∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑 → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ 𝜑 ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))) | |
8 | 6, 7 | mp1i 13 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ 𝜑 ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))) |
9 | 5, 8 | syl5bb 286 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))) |
10 | 1, 9 | mpbid 235 | 1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∃wrex 3055 ∃!wreu 3056 ∃*wrmo 3057 class class class wbr 5031 (class class class)co 7173 1c1 10619 + caddc 10621 ≤ cle 10757 ℕcn 11719 2c2 11774 4c4 11776 mod cmo 13331 ↑cexp 13524 ℙcprime 16115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 ax-mulf 10698 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-ofr 7429 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-tpos 7924 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-oadd 8138 df-er 8323 df-ec 8325 df-qs 8329 df-map 8442 df-pm 8443 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-sup 8982 df-inf 8983 df-oi 9050 df-dju 9406 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-xnn0 12052 df-z 12066 df-dec 12183 df-uz 12328 df-q 12434 df-rp 12476 df-fz 12985 df-fzo 13128 df-fl 13256 df-mod 13332 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-dvds 15703 df-gcd 15941 df-prm 16116 df-phi 16206 df-pc 16277 df-gz 16369 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-hom 16695 df-cco 16696 df-0g 16821 df-gsum 16822 df-prds 16827 df-pws 16829 df-imas 16887 df-qus 16888 df-mre 16963 df-mrc 16964 df-acs 16966 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-mhm 18075 df-submnd 18076 df-grp 18225 df-minusg 18226 df-sbg 18227 df-mulg 18346 df-subg 18397 df-nsg 18398 df-eqg 18399 df-ghm 18477 df-cntz 18568 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-srg 19378 df-ring 19421 df-cring 19422 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-invr 19547 df-dvr 19558 df-rnghom 19592 df-drng 19626 df-field 19627 df-subrg 19655 df-lmod 19758 df-lss 19826 df-lsp 19866 df-sra 20066 df-rgmod 20067 df-lidl 20068 df-rsp 20069 df-2idl 20127 df-nzr 20153 df-rlreg 20178 df-domn 20179 df-idom 20180 df-cnfld 20221 df-zring 20293 df-zrh 20327 df-zn 20330 df-assa 20672 df-asp 20673 df-ascl 20674 df-psr 20725 df-mvr 20726 df-mpl 20727 df-opsr 20729 df-evls 20889 df-evl 20890 df-psr1 20958 df-vr1 20959 df-ply1 20960 df-coe1 20961 df-evl1 21089 df-mdeg 24808 df-deg1 24809 df-mon1 24886 df-uc1p 24887 df-q1p 24888 df-r1p 24889 df-lgs 26034 |
This theorem is referenced by: 2sqreuopnn 26202 |
Copyright terms: Public domain | W3C validator |