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Mirrors > Home > MPE Home > Th. List > 2sqreu | 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 nonnegative integers. See 2sqnn0 26575 for the existence of such a decomposition. (Contributed by AV, 4-Jun-2023.) (Revised by AV, 25-Jun-2023.) |
Ref | Expression |
---|---|
2sqreu.1 | ⊢ (𝜑 ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
Ref | Expression |
---|---|
2sqreu | ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreulem1 26583 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
2 | 2sqreu.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 3324 | . . . 4 ⊢ (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 𝜑) |
5 | 4 | reubii 3324 | . . 3 ⊢ (∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 𝜑) |
6 | 2 | 2sqreulem4 26591 | . . . 4 ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
7 | 2reu1 3831 | . . . 4 ⊢ (∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 → (∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 𝜑 ↔ (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑))) | |
8 | 6, 7 | mp1i 13 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 𝜑 ↔ (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑))) |
9 | 5, 8 | syl5bb 283 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 𝜑 ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 𝜑))) |
10 | 1, 9 | mpbid 231 | 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 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∃!wreu 3066 ∃*wrmo 3067 class class class wbr 5075 (class class class)co 7269 1c1 10861 + caddc 10863 ≤ cle 10999 2c2 12017 4c4 12019 ℕ0cn0 12222 mod cmo 13578 ↑cexp 13771 ℙcprime 16365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 ax-addf 10939 ax-mulf 10940 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-se 5542 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-isom 6437 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-ofr 7526 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-tpos 8031 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-oadd 8290 df-er 8487 df-ec 8489 df-qs 8493 df-map 8606 df-pm 8607 df-ixp 8675 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-fsupp 9118 df-sup 9190 df-inf 9191 df-oi 9258 df-dju 9648 df-card 9686 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-7 12030 df-8 12031 df-9 12032 df-n0 12223 df-xnn0 12295 df-z 12309 df-dec 12427 df-uz 12572 df-q 12678 df-rp 12720 df-fz 13229 df-fzo 13372 df-fl 13501 df-mod 13579 df-seq 13711 df-exp 13772 df-hash 14034 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-dvds 15953 df-gcd 16191 df-prm 16366 df-phi 16456 df-pc 16527 df-gz 16620 df-struct 16837 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-ress 16931 df-plusg 16964 df-mulr 16965 df-starv 16966 df-sca 16967 df-vsca 16968 df-ip 16969 df-tset 16970 df-ple 16971 df-ds 16973 df-unif 16974 df-hom 16975 df-cco 16976 df-0g 17141 df-gsum 17142 df-prds 17147 df-pws 17149 df-imas 17208 df-qus 17209 df-mre 17284 df-mrc 17285 df-acs 17287 df-mgm 18315 df-sgrp 18364 df-mnd 18375 df-mhm 18419 df-submnd 18420 df-grp 18569 df-minusg 18570 df-sbg 18571 df-mulg 18690 df-subg 18741 df-nsg 18742 df-eqg 18743 df-ghm 18821 df-cntz 18912 df-cmn 19377 df-abl 19378 df-mgp 19710 df-ur 19727 df-srg 19731 df-ring 19774 df-cring 19775 df-oppr 19851 df-dvdsr 19872 df-unit 19873 df-invr 19903 df-dvr 19914 df-rnghom 19948 df-drng 19982 df-field 19983 df-subrg 20011 df-lmod 20114 df-lss 20183 df-lsp 20223 df-sra 20423 df-rgmod 20424 df-lidl 20425 df-rsp 20426 df-2idl 20492 df-nzr 20518 df-rlreg 20543 df-domn 20544 df-idom 20545 df-cnfld 20587 df-zring 20660 df-zrh 20694 df-zn 20697 df-assa 21049 df-asp 21050 df-ascl 21051 df-psr 21101 df-mvr 21102 df-mpl 21103 df-opsr 21105 df-evls 21271 df-evl 21272 df-psr1 21340 df-vr1 21341 df-ply1 21342 df-coe1 21343 df-evl1 21471 df-mdeg 25206 df-deg1 25207 df-mon1 25284 df-uc1p 25285 df-q1p 25286 df-r1p 25287 df-lgs 26432 |
This theorem is referenced by: 2sqreuop 26599 |
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