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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 26013 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 26023 | . 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 226 | . . . . 5 ⊢ ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ 𝜑) |
4 | 3 | reubii 3390 | . . . 4 ⊢ (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ 𝜑) |
5 | 4 | reubii 3390 | . . 3 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ 𝜑) |
6 | 2 | 2sqreunnlem2 26029 | . . . 4 ⊢ ∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑 |
7 | 2reu1 3878 | . . . 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 285 | . 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 234 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 ∃!wreu 3139 ∃*wrmo 3140 class class class wbr 5063 (class class class)co 7153 1c1 10535 + caddc 10537 ≤ cle 10673 ℕcn 11635 2c2 11690 4c4 11692 mod cmo 13235 ↑cexp 13427 ℙcprime 16011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 ax-pre-sup 10612 ax-addf 10613 ax-mulf 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-iin 4919 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-of 7406 df-ofr 7407 df-om 7578 df-1st 7686 df-2nd 7687 df-supp 7828 df-tpos 7889 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-oadd 8103 df-er 8286 df-ec 8288 df-qs 8292 df-map 8405 df-pm 8406 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-fsupp 8831 df-sup 8903 df-inf 8904 df-oi 8971 df-dju 9327 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-div 11295 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-xnn0 11966 df-z 11980 df-dec 12097 df-uz 12242 df-q 12347 df-rp 12388 df-fz 12891 df-fzo 13032 df-fl 13160 df-mod 13236 df-seq 13368 df-exp 13428 df-hash 13689 df-cj 14454 df-re 14455 df-im 14456 df-sqrt 14590 df-abs 14591 df-dvds 15604 df-gcd 15840 df-prm 16012 df-phi 16099 df-pc 16170 df-gz 16262 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-starv 16576 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-unif 16584 df-hom 16585 df-cco 16586 df-0g 16711 df-gsum 16712 df-prds 16717 df-pws 16719 df-imas 16777 df-qus 16778 df-mre 16853 df-mrc 16854 df-acs 16856 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-mhm 17952 df-submnd 17953 df-grp 18102 df-minusg 18103 df-sbg 18104 df-mulg 18221 df-subg 18272 df-nsg 18273 df-eqg 18274 df-ghm 18352 df-cntz 18443 df-cmn 18904 df-abl 18905 df-mgp 19236 df-ur 19248 df-srg 19252 df-ring 19295 df-cring 19296 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19500 df-field 19501 df-subrg 19529 df-lmod 19632 df-lss 19700 df-lsp 19740 df-sra 19940 df-rgmod 19941 df-lidl 19942 df-rsp 19943 df-2idl 20001 df-nzr 20027 df-rlreg 20052 df-domn 20053 df-idom 20054 df-assa 20081 df-asp 20082 df-ascl 20083 df-psr 20132 df-mvr 20133 df-mpl 20134 df-opsr 20136 df-evls 20282 df-evl 20283 df-psr1 20344 df-vr1 20345 df-ply1 20346 df-coe1 20347 df-evl1 20475 df-cnfld 20542 df-zring 20614 df-zrh 20647 df-zn 20650 df-mdeg 24647 df-deg1 24648 df-mon1 24722 df-uc1p 24723 df-q1p 24724 df-r1p 24725 df-lgs 25869 |
This theorem is referenced by: 2sqreuopnn 26037 |
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