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Mirrors > Home > MPE Home > Th. List > 2sqreunnltb | Structured version Visualization version GIF version |
Description: There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. (Contributed by AV, 11-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 25-Jun-2023.) |
Ref | Expression |
---|---|
2sqreult.1 | ⊢ (𝜑 ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
Ref | Expression |
---|---|
2sqreunnltb | ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreunnltblem 26023 | . 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) | |
2 | 2sqreult.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 26027 | . . . 4 ⊢ ∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑 |
7 | 2reu1 3874 | . . . 4 ⊢ (∀𝑎 ∈ ℕ ∃*𝑏 ∈ ℕ 𝜑 → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ 𝜑 ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑))) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ 𝜑 ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑)) |
9 | 5, 8 | bitri 277 | . 2 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃!𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝜑 ∧ ∃!𝑏 ∈ ℕ ∃𝑎 ∈ ℕ 𝜑)) |
10 | 1, 9 | syl6bb 289 | 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 5059 (class class class)co 7149 1c1 10531 + caddc 10533 < clt 10668 ℕcn 11631 2c2 11686 4c4 11688 mod cmo 13234 ↑cexp 13426 ℙcprime 16008 |
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 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
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 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-ofr 7403 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-ec 8284 df-qs 8288 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-inf 8900 df-oi 8967 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-dvds 15601 df-gcd 15837 df-prm 16009 df-phi 16096 df-pc 16167 df-gz 16259 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-hom 16582 df-cco 16583 df-0g 16708 df-gsum 16709 df-prds 16714 df-pws 16716 df-imas 16774 df-qus 16775 df-mre 16850 df-mrc 16851 df-acs 16853 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-submnd 17950 df-grp 18099 df-minusg 18100 df-sbg 18101 df-mulg 18218 df-subg 18269 df-nsg 18270 df-eqg 18271 df-ghm 18349 df-cntz 18440 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-dvr 19426 df-rnghom 19460 df-drng 19497 df-field 19498 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-nzr 20024 df-rlreg 20049 df-domn 20050 df-idom 20051 df-assa 20078 df-asp 20079 df-ascl 20080 df-psr 20129 df-mvr 20130 df-mpl 20131 df-opsr 20133 df-evls 20279 df-evl 20280 df-psr1 20341 df-vr1 20342 df-ply1 20343 df-coe1 20344 df-evl1 20472 df-cnfld 20539 df-zring 20611 df-zrh 20644 df-zn 20647 df-mdeg 24645 df-deg1 24646 df-mon1 24720 df-uc1p 24721 df-q1p 24722 df-r1p 24723 df-lgs 25867 |
This theorem is referenced by: 2sqreuopnnltb 26039 sq2reunnltb 30244 |
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