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Mirrors > Home > MPE Home > Th. List > 2sqreuopb | 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. Alternate ordered pair variant of 2sqreunnltb 26590. (Contributed by AV, 3-Jul-2023.) |
Ref | Expression |
---|---|
2sqreuopb | ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = 〈𝑎, 𝑏〉 ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreuopnnltb 26596 | . 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃))) | |
2 | breq12 5083 | . . . 4 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝑎 < 𝑏 ↔ (1st ‘𝑝) < (2nd ‘𝑝))) | |
3 | simpl 482 | . . . . . . 7 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → 𝑎 = (1st ‘𝑝)) | |
4 | 3 | oveq1d 7283 | . . . . . 6 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝑎↑2) = ((1st ‘𝑝)↑2)) |
5 | simpr 484 | . . . . . . 7 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → 𝑏 = (2nd ‘𝑝)) | |
6 | 5 | oveq1d 7283 | . . . . . 6 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝑏↑2) = ((2nd ‘𝑝)↑2)) |
7 | 4, 6 | oveq12d 7286 | . . . . 5 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → ((𝑎↑2) + (𝑏↑2)) = (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2))) |
8 | 7 | eqeq1d 2741 | . . . 4 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃)) |
9 | 2, 8 | anbi12d 630 | . . 3 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃))) |
10 | 9 | opreuopreu 7862 | . 2 ⊢ (∃!𝑝 ∈ (ℕ × ℕ)((1st ‘𝑝) < (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃) ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = 〈𝑎, 𝑏〉 ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
11 | 1, 10 | bitrdi 286 | 1 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑝 ∈ (ℕ × ℕ)∃𝑎∃𝑏(𝑝 = 〈𝑎, 𝑏〉 ∧ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ∃wex 1785 ∈ wcel 2109 ∃!wreu 3067 〈cop 4572 class class class wbr 5078 × cxp 5586 ‘cfv 6430 (class class class)co 7268 1st c1st 7815 2nd c2nd 7816 1c1 10856 + caddc 10858 < clt 10993 ℕcn 11956 2c2 12011 4c4 12013 mod cmo 13570 ↑cexp 13763 ℙcprime 16357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-ofr 7525 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-oadd 8285 df-er 8472 df-ec 8474 df-qs 8478 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-sup 9162 df-inf 9163 df-oi 9230 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-xnn0 12289 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-fz 13222 df-fzo 13365 df-fl 13493 df-mod 13571 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-dvds 15945 df-gcd 16183 df-prm 16358 df-phi 16448 df-pc 16519 df-gz 16612 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-0g 17133 df-gsum 17134 df-prds 17139 df-pws 17141 df-imas 17200 df-qus 17201 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-mulg 18682 df-subg 18733 df-nsg 18734 df-eqg 18735 df-ghm 18813 df-cntz 18904 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-srg 19723 df-ring 19766 df-cring 19767 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-rnghom 19940 df-drng 19974 df-field 19975 df-subrg 20003 df-lmod 20106 df-lss 20175 df-lsp 20215 df-sra 20415 df-rgmod 20416 df-lidl 20417 df-rsp 20418 df-2idl 20484 df-nzr 20510 df-rlreg 20535 df-domn 20536 df-idom 20537 df-cnfld 20579 df-zring 20652 df-zrh 20686 df-zn 20689 df-assa 21041 df-asp 21042 df-ascl 21043 df-psr 21093 df-mvr 21094 df-mpl 21095 df-opsr 21097 df-evls 21263 df-evl 21264 df-psr1 21332 df-vr1 21333 df-ply1 21334 df-coe1 21335 df-evl1 21463 df-mdeg 25198 df-deg1 25199 df-mon1 25276 df-uc1p 25277 df-q1p 25278 df-r1p 25279 df-lgs 26424 |
This theorem is referenced by: (None) |
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