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Mirrors > Home > MPE Home > Th. List > 2sqreunnltblem | Structured version Visualization version GIF version |
Description: Lemma for 2sqreunnltb 26339. (Contributed by AV, 11-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 18-Jun-2023.) |
Ref | Expression |
---|---|
2sqreunnltblem | ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreunnltlem 26328 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
2 | 1 | ex 416 | . 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
3 | 2reu2rex 3336 | . . . . 5 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
4 | eqeq2 2749 | . . . . . . . . 9 ⊢ (𝑃 = 2 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑏↑2)) = 2)) | |
5 | 4 | adantr 484 | . . . . . . . 8 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑏↑2)) = 2)) |
6 | nnnn0 12094 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ0) | |
7 | nnnn0 12094 | . . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ0) | |
8 | 2sq2 26311 | . . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1))) | |
9 | 6, 7, 8 | syl2an 599 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1))) |
10 | breq12 5055 | . . . . . . . . . . 11 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 ↔ 1 < 1)) | |
11 | 1re 10830 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ | |
12 | 11 | ltnri 10938 | . . . . . . . . . . . 12 ⊢ ¬ 1 < 1 |
13 | 12 | pm2.21i 119 | . . . . . . . . . . 11 ⊢ (1 < 1 → (𝑃 mod 4) = 1) |
14 | 10, 13 | syl6bi 256 | . . . . . . . . . 10 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 → (𝑃 mod 4) = 1)) |
15 | 9, 14 | syl6bi 256 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1))) |
16 | 15 | adantl 485 | . . . . . . . 8 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 2 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1))) |
17 | 5, 16 | sylbid 243 | . . . . . . 7 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1))) |
18 | 17 | impcomd 415 | . . . . . 6 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1)) |
19 | 18 | rexlimdvva 3210 | . . . . 5 ⊢ (𝑃 = 2 → (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1)) |
20 | 3, 19 | syl5 34 | . . . 4 ⊢ (𝑃 = 2 → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1)) |
21 | 20 | a1d 25 | . . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))) |
22 | nnssz 12194 | . . . . . . . . 9 ⊢ ℕ ⊆ ℤ | |
23 | id 22 | . . . . . . . . . . . . . 14 ⊢ (((𝑎↑2) + (𝑏↑2)) = 𝑃 → ((𝑎↑2) + (𝑏↑2)) = 𝑃) | |
24 | 23 | eqcomd 2743 | . . . . . . . . . . . . 13 ⊢ (((𝑎↑2) + (𝑏↑2)) = 𝑃 → 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
25 | 24 | adantl 485 | . . . . . . . . . . . 12 ⊢ ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
26 | 25 | reximi 3163 | . . . . . . . . . . 11 ⊢ (∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
27 | 26 | reximi 3163 | . . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
28 | ssrexv 3965 | . . . . . . . . . . . 12 ⊢ (ℕ ⊆ ℤ → (∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))) | |
29 | 22, 28 | ax-mp 5 | . . . . . . . . . . 11 ⊢ (∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
30 | 29 | reximi 3163 | . . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
31 | 3, 27, 30 | 3syl 18 | . . . . . . . . 9 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
32 | ssrexv 3965 | . . . . . . . . 9 ⊢ (ℕ ⊆ ℤ → (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))) | |
33 | 22, 31, 32 | mpsyl 68 | . . . . . . . 8 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
34 | 33 | adantl 485 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2))) |
35 | 2sqb 26310 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1))) | |
36 | 35 | adantr 484 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1))) |
37 | 34, 36 | mpbid 235 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (𝑃 = 2 ∨ (𝑃 mod 4) = 1)) |
38 | 37 | ord 864 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (¬ 𝑃 = 2 → (𝑃 mod 4) = 1)) |
39 | 38 | expcom 417 | . . . 4 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → (𝑃 mod 4) = 1))) |
40 | 39 | com13 88 | . . 3 ⊢ (¬ 𝑃 = 2 → (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))) |
41 | 21, 40 | pm2.61i 185 | . 2 ⊢ (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1)) |
42 | 2, 41 | impbid 215 | 1 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∃wrex 3059 ∃!wreu 3060 ⊆ wss 3863 class class class wbr 5050 (class class class)co 7210 1c1 10727 + caddc 10729 < clt 10864 ℕcn 11827 2c2 11882 4c4 11884 ℕ0cn0 12087 ℤcz 12173 mod cmo 13439 ↑cexp 13632 ℙcprime 16225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 ax-addf 10805 ax-mulf 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-se 5507 df-we 5508 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 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-isom 6386 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-of 7466 df-ofr 7467 df-om 7642 df-1st 7758 df-2nd 7759 df-supp 7901 df-tpos 7965 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-2o 8200 df-oadd 8203 df-er 8388 df-ec 8390 df-qs 8394 df-map 8507 df-pm 8508 df-ixp 8576 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-fsupp 8983 df-sup 9055 df-inf 9056 df-oi 9123 df-dju 9514 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-xnn0 12160 df-z 12174 df-dec 12291 df-uz 12436 df-q 12542 df-rp 12584 df-fz 13093 df-fzo 13236 df-fl 13364 df-mod 13440 df-seq 13572 df-exp 13633 df-hash 13894 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-dvds 15813 df-gcd 16051 df-prm 16226 df-phi 16316 df-pc 16387 df-gz 16480 df-struct 16697 df-sets 16714 df-slot 16732 df-ndx 16742 df-base 16758 df-ress 16782 df-plusg 16812 df-mulr 16813 df-starv 16814 df-sca 16815 df-vsca 16816 df-ip 16817 df-tset 16818 df-ple 16819 df-ds 16821 df-unif 16822 df-hom 16823 df-cco 16824 df-0g 16943 df-gsum 16944 df-prds 16949 df-pws 16951 df-imas 17010 df-qus 17011 df-mre 17086 df-mrc 17087 df-acs 17089 df-mgm 18111 df-sgrp 18160 df-mnd 18171 df-mhm 18215 df-submnd 18216 df-grp 18365 df-minusg 18366 df-sbg 18367 df-mulg 18486 df-subg 18537 df-nsg 18538 df-eqg 18539 df-ghm 18617 df-cntz 18708 df-cmn 19169 df-abl 19170 df-mgp 19502 df-ur 19514 df-srg 19518 df-ring 19561 df-cring 19562 df-oppr 19638 df-dvdsr 19656 df-unit 19657 df-invr 19687 df-dvr 19698 df-rnghom 19732 df-drng 19766 df-field 19767 df-subrg 19795 df-lmod 19898 df-lss 19966 df-lsp 20006 df-sra 20206 df-rgmod 20207 df-lidl 20208 df-rsp 20209 df-2idl 20267 df-nzr 20293 df-rlreg 20318 df-domn 20319 df-idom 20320 df-cnfld 20361 df-zring 20433 df-zrh 20467 df-zn 20470 df-assa 20812 df-asp 20813 df-ascl 20814 df-psr 20865 df-mvr 20866 df-mpl 20867 df-opsr 20869 df-evls 21029 df-evl 21030 df-psr1 21098 df-vr1 21099 df-ply1 21100 df-coe1 21101 df-evl1 21229 df-mdeg 24947 df-deg1 24948 df-mon1 25025 df-uc1p 25026 df-q1p 25027 df-r1p 25028 df-lgs 26173 |
This theorem is referenced by: 2sqreunnltb 26339 |
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