2sqreunnltblem - Metamath Proof Explorer

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Theorem 2sqreunnltblem 27432

Description: Lemma for 2sqreunnltb 27442. (Contributed by AV, 11-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 18-Jun-2023.)

**Assertion**
Ref	Expression
2sqreunnltblem	⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))

Distinct variable group: 𝑃,𝑎,𝑏

**Proof of Theorem 2sqreunnltblem**
Step	Hyp	Ref	Expression
1		2sqreunnltlem 27431	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
2	1	ex 412	. 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
3		2reu2rex 3355	. . . . 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 480	. . . . . . . 8 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑏↑2)) = 2))
6		nnnn0 12439	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ₀)
7		nnnn0 12439	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
8		2sq2 27414	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1)))
9	6, 7, 8	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1)))
10		breq12 5091	. . . . . . . . . . 11 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 ↔ 1 < 1))
11		1re 11139	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
12	11	ltnri 11250	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
13	12	pm2.21i 119	. . . . . . . . . . 11 ⊢ (1 < 1 → (𝑃 mod 4) = 1)
14	10, 13	biimtrdi 253	. . . . . . . . . 10 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 → (𝑃 mod 4) = 1))
15	9, 14	biimtrdi 253	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1)))
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 2 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1)))
17	5, 16	sylbid 240	. . . . . . 7 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1)))
18	17	impcomd 411	. . . . . 6 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))
19	18	rexlimdvva 3195	. . . . 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 12541	. . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → 𝑃 = ((𝑎↑2) + (𝑏↑2)))
26	25	reximi 3076	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
27	26	reximi 3076	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
28		ssrexv 3992	. . . . . . . . . . . 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 3076	. . . . . . . . . 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 3992	. . . . . . . . 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 481	. . . . . . 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 27413	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1)))
36	35	adantr 480	. . . . . . 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 232	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (𝑃 = 2 ∨ (𝑃 mod 4) = 1))
38	37	ord 865	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (¬ 𝑃 = 2 → (𝑃 mod 4) = 1))
39	38	expcom 413	. . . 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 182	. 2 ⊢ (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))
42	2, 41	impbid 212	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 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 ∃!wreu 3341 ⊆ wss 3890 class class class wbr 5086 (class class class)co 7362 1c1 11034 + caddc 11036 < clt 11174 ℕcn 12169 2c2 12231 4c4 12233 ℕ₀cn0 12432 ℤcz 12519 mod cmo 13823 ↑cexp 14018 ℙcprime 16635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-ofr 7627 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-gcd 16459 df-prm 16636 df-phi 16731 df-pc 16803 df-gz 16896 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-imas 17467 df-qus 17468 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-nsg 19095 df-eqg 19096 df-ghm 19183 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-srg 20163 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-nzr 20485 df-subrng 20518 df-subrg 20542 df-rlreg 20666 df-domn 20667 df-idom 20668 df-drng 20703 df-field 20704 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-lidl 21202 df-rsp 21203 df-2idl 21244 df-cnfld 21349 df-zring 21441 df-zrh 21497 df-zn 21500 df-assa 21847 df-asp 21848 df-ascl 21849 df-psr 21903 df-mvr 21904 df-mpl 21905 df-opsr 21907 df-evls 22066 df-evl 22067 df-psr1 22157 df-vr1 22158 df-ply1 22159 df-coe1 22160 df-evl1 22295 df-mdeg 26034 df-deg1 26035 df-mon1 26110 df-uc1p 26111 df-q1p 26112 df-r1p 26113 df-lgs 27276

This theorem is referenced by: 2sqreunnltb 27442

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