2sqreunnltblem - Metamath Proof Explorer

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Theorem 2sqreunnltblem 27191

Description: Lemma for 2sqreunnltb 27201. (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 27190	. . 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 3389	. . . . 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 2743	. . . . . . . . 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 12484	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ₀)
7		nnnn0 12484	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
8		2sq2 27173	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1)))
9	6, 7, 8	syl2an 595	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1)))
10		breq12 5153	. . . . . . . . . . 11 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 ↔ 1 < 1))
11		1re 11219	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
12	11	ltnri 11328	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
13	12	pm2.21i 119	. . . . . . . . . . 11 ⊢ (1 < 1 → (𝑃 mod 4) = 1)
14	10, 13	syl6bi 253	. . . . . . . . . 10 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 → (𝑃 mod 4) = 1))
15	9, 14	syl6bi 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 239	. . . . . . 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 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 12585	. . . . . . . . 9 ⊢ ℕ ⊆ ℤ
23		id 22	. . . . . . . . . . . . . 14 ⊢ (((𝑎↑2) + (𝑏↑2)) = 𝑃 → ((𝑎↑2) + (𝑏↑2)) = 𝑃)
24	23	eqcomd 2737	. . . . . . . . . . . . 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 3083	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
27	26	reximi 3083	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
28		ssrexv 4051	. . . . . . . . . . . 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 3083	. . . . . . . . . 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 4051	. . . . . . . . 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 27172	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (𝑃 = 2 ∨ (𝑃 mod 4) = 1))
38	37	ord 861	. . . . 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 211	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 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ∃!wreu 3373 ⊆ wss 3948 class class class wbr 5148 (class class class)co 7412 1c1 11115 + caddc 11117 < clt 11253 ℕcn 12217 2c2 12272 4c4 12274 ℕ₀cn0 12477 ℤcz 12563 mod cmo 13839 ↑cexp 14032 ℙcprime 16613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-ec 8709 df-qs 8713 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-dvds 16203 df-gcd 16441 df-prm 16614 df-phi 16704 df-pc 16775 df-gz 16868 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-imas 17459 df-qus 17460 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-nzr 20405 df-subrng 20435 df-subrg 20460 df-drng 20503 df-field 20504 df-lmod 20617 df-lss 20688 df-lsp 20728 df-sra 20931 df-rgmod 20932 df-lidl 20933 df-rsp 20934 df-2idl 21007 df-rlreg 21100 df-domn 21101 df-idom 21102 df-cnfld 21146 df-zring 21219 df-zrh 21273 df-zn 21276 df-assa 21628 df-asp 21629 df-ascl 21630 df-psr 21682 df-mvr 21683 df-mpl 21684 df-opsr 21686 df-evls 21855 df-evl 21856 df-psr1 21924 df-vr1 21925 df-ply1 21926 df-coe1 21927 df-evl1 22056 df-mdeg 25806 df-deg1 25807 df-mon1 25884 df-uc1p 25885 df-q1p 25886 df-r1p 25887 df-lgs 27035

This theorem is referenced by: 2sqreunnltb 27201

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