2sqreunnltblem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2sqreunnltblem		Structured version Visualization version GIF version

Theorem 2sqreunnltblem 27805

Description: Lemma for 2sqreunnltb 27815. (Contributed by AV, 11-Jun-2023.) The prime need 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 27804	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
2	1	ex 418	. 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
3		2reu2rex 3381	. . . . 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 2775	. . . . . . . . 9 ⊢ (𝑃 = 2 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑏↑2)) = 2))
5	4	adantr 486	. . . . . . . 8 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑏↑2)) = 2))
6		nnnn0 12637	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ₀)
7		nnnn0 12637	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ₀)
8		2sq2 27787	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1)))
9	6, 7, 8	syl2an 608	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 ↔ (𝑎 = 1 ∧ 𝑏 = 1)))
10		breq12 5111	. . . . . . . . . . 11 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 ↔ 1 < 1))
11		1re 11332	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
12	11	ltnri 11443	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
13	12	pm2.21i 120	. . . . . . . . . . 11 ⊢ (1 < 1 → (𝑃 mod 4) = 1)
14	10, 13	biimtrdi 256	. . . . . . . . . 10 ⊢ ((𝑎 = 1 ∧ 𝑏 = 1) → (𝑎 < 𝑏 → (𝑃 mod 4) = 1))
15	9, 14	biimtrdi 256	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 2 → (𝑎 < 𝑏 → (𝑃 mod 4) = 1)))
16	15	adantl 487	. . . . . . . 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 417	. . . . . 6 ⊢ ((𝑃 = 2 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))
19	18	rexlimdvva 3223	. . . . 5 ⊢ (𝑃 = 2 → (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))
20	3, 19	syl5 35	. . . 4 ⊢ (𝑃 = 2 → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1))
21	20	a1d 26	. . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ ℙ → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 mod 4) = 1)))
22		nnssz 12739	. . . . . . . . 9 ⊢ ℕ ⊆ ℤ
23		id 23	. . . . . . . . . . . . . 14 ⊢ (((𝑎↑2) + (𝑏↑2)) = 𝑃 → ((𝑎↑2) + (𝑏↑2)) = 𝑃)
24	23	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (((𝑎↑2) + (𝑏↑2)) = 𝑃 → 𝑃 = ((𝑎↑2) + (𝑏↑2)))
25	24	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → 𝑃 = ((𝑎↑2) + (𝑏↑2)))
26	25	reximi 3104	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
27	26	reximi 3104	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
28		ssrexv 3983	. . . . . . . . . . . 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 3104	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
31	3, 27, 30	3syl 19	. . . . . . . . 9 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
32		ssrexv 3983	. . . . . . . . 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 69	. . . . . . . 8 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
34	33	adantl 487	. . . . . . 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 27786	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ (𝑃 = 2 ∨ (𝑃 mod 4) = 1)))
36	35	adantr 486	. . . . . . 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 878	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (¬ 𝑃 = 2 → (𝑃 mod 4) = 1))
39	38	expcom 419	. . . 4 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → (𝑃 mod 4) = 1)))
40	39	com13 89	. . 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 184	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ∃wrex 3090 ∃!wreu 3367 ⊆ wss 3900 class class class wbr 5106 (class class class)co 7440 1c1 11225 + caddc 11227 < clt 11367 ℕcn 12359 2c2 12421 4c4 12423 ℕ₀cn0 12630 ℤcz 12717 mod cmo 14033 ↑cexp 14228 ℙcprime 16872

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302 ax-addf 11303 ax-mulf 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-ofr 7715 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-oadd 8505 df-er 8742 df-ec 8744 df-qs 8748 df-map 8874 df-pm 8875 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-inf 9459 df-oi 9528 df-dju 10006 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-xnn0 12704 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-fz 13664 df-fzo 13813 df-fl 13956 df-mod 14034 df-seq 14169 df-exp 14229 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-dvds 16449 df-gcd 16691 df-prm 16873 df-phi 16969 df-pc 17041 df-gz 17134 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-imas 17706 df-qus 17707 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-submnd 19005 df-grp 19173 df-minusg 19174 df-sbg 19175 df-mulg 19304 df-subg 19359 df-nsg 19360 df-eqg 19361 df-ghm 19454 df-cntz 19558 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-srg 20440 df-ring 20488 df-cring 20489 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-rhm 20729 df-nzr 20790 df-subrng 20825 df-subrg 20849 df-rlreg 20973 df-domn 20974 df-idom 20975 df-drng 21009 df-field 21010 df-lmod 21164 df-lss 21234 df-lsp 21274 df-sra 21475 df-rgmod 21476 df-lidl 21513 df-rsp 21514 df-2idl 21570 df-cnfld 21706 df-zring 21780 df-zrh 21836 df-zn 21839 df-assa 22188 df-asp 22189 df-ascl 22190 df-psr 22244 df-mvr 22245 df-mpl 22246 df-opsr 22248 df-evls 22410 df-evl 22411 df-psr1 22525 df-vr1 22526 df-ply1 22527 df-coe1 22528 df-evl1 22661 df-mdeg 26400 df-deg1 26401 df-mon1 26476 df-uc1p 26477 df-q1p 26478 df-r1p 26479 df-lgs 27649

This theorem is used by: 2sqreunnltb 27815

W3C validator