2lgslem1 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 2lgslem1		GIF version

Theorem 2lgslem1 15358

Description: Lemma 1 for 2lgs 15371. (Contributed by AV, 19-Jun-2021.)

**Assertion**
Ref	Expression
2lgslem1	⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))}) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))))

Distinct variable group: 𝑃,𝑖,𝑥

Proof of Theorem 2lgslem1 Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2lgslem1a 15355	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))} = {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)})
2	1	fveq2d 5563	. 2 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))}) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}))
3		prmz 12290	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
4		4nn 9157	. . . . . . . 8 ⊢ 4 ∈ ℕ
5		znq 9701	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑃 / 4) ∈ ℚ)
6	3, 4, 5	sylancl 413	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 / 4) ∈ ℚ)
7	6	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (𝑃 / 4) ∈ ℚ)
8	7	flqcld 10370	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ∈ ℤ)
9	8	peano2zd 9454	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((⌊‘(𝑃 / 4)) + 1) ∈ ℤ)
10		nnoddn2prmb 12442	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃))
11		oddprm 12439	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
12	10, 11	sylbir 135	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ ℕ)
13	12	nnzd 9450	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ ℤ)
14	9, 13	fzfigd 10526	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ∈ Fin)
15	14	mptexd 5790	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)) ∈ V)
16		eqid 2196	. . . . 5 ⊢ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) = (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))
17		eqid 2196	. . . . 5 ⊢ (𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)) = (𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2))
18	16, 17	2lgslem1b 15356	. . . 4 ⊢ (𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)):(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}
19		f1oeq1 5493	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)) → (𝑓:(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)} ↔ (𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)):(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}))
20	19	spcegv 2852	. . . 4 ⊢ ((𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)) ∈ V → ((𝑦 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ↦ (𝑦 · 2)):(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)} → ∃𝑓 𝑓:(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}))
21	15, 18, 20	mpisyl 1457	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ∃𝑓 𝑓:(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)})
22		fihasheqf1oi 10882	. . . . 5 ⊢ (((((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ∈ Fin ∧ 𝑓:(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}) → (♯‘(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}))
23	22	ex 115	. . . 4 ⊢ ((((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ∈ Fin → (𝑓:(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)} → (♯‘(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)})))
24	23	exlimdv 1833	. . 3 ⊢ ((((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2)) ∈ Fin → (∃𝑓 𝑓:(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)} → (♯‘(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)})))
25	14, 21, 24	sylc 62	. 2 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (♯‘(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))𝑥 = (𝑖 · 2)}))
26	6	flqcld 10370	. . . . 5 ⊢ (𝑃 ∈ ℙ → (⌊‘(𝑃 / 4)) ∈ ℤ)
27	26	adantr 276	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ∈ ℤ)
28		oddm1d2 12060	. . . . . 6 ⊢ (𝑃 ∈ ℤ → (¬ 2 ∥ 𝑃 ↔ ((𝑃 − 1) / 2) ∈ ℤ))
29	3, 28	syl 14	. . . . 5 ⊢ (𝑃 ∈ ℙ → (¬ 2 ∥ 𝑃 ↔ ((𝑃 − 1) / 2) ∈ ℤ))
30	29	biimpa 296	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ ℤ)
31		2lgslem1c 15357	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
32		eluz2 9610	. . . 4 ⊢ (((𝑃 − 1) / 2) ∈ (ℤ_≥‘(⌊‘(𝑃 / 4))) ↔ ((⌊‘(𝑃 / 4)) ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℤ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
33	27, 30, 31, 32	syl3anbrc 1183	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((𝑃 − 1) / 2) ∈ (ℤ_≥‘(⌊‘(𝑃 / 4))))
34		hashfzp1 10919	. . 3 ⊢ (((𝑃 − 1) / 2) ∈ (ℤ_≥‘(⌊‘(𝑃 / 4))) → (♯‘(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))))
35	33, 34	syl 14	. 2 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (♯‘(((⌊‘(𝑃 / 4)) + 1)...((𝑃 − 1) / 2))) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))))
36	2, 25, 35	3eqtr2d 2235	1 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))}) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1506 ∈ wcel 2167 ∃wrex 2476 {crab 2479 Vcvv 2763 ∖ cdif 3154 {csn 3623 class class class wbr 4034 ↦ cmpt 4095 –1-1-onto→wf1o 5258 ‘cfv 5259 (class class class)co 5923 Fincfn 6801 1c1 7883 + caddc 7885 · cmul 7887 < clt 8064 ≤ cle 8065 − cmin 8200 / cdiv 8702 ℕcn 8993 2c2 9044 4c4 9046 ℤcz 9329 ℤ_≥cuz 9604 ℚcq 9696 ...cfz 10086 ⌊cfl 10361 mod cmo 10417 ♯chash 10870 ∥ cdvds 11955 ℙcprime 12286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7973 ax-resscn 7974 ax-1cn 7975 ax-1re 7976 ax-icn 7977 ax-addcl 7978 ax-addrcl 7979 ax-mulcl 7980 ax-mulrcl 7981 ax-addcom 7982 ax-mulcom 7983 ax-addass 7984 ax-mulass 7985 ax-distr 7986 ax-i2m1 7987 ax-0lt1 7988 ax-1rid 7989 ax-0id 7990 ax-rnegex 7991 ax-precex 7992 ax-cnre 7993 ax-pre-ltirr 7994 ax-pre-ltwlin 7995 ax-pre-lttrn 7996 ax-pre-apti 7997 ax-pre-ltadd 7998 ax-pre-mulgt0 7999 ax-pre-mulext 8000 ax-arch 8001 ax-caucvg 8002

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5878 df-ov 5926 df-oprab 5927 df-mpo 5928 df-1st 6200 df-2nd 6201 df-recs 6365 df-frec 6451 df-1o 6476 df-2o 6477 df-er 6594 df-en 6802 df-dom 6803 df-fin 6804 df-pnf 8066 df-mnf 8067 df-xr 8068 df-ltxr 8069 df-le 8070 df-sub 8202 df-neg 8203 df-reap 8605 df-ap 8612 df-div 8703 df-inn 8994 df-2 9052 df-3 9053 df-4 9054 df-n0 9253 df-z 9330 df-uz 9605 df-q 9697 df-rp 9732 df-fz 10087 df-fl 10363 df-mod 10418 df-seqfrec 10543 df-exp 10634 df-ihash 10871 df-cj 11010 df-re 11011 df-im 11012 df-rsqrt 11166 df-abs 11167 df-dvds 11956 df-prm 12287

This theorem is referenced by: 2lgs 15371

W3C validator