gausslemma2dlem7 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem7 15867

Description: Lemma 7 for gausslemma2d 15868. (Contributed by AV, 13-Jul-2021.)

**Hypotheses**
Ref	Expression
gausslemma2d.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2d.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2d.r	⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
gausslemma2d.m	⊢ 𝑀 = (⌊‘(𝑃 / 4))
gausslemma2d.n	⊢ 𝑁 = (𝐻 − 𝑀)

**Assertion**
Ref	Expression
gausslemma2dlem7	⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑥,𝑀 Allowed substitution hints: 𝑅(𝑥) 𝑁(𝑥)

**Proof of Theorem gausslemma2dlem7**
Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		gausslemma2d.h	. . 3 ⊢ 𝐻 = ((𝑃 − 1) / 2)
3		gausslemma2d.r	. . 3 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
4		gausslemma2d.m	. . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
5		gausslemma2d.n	. . 3 ⊢ 𝑁 = (𝐻 − 𝑀)
6	1, 2, 3, 4, 5	gausslemma2dlem6 15866	. 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃))
7	1, 2	gausslemma2dlem0b 15849	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ)
8	7	nnnn0d 9498	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
9	8	faccld 11042	. . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ)
10	9	nncnd 9200	. . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ)
11	10	mullidd 8240	. . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻))
12	11	eqcomd 2237	. . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻)))
13	12	oveq1d 6043	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃))
14	13	eqeq1d 2240	. . . 4 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ ((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃)))
15		1zzd 9549	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16		neg1z 9554	. . . . . . 7 ⊢ -1 ∈ ℤ
17	1, 4, 2, 5	gausslemma2dlem0h 15855	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18		zexpcl 10860	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
19	16, 17, 18	sylancr 414	. . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
20		2z 9550	. . . . . . 7 ⊢ 2 ∈ ℤ
21		zexpcl 10860	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ₀) → (2↑𝐻) ∈ ℤ)
22	20, 8, 21	sylancr 414	. . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
23	19, 22	zmulcld 9651	. . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
24	9	nnzd 9644	. . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ)
25	1	gausslemma2dlem0a 15848	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26	1, 2	gausslemma2dlem0c 15850	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1)
27		cncongrcoprm 12739	. . . . 5 ⊢ (((1 ∈ ℤ ∧ ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ ∧ (!‘𝐻) ∈ ℤ) ∧ (𝑃 ∈ ℕ ∧ ((!‘𝐻) gcd 𝑃) = 1)) → (((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)))
28	15, 23, 24, 25, 26, 27	syl32anc 1282	. . . 4 ⊢ (𝜑 → (((1 · (!‘𝐻)) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)))
29	14, 28	bitrd 188	. . 3 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) ↔ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)))
30		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃))
31		nnq 9910	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
32	25, 31	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℚ)
33	1	eldifad 3212	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
34		prmgt1 12765	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
35	33, 34	syl 14	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑃)
36		q1mod 10662	. . . . . . 7 ⊢ ((𝑃 ∈ ℚ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
37	32, 35, 36	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (1 mod 𝑃) = 1)
38	37	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (1 mod 𝑃) = 1)
39	30, 38	eqtr3d 2266	. . . 4 ⊢ ((𝜑 ∧ (1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)
40	39	ex 115	. . 3 ⊢ (𝜑 → ((1 mod 𝑃) = (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1))
41	29, 40	sylbid 150	. 2 ⊢ (𝜑 → (((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1))
42	6, 41	mpd 13	1 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2202 ∖ cdif 3198 ifcif 3607 {csn 3673 class class class wbr 4093 ↦ cmpt 4155 ‘cfv 5333 (class class class)co 6028 1c1 8076 · cmul 8080 < clt 8257 − cmin 8393 -cneg 8394 / cdiv 8895 ℕcn 9186 2c2 9237 4c4 9239 ℕ₀cn0 9445 ℤcz 9522 ℚcq 9896 ...cfz 10286 ⌊cfl 10572 mod cmo 10628 ↑cexp 10844 !cfa 11031 gcd cgcd 12585 ℙcprime 12740

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-tp 3681 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-5 9248 df-6 9249 df-n0 9446 df-z 9523 df-uz 9799 df-q 9897 df-rp 9932 df-ioo 10170 df-fz 10287 df-fzo 10421 df-fl 10574 df-mod 10629 df-seqfrec 10754 df-exp 10845 df-fac 11032 df-ihash 11082 df-cj 11463 df-re 11464 df-im 11465 df-rsqrt 11619 df-abs 11620 df-clim 11900 df-proddc 12173 df-dvds 12410 df-gcd 12586 df-prm 12741

This theorem is referenced by: gausslemma2d 15868

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