gausslemma2dlem7 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem7 15276

Description: Lemma 7 for gausslemma2d 15277. (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 15275	. 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃))
7	1, 2	gausslemma2dlem0b 15258	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ)
8	7	nnnn0d 9299	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
9	8	faccld 10813	. . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ)
10	9	nncnd 9001	. . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ)
11	10	mullidd 8042	. . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻))
12	11	eqcomd 2202	. . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻)))
13	12	oveq1d 5937	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃))
14	13	eqeq1d 2205	. . . 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 9350	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16		neg1z 9355	. . . . . . 7 ⊢ -1 ∈ ℤ
17	1, 4, 2, 5	gausslemma2dlem0h 15264	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18		zexpcl 10631	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
19	16, 17, 18	sylancr 414	. . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
20		2z 9351	. . . . . . 7 ⊢ 2 ∈ ℤ
21		zexpcl 10631	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ₀) → (2↑𝐻) ∈ ℤ)
22	20, 8, 21	sylancr 414	. . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
23	19, 22	zmulcld 9451	. . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
24	9	nnzd 9444	. . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ)
25	1	gausslemma2dlem0a 15257	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26	1, 2	gausslemma2dlem0c 15259	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1)
27		cncongrcoprm 12250	. . . . 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 1257	. . . 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 9704	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
32	25, 31	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℚ)
33	1	eldifad 3168	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
34		prmgt1 12276	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
35	33, 34	syl 14	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑃)
36		q1mod 10433	. . . . . . 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 2231	. . . 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 1364 ∈ wcel 2167 ∖ cdif 3154 ifcif 3561 {csn 3622 class class class wbr 4033 ↦ cmpt 4094 ‘cfv 5258 (class class class)co 5922 1c1 7878 · cmul 7882 < clt 8059 − cmin 8195 -cneg 8196 / cdiv 8696 ℕcn 8987 2c2 9038 4c4 9040 ℕ₀cn0 9246 ℤcz 9323 ℚcq 9690 ...cfz 10080 ⌊cfl 10343 mod cmo 10399 ↑cexp 10615 !cfa 10802 gcd cgcd 12085 ℙcprime 12251

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 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7968 ax-resscn 7969 ax-1cn 7970 ax-1re 7971 ax-icn 7972 ax-addcl 7973 ax-addrcl 7974 ax-mulcl 7975 ax-mulrcl 7976 ax-addcom 7977 ax-mulcom 7978 ax-addass 7979 ax-mulass 7980 ax-distr 7981 ax-i2m1 7982 ax-0lt1 7983 ax-1rid 7984 ax-0id 7985 ax-rnegex 7986 ax-precex 7987 ax-cnre 7988 ax-pre-ltirr 7989 ax-pre-ltwlin 7990 ax-pre-lttrn 7991 ax-pre-apti 7992 ax-pre-ltadd 7993 ax-pre-mulgt0 7994 ax-pre-mulext 7995 ax-arch 7996 ax-caucvg 7997

This theorem depends on definitions: df-bi 117 df-stab 832 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 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-tp 3630 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-2o 6475 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7048 df-pnf 8061 df-mnf 8062 df-xr 8063 df-ltxr 8064 df-le 8065 df-sub 8197 df-neg 8198 df-reap 8599 df-ap 8606 df-div 8697 df-inn 8988 df-2 9046 df-3 9047 df-4 9048 df-5 9049 df-6 9050 df-n0 9247 df-z 9324 df-uz 9599 df-q 9691 df-rp 9726 df-ioo 9964 df-fz 10081 df-fzo 10215 df-fl 10345 df-mod 10400 df-seqfrec 10525 df-exp 10616 df-fac 10803 df-ihash 10853 df-cj 10992 df-re 10993 df-im 10994 df-rsqrt 11148 df-abs 11149 df-clim 11428 df-proddc 11700 df-dvds 11937 df-gcd 12086 df-prm 12252

This theorem is referenced by: gausslemma2d 15277

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