gausslemma2dlem7 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem7 15184

Description: Lemma 7 for gausslemma2d 15185. (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 15183	. 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃))
7	1, 2	gausslemma2dlem0b 15166	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ)
8	7	nnnn0d 9293	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
9	8	faccld 10807	. . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ)
10	9	nncnd 8996	. . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ)
11	10	mullidd 8037	. . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻))
12	11	eqcomd 2199	. . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻)))
13	12	oveq1d 5933	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃))
14	13	eqeq1d 2202	. . . 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 9344	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16		neg1z 9349	. . . . . . 7 ⊢ -1 ∈ ℤ
17	1, 4, 2, 5	gausslemma2dlem0h 15172	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18		zexpcl 10625	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
19	16, 17, 18	sylancr 414	. . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
20		2z 9345	. . . . . . 7 ⊢ 2 ∈ ℤ
21		zexpcl 10625	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ₀) → (2↑𝐻) ∈ ℤ)
22	20, 8, 21	sylancr 414	. . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
23	19, 22	zmulcld 9445	. . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
24	9	nnzd 9438	. . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ)
25	1	gausslemma2dlem0a 15165	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26	1, 2	gausslemma2dlem0c 15167	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1)
27		cncongrcoprm 12244	. . . . 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 9698	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
32	25, 31	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℚ)
33	1	eldifad 3164	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
34		prmgt1 12270	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
35	33, 34	syl 14	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑃)
36		q1mod 10427	. . . . . . 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 2228	. . . 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 2164 ∖ cdif 3150 ifcif 3557 {csn 3618 class class class wbr 4029 ↦ cmpt 4090 ‘cfv 5254 (class class class)co 5918 1c1 7873 · cmul 7877 < clt 8054 − cmin 8190 -cneg 8191 / cdiv 8691 ℕcn 8982 2c2 9033 4c4 9035 ℕ₀cn0 9240 ℤcz 9317 ℚcq 9684 ...cfz 10074 ⌊cfl 10337 mod cmo 10393 ↑cexp 10609 !cfa 10796 gcd cgcd 12079 ℙcprime 12245

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-2o 6470 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-5 9044 df-6 9045 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-ioo 9958 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-proddc 11694 df-dvds 11931 df-gcd 12080 df-prm 12246

This theorem is referenced by: gausslemma2d 15185

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