gausslemma2dlem7 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem7 15595

Description: Lemma 7 for gausslemma2d 15596. (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 15594	. 2 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃))
7	1, 2	gausslemma2dlem0b 15577	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℕ)
8	7	nnnn0d 9361	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
9	8	faccld 10894	. . . . . . . . 9 ⊢ (𝜑 → (!‘𝐻) ∈ ℕ)
10	9	nncnd 9063	. . . . . . . 8 ⊢ (𝜑 → (!‘𝐻) ∈ ℂ)
11	10	mullidd 8103	. . . . . . 7 ⊢ (𝜑 → (1 · (!‘𝐻)) = (!‘𝐻))
12	11	eqcomd 2212	. . . . . 6 ⊢ (𝜑 → (!‘𝐻) = (1 · (!‘𝐻)))
13	12	oveq1d 5969	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((1 · (!‘𝐻)) mod 𝑃))
14	13	eqeq1d 2215	. . . 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 9412	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
16		neg1z 9417	. . . . . . 7 ⊢ -1 ∈ ℤ
17	1, 4, 2, 5	gausslemma2dlem0h 15583	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18		zexpcl 10712	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
19	16, 17, 18	sylancr 414	. . . . . 6 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
20		2z 9413	. . . . . . 7 ⊢ 2 ∈ ℤ
21		zexpcl 10712	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐻 ∈ ℕ₀) → (2↑𝐻) ∈ ℤ)
22	20, 8, 21	sylancr 414	. . . . . 6 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
23	19, 22	zmulcld 9514	. . . . 5 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
24	9	nnzd 9507	. . . . 5 ⊢ (𝜑 → (!‘𝐻) ∈ ℤ)
25	1	gausslemma2dlem0a 15576	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26	1, 2	gausslemma2dlem0c 15578	. . . . 5 ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1)
27		cncongrcoprm 12478	. . . . 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 1258	. . . 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 9767	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
32	25, 31	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℚ)
33	1	eldifad 3179	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
34		prmgt1 12504	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
35	33, 34	syl 14	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑃)
36		q1mod 10514	. . . . . . 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 2241	. . . 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 1373 ∈ wcel 2177 ∖ cdif 3165 ifcif 3573 {csn 3635 class class class wbr 4048 ↦ cmpt 4110 ‘cfv 5277 (class class class)co 5954 1c1 7939 · cmul 7943 < clt 8120 − cmin 8256 -cneg 8257 / cdiv 8758 ℕcn 9049 2c2 9100 4c4 9102 ℕ₀cn0 9308 ℤcz 9385 ℚcq 9753 ...cfz 10143 ⌊cfl 10424 mod cmo 10480 ↑cexp 10696 !cfa 10883 gcd cgcd 12324 ℙcprime 12479

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-mulrcl 8037 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-precex 8048 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 ax-pre-mulgt0 8055 ax-pre-mulext 8056 ax-arch 8057 ax-caucvg 8058

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-xor 1396 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-tp 3643 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-po 4348 df-iso 4349 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-isom 5286 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237 df-recs 6401 df-irdg 6466 df-frec 6487 df-1o 6512 df-2o 6513 df-oadd 6516 df-er 6630 df-en 6838 df-dom 6839 df-fin 6840 df-sup 7098 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-reap 8661 df-ap 8668 df-div 8759 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-5 9111 df-6 9112 df-n0 9309 df-z 9386 df-uz 9662 df-q 9754 df-rp 9789 df-ioo 10027 df-fz 10144 df-fzo 10278 df-fl 10426 df-mod 10481 df-seqfrec 10606 df-exp 10697 df-fac 10884 df-ihash 10934 df-cj 11203 df-re 11204 df-im 11205 df-rsqrt 11359 df-abs 11360 df-clim 11640 df-proddc 11912 df-dvds 12149 df-gcd 12325 df-prm 12480

This theorem is referenced by: gausslemma2d 15596

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