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Theorem eulerthlem1 12181

Description: Lemma for eulerth 12187. (Contributed by Mario Carneiro, 8-May-2015.)

**Hypotheses**
Ref	Expression
eulerthlem1.1	⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
eulerthlem1.2	⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
eulerthlem1.3	⊢ 𝑇 = (1...(ϕ‘𝑁))
eulerthlem1.4	⊢ (𝜑 → 𝐹:𝑇–1-1-onto→𝑆)
eulerthlem1.5	⊢ 𝐺 = (𝑥 ∈ 𝑇 ↦ ((𝐴 · (𝐹‘𝑥)) mod 𝑁))

**Assertion**
Ref	Expression
eulerthlem1	⊢ (𝜑 → 𝐺:𝑇⟶𝑆)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐹,𝑦 𝑥,𝐺,𝑦 𝑥,𝑁,𝑦 𝑥,𝑆 𝜑,𝑥,𝑦 𝑥,𝑇,𝑦 Allowed substitution hint: 𝑆(𝑦)

**Proof of Theorem eulerthlem1**
Step	Hyp	Ref	Expression
1		eulerthlem1.1	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
2	1	simp2d 1005	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
3	2	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐴 ∈ ℤ)
4		eulerthlem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑇–1-1-onto→𝑆)
5		f1of 5442	. . . . . . . . . 10 ⊢ (𝐹:𝑇–1-1-onto→𝑆 → 𝐹:𝑇⟶𝑆)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑇⟶𝑆)
7	6	ffvelrnda 5631	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ 𝑆)
8		oveq1 5860	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 gcd 𝑁) = ((𝐹‘𝑥) gcd 𝑁))
9	8	eqeq1d 2179	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘𝑥) gcd 𝑁) = 1))
10		eulerthlem1.2	. . . . . . . . 9 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
11	9, 10	elrab2 2889	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝑆 ↔ ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
12	7, 11	sylib 121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
13	12	simpld 111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ (0..^𝑁))
14		elfzoelz 10103	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0..^𝑁) → (𝐹‘𝑥) ∈ ℤ)
15	13, 14	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ ℤ)
16	3, 15	zmulcld 9340	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴 · (𝐹‘𝑥)) ∈ ℤ)
17	1	simp1d 1004	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
18	17	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑁 ∈ ℕ)
19		zmodfzo 10303	. . . 4 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
20	16, 18, 19	syl2anc 409	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
21		modgcd 11946	. . . . 5 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
22	16, 18, 21	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
23	17	nnzd 9333	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24	23	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑁 ∈ ℤ)
25	16, 24	gcdcomd 11929	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 · (𝐹‘𝑥)) gcd 𝑁) = (𝑁 gcd (𝐴 · (𝐹‘𝑥))))
26	23, 2	gcdcomd 11929	. . . . . . 7 ⊢ (𝜑 → (𝑁 gcd 𝐴) = (𝐴 gcd 𝑁))
27	1	simp3d 1006	. . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
28	26, 27	eqtrd 2203	. . . . . 6 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
29	28	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd 𝐴) = 1)
30	24, 15	gcdcomd 11929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐹‘𝑥)) = ((𝐹‘𝑥) gcd 𝑁))
31	12	simprd 113	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐹‘𝑥) gcd 𝑁) = 1)
32	30, 31	eqtrd 2203	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐹‘𝑥)) = 1)
33		rpmul 12052	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐹‘𝑥) ∈ ℤ) → (((𝑁 gcd 𝐴) = 1 ∧ (𝑁 gcd (𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1))
34	24, 3, 15, 33	syl3anc 1233	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝑁 gcd 𝐴) = 1 ∧ (𝑁 gcd (𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1))
35	29, 32, 34	mp2and 431	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1)
36	22, 25, 35	3eqtrd 2207	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1)
37		oveq1 5860	. . . . 5 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → (𝑦 gcd 𝑁) = (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁))
38	37	eqeq1d 2179	. . . 4 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
39	38, 10	elrab2 2889	. . 3 ⊢ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁) ∧ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
40	20, 36, 39	sylanbrc 415	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆)
41		eulerthlem1.5	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝑇 ↦ ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
42	40, 41	fmptd 5650	1 ⊢ (𝜑 → 𝐺:𝑇⟶𝑆)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 {crab 2452 ↦ cmpt 4050 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 0cc0 7774 1c1 7775 · cmul 7779 ℕcn 8878 ℤcz 9212 ...cfz 9965 ..^cfzo 10098 mod cmo 10278 gcd cgcd 11897 ϕcphi 12163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898

This theorem is referenced by: eulerthlemh 12185 eulerthlemth 12186

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