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Theorem eulerthlem1 12090

Description: Lemma for eulerth 12096. (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 995	. . . . . 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 5413	. . . . . . . . . 10 ⊢ (𝐹:𝑇–1-1-onto→𝑆 → 𝐹:𝑇⟶𝑆)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑇⟶𝑆)
7	6	ffvelrnda 5601	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ 𝑆)
8		oveq1 5828	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 gcd 𝑁) = ((𝐹‘𝑥) gcd 𝑁))
9	8	eqeq1d 2166	. . . . . . . . 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 2871	. . . . . . . 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 10039	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0..^𝑁) → (𝐹‘𝑥) ∈ ℤ)
15	13, 14	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ ℤ)
16	3, 15	zmulcld 9286	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴 · (𝐹‘𝑥)) ∈ ℤ)
17	1	simp1d 994	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
18	17	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑁 ∈ ℕ)
19		zmodfzo 10239	. . . 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 11866	. . . . 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 9279	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24	23	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑁 ∈ ℤ)
25	16, 24	gcdcomd 11849	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 · (𝐹‘𝑥)) gcd 𝑁) = (𝑁 gcd (𝐴 · (𝐹‘𝑥))))
26	23, 2	gcdcomd 11849	. . . . . . 7 ⊢ (𝜑 → (𝑁 gcd 𝐴) = (𝐴 gcd 𝑁))
27	1	simp3d 996	. . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
28	26, 27	eqtrd 2190	. . . . . 6 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
29	28	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd 𝐴) = 1)
30	24, 15	gcdcomd 11849	. . . . . 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 2190	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐹‘𝑥)) = 1)
33		rpmul 11966	. . . . . 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 1220	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝑁 gcd 𝐴) = 1 ∧ (𝑁 gcd (𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1))
35	29, 32, 34	mp2and 430	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1)
36	22, 25, 35	3eqtrd 2194	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1)
37		oveq1 5828	. . . . 5 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → (𝑦 gcd 𝑁) = (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁))
38	37	eqeq1d 2166	. . . 4 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
39	38, 10	elrab2 2871	. . 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 414	. 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 5620	1 ⊢ (𝜑 → 𝐺:𝑇⟶𝑆)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 {crab 2439 ↦ cmpt 4025 ⟶wf 5165 –1-1-onto→wf1o 5168 ‘cfv 5169 (class class class)co 5821 0cc0 7726 1c1 7727 · cmul 7731 ℕcn 8827 ℤcz 9161 ...cfz 9905 ..^cfzo 10034 mod cmo 10214 gcd cgcd 11821 ϕcphi 12073

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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-sup 6924 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-fz 9906 df-fzo 10035 df-fl 10162 df-mod 10215 df-seqfrec 10338 df-exp 10412 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-dvds 11677 df-gcd 11822

This theorem is referenced by: eulerthlemh 12094 eulerthlemth 12095

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