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Theorem eulerthlem1 12219

Description: Lemma for eulerth 12225. (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 1010	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
3	2	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐴 ∈ ℤ)
4		eulerthlem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑇–1-1-onto→𝑆)
5		f1of 5460	. . . . . . . . . 10 ⊢ (𝐹:𝑇–1-1-onto→𝑆 → 𝐹:𝑇⟶𝑆)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑇⟶𝑆)
7	6	ffvelcdmda 5650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ 𝑆)
8		oveq1 5879	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 gcd 𝑁) = ((𝐹‘𝑥) gcd 𝑁))
9	8	eqeq1d 2186	. . . . . . . . 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 2896	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝑆 ↔ ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
12	7, 11	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
13	12	simpld 112	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ (0..^𝑁))
14		elfzoelz 10142	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0..^𝑁) → (𝐹‘𝑥) ∈ ℤ)
15	13, 14	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ ℤ)
16	3, 15	zmulcld 9377	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴 · (𝐹‘𝑥)) ∈ ℤ)
17	1	simp1d 1009	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
18	17	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑁 ∈ ℕ)
19		zmodfzo 10342	. . . 4 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
20	16, 18, 19	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
21		modgcd 11984	. . . . 5 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
22	16, 18, 21	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
23	17	nnzd 9370	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24	23	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑁 ∈ ℤ)
25	16, 24	gcdcomd 11967	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 · (𝐹‘𝑥)) gcd 𝑁) = (𝑁 gcd (𝐴 · (𝐹‘𝑥))))
26	23, 2	gcdcomd 11967	. . . . . . 7 ⊢ (𝜑 → (𝑁 gcd 𝐴) = (𝐴 gcd 𝑁))
27	1	simp3d 1011	. . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
28	26, 27	eqtrd 2210	. . . . . 6 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
29	28	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd 𝐴) = 1)
30	24, 15	gcdcomd 11967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐹‘𝑥)) = ((𝐹‘𝑥) gcd 𝑁))
31	12	simprd 114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐹‘𝑥) gcd 𝑁) = 1)
32	30, 31	eqtrd 2210	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐹‘𝑥)) = 1)
33		rpmul 12090	. . . . . 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 1238	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝑁 gcd 𝐴) = 1 ∧ (𝑁 gcd (𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1))
35	29, 32, 34	mp2and 433	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1)
36	22, 25, 35	3eqtrd 2214	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1)
37		oveq1 5879	. . . . 5 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → (𝑦 gcd 𝑁) = (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁))
38	37	eqeq1d 2186	. . . 4 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
39	38, 10	elrab2 2896	. . 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 417	. 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 5669	1 ⊢ (𝜑 → 𝐺:𝑇⟶𝑆)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {crab 2459 ↦ cmpt 4063 ⟶wf 5211 –1-1-onto→wf1o 5214 ‘cfv 5215 (class class class)co 5872 0cc0 7808 1c1 7809 · cmul 7813 ℕcn 8915 ℤcz 9249 ...cfz 10004 ..^cfzo 10137 mod cmo 10317 gcd cgcd 11935 ϕcphi 12201

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 ax-arch 7927 ax-caucvg 7928

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-sup 6980 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-n0 9173 df-z 9250 df-uz 9525 df-q 9616 df-rp 9650 df-fz 10005 df-fzo 10138 df-fl 10265 df-mod 10318 df-seqfrec 10441 df-exp 10515 df-cj 10844 df-re 10845 df-im 10846 df-rsqrt 11000 df-abs 11001 df-dvds 11788 df-gcd 11936

This theorem is referenced by: eulerthlemh 12223 eulerthlemth 12224

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