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Theorem eulerthlemfi 12160

Description: Lemma for eulerth 12165. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)

**Hypotheses**
Ref	Expression
eulerth.1	⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
eulerth.2	⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}

**Assertion**
Ref	Expression
eulerthlemfi	⊢ (𝜑 → 𝑆 ∈ Fin)

Distinct variable group: 𝑦,𝑁 Allowed substitution hints: 𝜑(𝑦) 𝐴(𝑦) 𝑆(𝑦)

Proof of Theorem eulerthlemfi Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9202	. . 3 ⊢ 0 ∈ ℤ
2		eulerth.1	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
3	2	simp1d 999	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9312	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5		fzofig 10367	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
6	1, 4, 5	sylancr 411	. 2 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
7		eulerth.2	. . . 4 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
8		ssrab2 3227	. . . 4 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
9	7, 8	eqsstri 3174	. . 3 ⊢ 𝑆 ⊆ (0..^𝑁)
10	9	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (0..^𝑁))
11		elfzoelz 10082	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ)
12	11	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℤ)
13	4	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
14	12, 13	gcdcld 11901	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ₀)
15	14	nn0zd 9311	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
16		1zzd 9218	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℤ)
17		zdceq 9266	. . . . 5 ⊢ (((𝑗 gcd 𝑁) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑗 gcd 𝑁) = 1)
18	15, 16, 17	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → DECID (𝑗 gcd 𝑁) = 1)
19		oveq1 5849	. . . . . . . . 9 ⊢ (𝑦 = 𝑗 → (𝑦 gcd 𝑁) = (𝑗 gcd 𝑁))
20	19	eqeq1d 2174	. . . . . . . 8 ⊢ (𝑦 = 𝑗 → ((𝑦 gcd 𝑁) = 1 ↔ (𝑗 gcd 𝑁) = 1))
21	20, 7	elrab2 2885	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 1))
22	21	baibr 910	. . . . . 6 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 1 ↔ 𝑗 ∈ 𝑆))
23	22	dcbid 828	. . . . 5 ⊢ (𝑗 ∈ (0..^𝑁) → (DECID (𝑗 gcd 𝑁) = 1 ↔ DECID 𝑗 ∈ 𝑆))
24	23	adantl 275	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (DECID (𝑗 gcd 𝑁) = 1 ↔ DECID 𝑗 ∈ 𝑆))
25	18, 24	mpbid 146	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → DECID 𝑗 ∈ 𝑆)
26	25	ralrimiva 2539	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆)
27		ssfidc 6900	. 2 ⊢ (((0..^𝑁) ∈ Fin ∧ 𝑆 ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆) → 𝑆 ∈ Fin)
28	6, 10, 26, 27	syl3anc 1228	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 824 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ∀wral 2444 {crab 2448 ⊆ wss 3116 (class class class)co 5842 Fincfn 6706 0cc0 7753 1c1 7754 ℕcn 8857 ℤcz 9191 ..^cfzo 10077 gcd cgcd 11875

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-1o 6384 df-er 6501 df-en 6707 df-fin 6709 df-sup 6949 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-fz 9945 df-fzo 10078 df-fl 10205 df-mod 10258 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-dvds 11728 df-gcd 11876

This theorem is referenced by: eulerthlemh 12163 eulerth 12165

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