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Theorem eulerthlemfi 12594

Description: Lemma for eulerth 12599. 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 9390	. . 3 ⊢ 0 ∈ ℤ
2		eulerth.1	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
3	2	simp1d 1012	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9501	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5		fzofig 10584	. . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
6	1, 4, 5	sylancr 414	. 2 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
7		eulerth.2	. . . 4 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
8		ssrab2 3279	. . . 4 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
9	7, 8	eqsstri 3226	. . 3 ⊢ 𝑆 ⊆ (0..^𝑁)
10	9	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (0..^𝑁))
11		elfzoelz 10276	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ)
12	11	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℤ)
13	4	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
14	12, 13	gcdcld 12333	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ₀)
15	14	nn0zd 9500	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
16		1zzd 9406	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℤ)
17		zdceq 9455	. . . . 5 ⊢ (((𝑗 gcd 𝑁) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑗 gcd 𝑁) = 1)
18	15, 16, 17	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → DECID (𝑗 gcd 𝑁) = 1)
19		oveq1 5958	. . . . . . . . 9 ⊢ (𝑦 = 𝑗 → (𝑦 gcd 𝑁) = (𝑗 gcd 𝑁))
20	19	eqeq1d 2215	. . . . . . . 8 ⊢ (𝑦 = 𝑗 → ((𝑦 gcd 𝑁) = 1 ↔ (𝑗 gcd 𝑁) = 1))
21	20, 7	elrab2 2933	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 1))
22	21	baibr 922	. . . . . 6 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 1 ↔ 𝑗 ∈ 𝑆))
23	22	dcbid 840	. . . . 5 ⊢ (𝑗 ∈ (0..^𝑁) → (DECID (𝑗 gcd 𝑁) = 1 ↔ DECID 𝑗 ∈ 𝑆))
24	23	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (DECID (𝑗 gcd 𝑁) = 1 ↔ DECID 𝑗 ∈ 𝑆))
25	18, 24	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → DECID 𝑗 ∈ 𝑆)
26	25	ralrimiva 2580	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆)
27		ssfidc 7041	. 2 ⊢ (((0..^𝑁) ∈ Fin ∧ 𝑆 ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆) → 𝑆 ∈ Fin)
28	6, 10, 26, 27	syl3anc 1250	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 836 ∧ w3a 981 = wceq 1373 ∈ wcel 2177 ∀wral 2485 {crab 2489 ⊆ wss 3167 (class class class)co 5951 Fincfn 6834 0cc0 7932 1c1 7933 ℕcn 9043 ℤcz 9379 ..^cfzo 10271 gcd cgcd 12318

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 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051 ax-caucvg 8052

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 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 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-frec 6484 df-1o 6509 df-er 6627 df-en 6835 df-fin 6837 df-sup 7093 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-3 9103 df-4 9104 df-n0 9303 df-z 9380 df-uz 9656 df-q 9748 df-rp 9783 df-fz 10138 df-fzo 10272 df-fl 10420 df-mod 10475 df-seqfrec 10600 df-exp 10691 df-cj 11197 df-re 11198 df-im 11199 df-rsqrt 11353 df-abs 11354 df-dvds 12143 df-gcd 12319

This theorem is referenced by: eulerthlemh 12597 eulerth 12599

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