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Theorem eulerthlemfi 12758

Description: Lemma for eulerth 12763. 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 9465	. . 3 ⊢ 0 ∈ ℤ
2		eulerth.1	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
3	2	simp1d 1033	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9576	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5		fzofig 10662	. . 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 3309	. . . 4 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
9	7, 8	eqsstri 3256	. . 3 ⊢ 𝑆 ⊆ (0..^𝑁)
10	9	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (0..^𝑁))
11		elfzoelz 10351	. . . . . . . 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 12497	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ₀)
15	14	nn0zd 9575	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
16		1zzd 9481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℤ)
17		zdceq 9530	. . . . 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 6014	. . . . . . . . 9 ⊢ (𝑦 = 𝑗 → (𝑦 gcd 𝑁) = (𝑗 gcd 𝑁))
20	19	eqeq1d 2238	. . . . . . . 8 ⊢ (𝑦 = 𝑗 → ((𝑦 gcd 𝑁) = 1 ↔ (𝑗 gcd 𝑁) = 1))
21	20, 7	elrab2 2962	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 1))
22	21	baibr 925	. . . . . 6 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 1 ↔ 𝑗 ∈ 𝑆))
23	22	dcbid 843	. . . . 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 2603	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆)
27		ssfidc 7107	. 2 ⊢ (((0..^𝑁) ∈ Fin ∧ 𝑆 ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆) → 𝑆 ∈ Fin)
28	6, 10, 26, 27	syl3anc 1271	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 {crab 2512 ⊆ wss 3197 (class class class)co 6007 Fincfn 6895 0cc0 8007 1c1 8008 ℕcn 9118 ℤcz 9454 ..^cfzo 10346 gcd cgcd 12482

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-er 6688 df-en 6896 df-fin 6898 df-sup 7159 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-fz 10213 df-fzo 10347 df-fl 10498 df-mod 10553 df-seqfrec 10678 df-exp 10769 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-dvds 12307 df-gcd 12483

This theorem is referenced by: eulerthlemh 12761 eulerth 12763

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