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Theorem eulerthlemfi 12396

Description: Lemma for eulerth 12401. 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 9337	. . 3 ⊢ 0 ∈ ℤ
2		eulerth.1	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
3	2	simp1d 1011	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9447	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5		fzofig 10524	. . 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 3268	. . . 4 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
9	7, 8	eqsstri 3215	. . 3 ⊢ 𝑆 ⊆ (0..^𝑁)
10	9	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (0..^𝑁))
11		elfzoelz 10222	. . . . . . . 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 12135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ₀)
15	14	nn0zd 9446	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
16		1zzd 9353	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℤ)
17		zdceq 9401	. . . . 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 5929	. . . . . . . . 9 ⊢ (𝑦 = 𝑗 → (𝑦 gcd 𝑁) = (𝑗 gcd 𝑁))
20	19	eqeq1d 2205	. . . . . . . 8 ⊢ (𝑦 = 𝑗 → ((𝑦 gcd 𝑁) = 1 ↔ (𝑗 gcd 𝑁) = 1))
21	20, 7	elrab2 2923	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 1))
22	21	baibr 921	. . . . . 6 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 1 ↔ 𝑗 ∈ 𝑆))
23	22	dcbid 839	. . . . 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 2570	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆)
27		ssfidc 6998	. 2 ⊢ (((0..^𝑁) ∈ Fin ∧ 𝑆 ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆) → 𝑆 ∈ Fin)
28	6, 10, 26, 27	syl3anc 1249	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ∀wral 2475 {crab 2479 ⊆ wss 3157 (class class class)co 5922 Fincfn 6799 0cc0 7879 1c1 7880 ℕcn 8990 ℤcz 9326 ..^cfzo 10217 gcd cgcd 12120

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-1o 6474 df-er 6592 df-en 6800 df-fin 6802 df-sup 7050 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121

This theorem is referenced by: eulerthlemh 12399 eulerth 12401

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