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Theorem lgsquadlemsfi 15894

Description: Lemma for lgsquad 15899. 𝑆 is finite. (Contributed by Jim Kingdon, 16-Sep-2025.)

**Hypotheses**
Ref	Expression
lgseisen.1	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
lgseisen.2	⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
lgseisen.3	⊢ (𝜑 → 𝑃 ≠ 𝑄)
lgsquad.4	⊢ 𝑀 = ((𝑃 − 1) / 2)
lgsquad.5	⊢ 𝑁 = ((𝑄 − 1) / 2)
lgsquad.6	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}

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

Distinct variable groups: 𝑥,𝑦,𝑃 𝜑,𝑥,𝑦 𝑦,𝑀 𝑥,𝑁,𝑦 𝑥,𝑄,𝑦 𝑥,𝑆 𝑥,𝑀 𝑦,𝑆

**Proof of Theorem lgsquadlemsfi**
Step	Hyp	Ref	Expression
1		lgsquad.6	. 2 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}
2		1zzd 9567	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		lgseisen.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
4		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
5	3, 4	gausslemma2dlem0b 15869	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnzd 9662	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7	2, 6	fzfigd 10756	. 2 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
8		lgseisen.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
9		lgsquad.5	. . . . 5 ⊢ 𝑁 = ((𝑄 − 1) / 2)
10	8, 9	gausslemma2dlem0b 15869	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	nnzd 9662	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	2, 11	fzfigd 10756	. 2 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
13		elfznn 10351	. . . . . . 7 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
14	13	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
15	3	gausslemma2dlem0a 15868	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
16	15	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
17	14, 16	nnmulcld 9251	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
18	17	nnzd 9662	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℤ)
19		elfznn 10351	. . . . . . 7 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
20	19	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
21	8	gausslemma2dlem0a 15868	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
23	20, 22	nnmulcld 9251	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
24	23	nnzd 9662	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℤ)
25		zdclt 9618	. . . 4 ⊢ (((𝑦 · 𝑃) ∈ ℤ ∧ (𝑥 · 𝑄) ∈ ℤ) → DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
26	18, 24, 25	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
27	26	ralrimivva 2615	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑁)DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
28	1, 7, 12, 27	opabfi 7175	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 DECID wdc 842 = wceq 1398 ∈ wcel 2202 ≠ wne 2403 ∖ cdif 3198 {csn 3673 class class class wbr 4093 {copab 4154 (class class class)co 6028 Fincfn 6952 1c1 8093 · cmul 8097 < clt 8273 − cmin 8409 / cdiv 8911 ℕcn 9202 2c2 9253 ℤcz 9540 ...cfz 10305 ℙcprime 12759

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-1o 6625 df-2o 6626 df-er 6745 df-en 6953 df-fin 6955 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-n0 9462 df-z 9541 df-uz 9817 df-q 9915 df-rp 9950 df-fz 10306 df-seqfrec 10773 df-exp 10864 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-dvds 12429 df-prm 12760

This theorem is referenced by: lgsquadlemofi 15895 lgsquadlem1 15896 lgsquadlem2 15897 lgsquadlem3 15898

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