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Theorem lgsquadlemsfi 16316

Description: Lemma for lgsquad 16321. 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 9676	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		lgseisen.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
4		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
5	3, 4	gausslemma2dlem0b 16291	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnzd 9772	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7	2, 6	fzfigd 10882	. 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 16291	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	nnzd 9772	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	2, 11	fzfigd 10882	. 2 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
13		elfznn 10471	. . . . . . 7 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
14	13	ad2antll 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
15	3	gausslemma2dlem0a 16290	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
16	15	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
17	14, 16	nnmulcld 9356	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
18	17	nnzd 9772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℤ)
19		elfznn 10471	. . . . . . 7 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
20	19	ad2antrl 494	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
21	8	gausslemma2dlem0a 16290	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
23	20, 22	nnmulcld 9356	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
24	23	nnzd 9772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℤ)
25		zdclt 9727	. . . 4 ⊢ (((𝑦 · 𝑃) ∈ ℤ ∧ (𝑥 · 𝑄) ∈ ℤ) → DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
26	18, 24, 25	syl2anc 415	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
27	26	ralrimivva 2632	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑁)DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
28	1, 7, 12, 27	opabfi 7247	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∖ cdif 3217 {csn 3709 class class class wbr 4130 {copab 4191 (class class class)co 6085 Fincfn 7022 1c1 8181 · cmul 8185 < clt 8361 − cmin 8499 / cdiv 9005 ℕcn 9307 2c2 9358 ℤcz 9649 ...cfz 10422 ℙcprime 12903

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-2o 6688 df-er 6807 df-en 7023 df-fin 7025 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-fz 10423 df-seqfrec 10899 df-exp 10990 df-cj 11622 df-re 11623 df-im 11624 df-rsqrt 11779 df-abs 11780 df-dvds 12573 df-prm 12904

This theorem is used by: lgsquadlemofi 16317 lgsquadlem1 16318 lgsquadlem2 16319 lgsquadlem3 16320

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