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Theorem lgsquadlemsfi 15762

Description: Lemma for lgsquad 15767. 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 9481	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		lgseisen.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
4		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
5	3, 4	gausslemma2dlem0b 15737	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnzd 9576	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7	2, 6	fzfigd 10661	. 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 15737	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	nnzd 9576	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	2, 11	fzfigd 10661	. 2 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
13		elfznn 10258	. . . . . . 7 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
14	13	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
15	3	gausslemma2dlem0a 15736	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
16	15	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
17	14, 16	nnmulcld 9167	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
18	17	nnzd 9576	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℤ)
19		elfznn 10258	. . . . . . 7 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
20	19	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
21	8	gausslemma2dlem0a 15736	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
23	20, 22	nnmulcld 9167	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
24	23	nnzd 9576	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℤ)
25		zdclt 9532	. . . 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 2612	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑁)DECID (𝑦 · 𝑃) < (𝑥 · 𝑄))
28	1, 7, 12, 27	opabfi 7108	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 DECID wdc 839 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∖ cdif 3194 {csn 3666 class class class wbr 4083 {copab 4144 (class class class)co 6007 Fincfn 6895 1c1 8008 · cmul 8012 < clt 8189 − cmin 8325 / cdiv 8827 ℕcn 9118 2c2 9169 ℤcz 9454 ...cfz 10212 ℙcprime 12637

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-xor 1418 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-2o 6569 df-er 6688 df-en 6896 df-fin 6898 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-seqfrec 10678 df-exp 10769 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-dvds 12307 df-prm 12638

This theorem is referenced by: lgsquadlemofi 15763 lgsquadlem1 15764 lgsquadlem2 15765 lgsquadlem3 15766

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