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Theorem lgsquadlemsfi 15739

Description: Lemma for lgsquad 15744. 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 9461	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		lgseisen.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
4		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
5	3, 4	gausslemma2dlem0b 15714	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnzd 9556	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7	2, 6	fzfigd 10640	. 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 15714	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	nnzd 9556	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12	2, 11	fzfigd 10640	. 2 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
13		elfznn 10238	. . . . . . 7 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
14	13	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
15	3	gausslemma2dlem0a 15713	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
16	15	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
17	14, 16	nnmulcld 9147	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
18	17	nnzd 9556	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℤ)
19		elfznn 10238	. . . . . . 7 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
20	19	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
21	8	gausslemma2dlem0a 15713	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
23	20, 22	nnmulcld 9147	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
24	23	nnzd 9556	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℤ)
25		zdclt 9512	. . . 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 7088	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 4082 {copab 4143 (class class class)co 5994 Fincfn 6877 1c1 7988 · cmul 7992 < clt 8169 − cmin 8305 / cdiv 8807 ℕcn 9098 2c2 9149 ℤcz 9434 ...cfz 10192 ℙcprime 12615

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 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-iinf 4677 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103 ax-pre-mulgt0 8104 ax-pre-mulext 8105 ax-arch 8106 ax-caucvg 8107

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 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4381 df-po 4384 df-iso 4385 df-iord 4454 df-on 4456 df-ilim 4457 df-suc 4459 df-iom 4680 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-rn 4727 df-res 4728 df-ima 4729 df-iota 5274 df-fun 5316 df-fn 5317 df-f 5318 df-f1 5319 df-fo 5320 df-f1o 5321 df-fv 5322 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-1st 6276 df-2nd 6277 df-recs 6441 df-frec 6527 df-1o 6552 df-2o 6553 df-er 6670 df-en 6878 df-fin 6880 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-reap 8710 df-ap 8717 df-div 8808 df-inn 9099 df-2 9157 df-3 9158 df-4 9159 df-n0 9358 df-z 9435 df-uz 9711 df-q 9803 df-rp 9838 df-fz 10193 df-seqfrec 10657 df-exp 10748 df-cj 11339 df-re 11340 df-im 11341 df-rsqrt 11495 df-abs 11496 df-dvds 12285 df-prm 12616

This theorem is referenced by: lgsquadlemofi 15740 lgsquadlem1 15741 lgsquadlem2 15742 lgsquadlem3 15743

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