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Theorem lgsquadlemofi 15941

Description: Lemma for lgsquad 15945. There are finitely many members of 𝑆 with odd first part. (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
lgsquadlemofi	⊢ (𝜑 → {𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1^st ‘𝑧)} ∈ Fin)

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

**Proof of Theorem lgsquadlemofi**
Step	Hyp	Ref	Expression
1		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		lgseisen.2	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
3		lgseisen.3	. . 3 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
4		lgsquad.4	. . 3 ⊢ 𝑀 = ((𝑃 − 1) / 2)
5		lgsquad.5	. . 3 ⊢ 𝑁 = ((𝑄 − 1) / 2)
6		lgsquad.6	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}
7	1, 2, 3, 4, 5, 6	lgsquadlemsfi 15940	. 2 ⊢ (𝜑 → 𝑆 ∈ Fin)
8		2nn 9398	. . . . 5 ⊢ 2 ∈ ℕ
9		opabssxp 4823	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
10	6, 9	eqsstri 3269	. . . . . . . . 9 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
11	10	sseli 3233	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((1...𝑀) × (1...𝑁)))
12		xp1st 6358	. . . . . . . 8 ⊢ (𝑧 ∈ ((1...𝑀) × (1...𝑁)) → (1^st ‘𝑧) ∈ (1...𝑀))
13	11, 12	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ (1...𝑀))
14	13	elfzelzd 10359	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ ℤ)
15	14	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (1^st ‘𝑧) ∈ ℤ)
16		dvdsdc 12480	. . . . 5 ⊢ ((2 ∈ ℕ ∧ (1^st ‘𝑧) ∈ ℤ) → DECID 2 ∥ (1^st ‘𝑧))
17	8, 15, 16	sylancr 414	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → DECID 2 ∥ (1^st ‘𝑧))
18		dcn 850	. . . 4 ⊢ (DECID 2 ∥ (1^st ‘𝑧) → DECID ¬ 2 ∥ (1^st ‘𝑧))
19	17, 18	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → DECID ¬ 2 ∥ (1^st ‘𝑧))
20	19	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 DECID ¬ 2 ∥ (1^st ‘𝑧))
21	7, 20	ssfirab 7196	1 ⊢ (𝜑 → {𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1^st ‘𝑧)} ∈ Fin)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 842 = wceq 1398 ∈ wcel 2203 ≠ wne 2412 {crab 2524 ∖ cdif 3207 {csn 3688 class class class wbr 4108 {copab 4169 × cxp 4746 ‘cfv 5351 (class class class)co 6049 1^st c1st 6331 Fincfn 6974 1c1 8127 · cmul 8131 < clt 8307 − cmin 8443 / cdiv 8945 ℕcn 9236 2c2 9287 ℤcz 9576 ...cfz 10341 ∥ cdvds 12469 ℙcprime 12800

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244 ax-arch 8245 ax-caucvg 8246

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 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-1o 6646 df-2o 6647 df-er 6766 df-en 6975 df-fin 6977 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-n0 9496 df-z 9577 df-uz 9853 df-q 9951 df-rp 9986 df-fz 10342 df-fl 10629 df-mod 10684 df-seqfrec 10809 df-exp 10900 df-cj 11523 df-re 11524 df-im 11525 df-rsqrt 11679 df-abs 11680 df-dvds 12470 df-prm 12801

This theorem is referenced by: lgsquadlem1 15942 lgsquadlem2 15943

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