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Theorem lgsquadlemofi 15284

Description: Lemma for lgsquad 15288. 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 15283	. 2 ⊢ (𝜑 → 𝑆 ∈ Fin)
8		2nn 9149	. . . . 5 ⊢ 2 ∈ ℕ
9		opabssxp 4737	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
10	6, 9	eqsstri 3215	. . . . . . . . 9 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
11	10	sseli 3179	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((1...𝑀) × (1...𝑁)))
12		xp1st 6223	. . . . . . . 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 10098	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ ℤ)
15	14	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (1^st ‘𝑧) ∈ ℤ)
16		dvdsdc 11947	. . . . 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 843	. . . 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 2570	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 DECID ¬ 2 ∥ (1^st ‘𝑧))
21	7, 20	ssfirab 6995	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 835 = wceq 1364 ∈ wcel 2167 ≠ wne 2367 {crab 2479 ∖ cdif 3154 {csn 3622 class class class wbr 4033 {copab 4093 × cxp 4661 ‘cfv 5258 (class class class)co 5922 1^st c1st 6196 Fincfn 6799 1c1 7878 · cmul 7882 < clt 8059 − cmin 8195 / cdiv 8696 ℕcn 8987 2c2 9038 ℤcz 9323 ...cfz 10080 ∥ cdvds 11936 ℙcprime 12251

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7968 ax-resscn 7969 ax-1cn 7970 ax-1re 7971 ax-icn 7972 ax-addcl 7973 ax-addrcl 7974 ax-mulcl 7975 ax-mulrcl 7976 ax-addcom 7977 ax-mulcom 7978 ax-addass 7979 ax-mulass 7980 ax-distr 7981 ax-i2m1 7982 ax-0lt1 7983 ax-1rid 7984 ax-0id 7985 ax-rnegex 7986 ax-precex 7987 ax-cnre 7988 ax-pre-ltirr 7989 ax-pre-ltwlin 7990 ax-pre-lttrn 7991 ax-pre-apti 7992 ax-pre-ltadd 7993 ax-pre-mulgt0 7994 ax-pre-mulext 7995 ax-arch 7996 ax-caucvg 7997

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-1o 6474 df-2o 6475 df-er 6592 df-en 6800 df-fin 6802 df-pnf 8061 df-mnf 8062 df-xr 8063 df-ltxr 8064 df-le 8065 df-sub 8197 df-neg 8198 df-reap 8599 df-ap 8606 df-div 8697 df-inn 8988 df-2 9046 df-3 9047 df-4 9048 df-n0 9247 df-z 9324 df-uz 9599 df-q 9691 df-rp 9726 df-fz 10081 df-fl 10345 df-mod 10400 df-seqfrec 10525 df-exp 10616 df-cj 10992 df-re 10993 df-im 10994 df-rsqrt 11148 df-abs 11149 df-dvds 11937 df-prm 12252

This theorem is referenced by: lgsquadlem1 15285 lgsquadlem2 15286

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