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Theorem lgsquadlemofi 16207

Description: Lemma for lgsquad 16211. 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 16206	. 2 ⊢ (𝜑 → 𝑆 ∈ Fin)
8		2nn 9468	. . . . 5 ⊢ 2 ∈ ℕ
9		opabssxp 4849	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
10	6, 9	eqsstri 3280	. . . . . . . . 9 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
11	10	sseli 3244	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((1...𝑀) × (1...𝑁)))
12		xp1st 6399	. . . . . . . 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 10431	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ ℤ)
15	14	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (1^st ‘𝑧) ∈ ℤ)
16		dvdsdc 12567	. . . . 5 ⊢ ((2 ∈ ℕ ∧ (1^st ‘𝑧) ∈ ℤ) → DECID 2 ∥ (1^st ‘𝑧))
17	8, 15, 16	sylancr 418	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → DECID 2 ∥ (1^st ‘𝑧))
18		dcn 854	. . . 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 2623	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 DECID ¬ 2 ∥ (1^st ‘𝑧))
21	7, 20	ssfirab 7244	1 ⊢ (𝜑 → {𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1^st ‘𝑧)} ∈ Fin)

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 {crab 2532 ∖ cdif 3217 {csn 3709 class class class wbr 4130 {copab 4191 × cxp 4772 ‘cfv 5377 (class class class)co 6085 1^st c1st 6372 Fincfn 7022 1c1 8180 · cmul 8184 < clt 8360 − cmin 8497 / cdiv 9003 ℕcn 9305 2c2 9356 ℤcz 9646 ...cfz 10413 ∥ cdvds 12556 ℙcprime 12887

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 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

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 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-fz 10414 df-fl 10707 df-mod 10762 df-seqfrec 10887 df-exp 10978 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767 df-dvds 12557 df-prm 12888

This theorem is used by: lgsquadlem1 16208 lgsquadlem2 16209

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