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Theorem lgsquadlemofi 15192

Description: Lemma for lgsquad 15196. 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 15191	. 2 ⊢ (𝜑 → 𝑆 ∈ Fin)
8		2nn 9143	. . . . 5 ⊢ 2 ∈ ℕ
9		opabssxp 4733	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
10	6, 9	eqsstri 3211	. . . . . . . . 9 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
11	10	sseli 3175	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((1...𝑀) × (1...𝑁)))
12		xp1st 6218	. . . . . . . 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 10092	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ ℤ)
15	14	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (1^st ‘𝑧) ∈ ℤ)
16		dvdsdc 11941	. . . . 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 2567	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 DECID ¬ 2 ∥ (1^st ‘𝑧))
21	7, 20	ssfirab 6990	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 2164 ≠ wne 2364 {crab 2476 ∖ cdif 3150 {csn 3618 class class class wbr 4029 {copab 4089 × cxp 4657 ‘cfv 5254 (class class class)co 5918 1^st c1st 6191 Fincfn 6794 1c1 7873 · cmul 7877 < clt 8054 − cmin 8190 / cdiv 8691 ℕcn 8982 2c2 9033 ℤcz 9317 ...cfz 10074 ∥ cdvds 11930 ℙcprime 12245

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-1o 6469 df-2o 6470 df-er 6587 df-en 6795 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-prm 12246

This theorem is referenced by: lgsquadlem1 15193 lgsquadlem2 15194

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