4sqleminfi - Intuitionistic Logic Explorer

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Theorem 4sqleminfi 12720

Description: Lemma for 4sq 12733. 𝐴 ∩ ran 𝐹 is finite. (Contributed by Jim Kingdon, 24-May-2025.)

**Hypotheses**
Ref	Expression
4sqlemafi.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
4sqlemafi.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
4sqlemafi.a	⊢ 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
4sqlemffi.f	⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))

**Assertion**
Ref	Expression
4sqleminfi	⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ∈ Fin)

Distinct variable groups: 𝑚,𝑁,𝑢 𝑃,𝑚,𝑢 𝜑,𝑚,𝑢 𝑣,𝐴 𝜑,𝑣 Allowed substitution hints: 𝐴(𝑢,𝑚) 𝑃(𝑣) 𝐹(𝑣,𝑢,𝑚) 𝑁(𝑣)

Proof of Theorem 4sqleminfi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlemafi.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		4sqlemafi.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ)
3		4sqlemafi.a	. . 3 ⊢ 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
4	1, 2, 3	4sqlemafi 12718	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 = ((𝑚↑2) mod 𝑃))
6		elfzelz 10147	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
7	6	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑚 ∈ ℤ)
8		zsqcl 10755	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
9	7, 8	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → (𝑚↑2) ∈ ℤ)
10	2	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑃 ∈ ℕ)
11	9, 10	zmodcld 10490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℕ₀)
12	11	nn0zd 9493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℤ)
13	5, 12	eqeltrd 2282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 ∈ ℤ)
14	13	rexlimdva2 2626	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ ℤ))
15	14	abssdv 3267	. . . . . . . . 9 ⊢ (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ ℤ)
16	3, 15	eqsstrid 3239	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
17	16	sselda 3193	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ)
18	2	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℕ)
19	18	nnzd 9494	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℤ)
20		peano2zm 9410	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
21	19, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑃 − 1) ∈ ℤ)
22	16	sselda 3193	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
23	22	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
24	21, 23	zsubcld 9500	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → ((𝑃 − 1) − 𝑣) ∈ ℤ)
25		zdceq 9448	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ ((𝑃 − 1) − 𝑣) ∈ ℤ) → DECID 𝑥 = ((𝑃 − 1) − 𝑣))
26	17, 24, 25	syl2an2r 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → DECID 𝑥 = ((𝑃 − 1) − 𝑣))
27	26	ralrimiva 2579	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 DECID 𝑥 = ((𝑃 − 1) − 𝑣))
28		finexdc 6999	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑣 ∈ 𝐴 DECID 𝑥 = ((𝑃 − 1) − 𝑣)) → DECID ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
29	4, 27, 28	syl2an2r 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → DECID ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
30		4sqlemffi.f	. . . . . . 7 ⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))
31	30	elrnmpt 4927	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣)))
32	31	elv 2776	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
33	32	dcbii 842	. . . 4 ⊢ (DECID 𝑥 ∈ ran 𝐹 ↔ DECID ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
34	29, 33	sylibr 134	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → DECID 𝑥 ∈ ran 𝐹)
35	34	ralrimiva 2579	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ ran 𝐹)
36		infidc 7036	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ ran 𝐹) → (𝐴 ∩ ran 𝐹) ∈ Fin)
37	4, 35, 36	syl2anc 411	1 ⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 836 = wceq 1373 ∈ wcel 2176 {cab 2191 ∀wral 2484 ∃wrex 2485 Vcvv 2772 ∩ cin 3165 ↦ cmpt 4105 ran crn 4676 (class class class)co 5944 Fincfn 6827 0cc0 7925 1c1 7926 − cmin 8243 ℕcn 9036 2c2 9087 ℤcz 9372 ...cfz 10130 mod cmo 10467 ↑cexp 10683

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-1o 6502 df-er 6620 df-en 6828 df-fin 6830 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-fz 10131 df-fzo 10265 df-fl 10413 df-mod 10468 df-seqfrec 10593 df-exp 10684

This theorem is referenced by: 4sqlem11 12724 4sqlem12 12725

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