4sqleminfi - Intuitionistic Logic Explorer

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Theorem 4sqleminfi 12566

Description: Lemma for 4sq 12579. 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 12564	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 = ((𝑚↑2) mod 𝑃))
6		elfzelz 10100	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
7	6	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑚 ∈ ℤ)
8		zsqcl 10702	. . . . . . . . . . . . . . 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 10437	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℕ₀)
12	11	nn0zd 9446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℤ)
13	5, 12	eqeltrd 2273	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 ∈ ℤ)
14	13	rexlimdva2 2617	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ ℤ))
15	14	abssdv 3257	. . . . . . . . 9 ⊢ (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ ℤ)
16	3, 15	eqsstrid 3229	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
17	16	sselda 3183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ)
18	2	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℕ)
19	18	nnzd 9447	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℤ)
20		peano2zm 9364	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
21	19, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑃 − 1) ∈ ℤ)
22	16	sselda 3183	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
23	22	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
24	21, 23	zsubcld 9453	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → ((𝑃 − 1) − 𝑣) ∈ ℤ)
25		zdceq 9401	. . . . . . 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 2570	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 DECID 𝑥 = ((𝑃 − 1) − 𝑣))
28		finexdc 6963	. . . . 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 4915	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣)))
32	31	elv 2767	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
33	32	dcbii 841	. . . 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 2570	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ ran 𝐹)
36		infidc 7000	. 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 835 = wceq 1364 ∈ wcel 2167 {cab 2182 ∀wral 2475 ∃wrex 2476 Vcvv 2763 ∩ cin 3156 ↦ cmpt 4094 ran crn 4664 (class class class)co 5922 Fincfn 6799 0cc0 7879 1c1 7880 − cmin 8197 ℕcn 8990 2c2 9041 ℤcz 9326 ...cfz 10083 mod cmo 10414 ↑cexp 10630

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 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 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-er 6592 df-en 6800 df-fin 6802 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631

This theorem is referenced by: 4sqlem11 12570 4sqlem12 12571

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