4sqleminfi - Intuitionistic Logic Explorer

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Theorem 4sqleminfi 12535

Description: Lemma for 4sq 12548. 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 12533	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 = ((𝑚↑2) mod 𝑃))
6		elfzelz 10091	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
7	6	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑚 ∈ ℤ)
8		zsqcl 10681	. . . . . . . . . . . . . . 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 10416	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℕ₀)
12	11	nn0zd 9437	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℤ)
13	5, 12	eqeltrd 2270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 ∈ ℤ)
14	13	rexlimdva2 2614	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ ℤ))
15	14	abssdv 3253	. . . . . . . . 9 ⊢ (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ ℤ)
16	3, 15	eqsstrid 3225	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
17	16	sselda 3179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ)
18	2	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℕ)
19	18	nnzd 9438	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℤ)
20		peano2zm 9355	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
21	19, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑃 − 1) ∈ ℤ)
22	16	sselda 3179	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
23	22	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
24	21, 23	zsubcld 9444	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → ((𝑃 − 1) − 𝑣) ∈ ℤ)
25		zdceq 9392	. . . . . . 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 2567	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 DECID 𝑥 = ((𝑃 − 1) − 𝑣))
28		finexdc 6958	. . . . 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 4911	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣)))
32	31	elv 2764	. . . . 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 2567	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ ran 𝐹)
36		infidc 6993	. 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 2164 {cab 2179 ∀wral 2472 ∃wrex 2473 Vcvv 2760 ∩ cin 3152 ↦ cmpt 4090 ran crn 4660 (class class class)co 5918 Fincfn 6794 0cc0 7872 1c1 7873 − cmin 8190 ℕcn 8982 2c2 9033 ℤcz 9317 ...cfz 10074 mod cmo 10393 ↑cexp 10609

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

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 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-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-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610

This theorem is referenced by: 4sqlem11 12539 4sqlem12 12540

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