4sqleminfi - Intuitionistic Logic Explorer

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Theorem 4sqleminfi 13205

Description: Lemma for 4sq 13218. 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 13203	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 = ((𝑚↑2) mod 𝑃))
6		elfzelz 10445	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
7	6	ad2antlr 493	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑚 ∈ ℤ)
8		zsqcl 11068	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑃 ∈ ℕ)
11	9, 10	zmodcld 10803	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℕ₀)
12	11	nn0zd 9777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → ((𝑚↑2) mod 𝑃) ∈ ℤ)
13	5, 12	eqeltrd 2315	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑢 = ((𝑚↑2) mod 𝑃)) → 𝑢 ∈ ℤ)
14	13	rexlimdva2 2671	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ ℤ))
15	14	abssdv 3322	. . . . . . . . 9 ⊢ (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ ℤ)
16	3, 15	eqsstrid 3294	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
17	16	sselda 3248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ)
18	2	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℕ)
19	18	nnzd 9778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑃 ∈ ℤ)
20		peano2zm 9693	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
21	19, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑃 − 1) ∈ ℤ)
22	16	sselda 3248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
23	22	adantlr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℤ)
24	21, 23	zsubcld 9784	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → ((𝑃 − 1) − 𝑣) ∈ ℤ)
25		zdceq 9731	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ ((𝑃 − 1) − 𝑣) ∈ ℤ) → DECID 𝑥 = ((𝑃 − 1) − 𝑣))
26	17, 24, 25	syl2an2r 603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → DECID 𝑥 = ((𝑃 − 1) − 𝑣))
27	26	ralrimiva 2623	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 DECID 𝑥 = ((𝑃 − 1) − 𝑣))
28		finexdc 7211	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑣 ∈ 𝐴 DECID 𝑥 = ((𝑃 − 1) − 𝑣)) → DECID ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
29	4, 27, 28	syl2an2r 603	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → DECID ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
30		4sqlemffi.f	. . . . . . 7 ⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))
31	30	elrnmpt 5033	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣)))
32	31	elv 2825	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑣 ∈ 𝐴 𝑥 = ((𝑃 − 1) − 𝑣))
33	32	dcbii 852	. . . 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 2623	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ ran 𝐹)
36		infidc 7252	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ ran 𝐹) → (𝐴 ∩ ran 𝐹) ∈ Fin)
37	4, 35, 36	syl2anc 415	1 ⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ∈ Fin)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 846 = wceq 1402 ∈ wcel 2209 {cab 2224 ∀wral 2528 ∃wrex 2529 Vcvv 2821 ∩ cin 3219 ↦ cmpt 4193 ran crn 4777 (class class class)co 6088 Fincfn 7026 0cc0 8185 1c1 8186 − cmin 8504 ℕcn 9313 2c2 9364 ℤcz 9655 ...cfz 10428 mod cmo 10780 ↑cexp 10996

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-1o 6691 df-er 6811 df-en 7027 df-fin 7029 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997

This theorem is used by: 4sqlem11 13209 4sqlem12 13210

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