eqrabdioph - Mathbox for Stefan O'Rear

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Theorem eqrabdioph 42261

Description: Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)

**Assertion**
Ref	Expression
eqrabdioph	⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))

Distinct variable group: 𝑡,𝑁 Allowed substitution hints: 𝐴(𝑡) 𝐵(𝑡)

**Proof of Theorem eqrabdioph**
Step	Hyp	Ref	Expression
1		nfmpt1 5251	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)
2	1	nfel1 2909	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
3		nfmpt1 5251	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵)
4	3	nfel1 2909	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))
5	2, 4	nfan 1894	. . . . 5 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)))
6		mzpf 42220	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ)
7	6	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ)
8		zex 12595	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
9		nn0ssz 12609	. . . . . . . . . . . . 13 ⊢ ℕ₀ ⊆ ℤ
10		mapss 8904	. . . . . . . . . . . . 13 ⊢ ((ℤ ∈ V ∧ ℕ₀ ⊆ ℤ) → (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁)))
11	8, 9, 10	mp2an 690	. . . . . . . . . . . 12 ⊢ (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁))
12	11	sseli 3968	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) → 𝑡 ∈ (ℤ ↑_m (1...𝑁)))
13	12	adantl 480	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑡 ∈ (ℤ ↑_m (1...𝑁)))
14		mptfcl 42204	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) → 𝐴 ∈ ℤ))
15	7, 13, 14	sylc 65	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐴 ∈ ℤ)
16	15	zcnd 12695	. . . . . . . 8 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐴 ∈ ℂ)
17		mzpf 42220	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ)
18	17	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ)
19		mptfcl 42204	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) → 𝐵 ∈ ℤ))
20	18, 13, 19	sylc 65	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐵 ∈ ℤ)
21	20	zcnd 12695	. . . . . . . 8 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐵 ∈ ℂ)
22	16, 21	subeq0ad 11609	. . . . . . 7 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
23	22	bicomd 222	. . . . . 6 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0))
24	23	ex 411	. . . . 5 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) → (𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0)))
25	5, 24	ralrimi 3245	. . . 4 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0))
26		rabbi 3450	. . . 4 ⊢ (∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0) ↔ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0})
27	25, 26	sylib 217	. . 3 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0})
28	27	3adant1 1127	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0})
29		simp1 1133	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ₀)
30		mzpsubmpt 42227	. . . 4 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘(1...𝑁)))
31	30	3adant1 1127	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘(1...𝑁)))
32		eq0rabdioph 42260	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0} ∈ (Dioph‘𝑁))
33	29, 31, 32	syl2anc 582	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0} ∈ (Dioph‘𝑁))
34	28, 33	eqeltrd 2825	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 {crab 3419 Vcvv 3463 ⊆ wss 3940 ↦ cmpt 5226 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 ↑_m cmap 8841 0cc0 11136 1c1 11137 − cmin 11472 ℕ₀cn0 12500 ℤcz 12586 ...cfz 13514 mzPolycmzp 42206 Diophcdioph 42239

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-mzpcl 42207 df-mzp 42208 df-dioph 42240

This theorem is referenced by: elnn0rabdioph 42287 dvdsrabdioph 42294 rmydioph 42499 rmxdioph 42501 expdiophlem2 42507 expdioph 42508

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