Theorem eqrabdioph 43622

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 5204	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
2	1	nfel1 2938	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
3		nfmpt1 5204	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵)
4	3	nfel1 2938	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))
5	2, 4	nfan 1932	. . . . 5 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)))
6		mzpf 43581	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
7	6	ad2antrr 739	. . . . . . . . . 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 12624	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
9		nn0ssz 12638	. . . . . . . . . . . . 13 ⊢ ℕ0 ⊆ ℤ
10		mapss 8896	. . . . . . . . . . . . 13 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁)))
11	8, 9, 10	mp2an 705	. . . . . . . . . . . 12 ⊢ (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁))
12	11	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
13	12	adantl 487	. . . . . . . . . 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 43565	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) → 𝐴 ∈ ℤ))
15	7, 13, 14	sylc 66	. . . . . . . . 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 12726	. . . . . . . 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 43581	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵):(ℤ ↑m (1...𝑁))⟶ℤ)
18	17	ad2antlr 740	. . . . . . . . . 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 43565	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵):(ℤ ↑m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) → 𝐵 ∈ ℤ))
20	18, 13, 19	sylc 66	. . . . . . . . 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 12726	. . . . . . . 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 11601	. . . . . . 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 226	. . . . . 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 418	. . . . 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 3260	. . . 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 3441	. . . 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 221	. . 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 1148	. 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 1154	. . 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 43588	. . . 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 1148	. . 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 43621	. . 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 596	. 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 2860	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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3899   ↦ cmpt 5186  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ↑_m cmap 8826  0cc0 11124  1c1 11125   − cmin 11465  ℕ₀cn0 12528  ℤcz 12615  ...cfz 13561  mzPolycmzp 43567  Diophcdioph 43600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-mzpcl 43568  df-mzp 43569  df-dioph 43601

This theorem is used by:  elnn0rabdioph  43644  dvdsrabdioph  43651  rmydioph  43855  rmxdioph  43857  expdiophlem2  43863  expdioph  43864