Theorem eldioph3 43216

Description: Inference version of eldioph3b 43215 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Assertion

Ref	Expression
eldioph3	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))

Distinct variable groups:   𝑡,𝑁,𝑢   𝑡,𝑃,𝑢

Proof of Theorem eldioph3

Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2		simpr 485	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3		eqidd 2741	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)})
4		fveq1 6833	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
5	4	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑏) = 0 ↔ (𝑃‘𝑏) = 0))
6	5	anbi2d 636	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
7	6	rexbidv 3164	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
8	7	abbidv 2806	. . . . 5 ⊢ (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)})
9		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
10	9	anbi1d 637	. . . . . . . 8 ⊢ (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
11	10	rexbidv 3164	. . . . . . 7 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
12		reseq1 5932	. . . . . . . . . 10 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
13	12	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14		fveqeq2 6843	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → ((𝑃‘𝑏) = 0 ↔ (𝑃‘𝑢) = 0))
15	13, 14	anbi12d 638	. . . . . . . 8 ⊢ (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
16	15	cbvrexvw 3219	. . . . . . 7 ⊢ (∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0))
17	11, 16	bitrdi 288	. . . . . 6 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
18	17	cbvabv 2810	. . . . 5 ⊢ {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}
19	8, 18	eqtrdi 2791	. . . 4 ⊢ (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)})
20	19	rspceeqv 3590	. . 3 ⊢ ((𝑃 ∈ (mzPoly‘ℕ) ∧ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)})
21	2, 3, 20	syl2anc 590	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)})
22		eldioph3b 43215	. 2 ⊢ ({𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)}))
23	1, 21, 22	sylanbrc 589	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  ∃wrex 3064   ↾ cres 5627  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  0cc0 11036  1c1 11037  ℕcn 12172  ℕ₀cn0 12435  ...cfz 13459  mzPolycmzp 43172  Diophcdioph 43205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-hash 14291  df-mzpcl 43173  df-mzp 43174  df-dioph 43206

This theorem is referenced by:  diophrex  43225