Theorem eldioph3 42217

Description: Inference version of eldioph3b 42216 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 481	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2		simpr 483	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3		eqidd 2729	. . 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 6901	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
5	4	eqeq1d 2730	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑏) = 0 ↔ (𝑃‘𝑏) = 0))
6	5	anbi2d 628	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
7	6	rexbidv 3176	. . . . . 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 2797	. . . . 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 2732	. . . . . . . . 9 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
10	9	anbi1d 629	. . . . . . . 8 ⊢ (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
11	10	rexbidv 3176	. . . . . . 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 5983	. . . . . . . . . 10 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
13	12	eqeq2d 2739	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14		fveqeq2 6911	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → ((𝑃‘𝑏) = 0 ↔ (𝑃‘𝑢) = 0))
15	13, 14	anbi12d 630	. . . . . . . 8 ⊢ (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
16	15	cbvrexvw 3233	. . . . . . 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 286	. . . . . 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 2801	. . . . 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 2784	. . . 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 3633	. . 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 582	. 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 42216	. 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 581	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 394   = wceq 1533   ∈ wcel 2098  {cab 2705  ∃wrex 3067   ↾ cres 5684  ‘cfv 6553  (class class class)co 7426   ↑_m cmap 8851  0cc0 11146  1c1 11147  ℕcn 12250  ℕ₀cn0 12510  ...cfz 13524  mzPolycmzp 42173  Diophcdioph 42206

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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-oadd 8497  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-dju 9932  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-hash 14330  df-mzpcl 42174  df-mzp 42175  df-dioph 42207

This theorem is referenced by:  diophrex  42226