Theorem eldioph3 40232

Description: Inference version of eldioph3b 40231 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 486	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2		simpr 488	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3		eqidd 2737	. . 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 6694	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
5	4	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑏) = 0 ↔ (𝑃‘𝑏) = 0))
6	5	anbi2d 632	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
7	6	rexbidv 3206	. . . . . 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 2800	. . . . 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 2740	. . . . . . . . 9 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
10	9	anbi1d 633	. . . . . . . 8 ⊢ (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
11	10	rexbidv 3206	. . . . . . 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 5830	. . . . . . . . . 10 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
13	12	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14		fveqeq2 6704	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → ((𝑃‘𝑏) = 0 ↔ (𝑃‘𝑢) = 0))
15	13, 14	anbi12d 634	. . . . . . . 8 ⊢ (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
16	15	cbvrexvw 3349	. . . . . . 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 290	. . . . . 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 2804	. . . . 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 2787	. . . 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 3542	. . 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 587	. 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 40231	. 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 586	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 399   = wceq 1543   ∈ wcel 2112  {cab 2714  ∃wrex 3052   ↾ cres 5538  ‘cfv 6358  (class class class)co 7191   ↑_m cmap 8486  0cc0 10694  1c1 10695  ℕcn 11795  ℕ₀cn0 12055  ...cfz 13060  mzPolycmzp 40188  Diophcdioph 40221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-inf2 9234  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-of 7447  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-oadd 8184  df-er 8369  df-map 8488  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-dju 9482  df-card 9520  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-n0 12056  df-z 12142  df-uz 12404  df-fz 13061  df-hash 13862  df-mzpcl 40189  df-mzp 40190  df-dioph 40222

This theorem is referenced by:  diophrex  40241