Theorem eldioph3 43198

Description: Inference version of eldioph3b 43197 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 482	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑁 ∈ ℕ0)
2		simpr 484	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → 𝑃 ∈ (mzPoly‘ℕ))
3		eqidd 2738	. . 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 6840	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
5	4	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑏) = 0 ↔ (𝑃‘𝑏) = 0))
6	5	anbi2d 631	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
7	6	rexbidv 3162	. . . . . 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 2803	. . . . 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 2741	. . . . . . . . 9 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
10	9	anbi1d 632	. . . . . . . 8 ⊢ (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)))
11	10	rexbidv 3162	. . . . . . 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 5939	. . . . . . . . . 10 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
13	12	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
14		fveqeq2 6850	. . . . . . . . 9 ⊢ (𝑏 = 𝑢 → ((𝑃‘𝑏) = 0 ↔ (𝑃‘𝑢) = 0))
15	13, 14	anbi12d 633	. . . . . . . 8 ⊢ (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
16	15	cbvrexvw 3217	. . . . . . 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 287	. . . . . 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 2807	. . . . 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 2788	. . . 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 3588	. . 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 585	. 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 43197	. 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 584	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 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ↾ cres 5633  ‘cfv 6499  (class class class)co 7367   ↑_m cmap 8773  0cc0 11038  1c1 11039  ℕcn 12174  ℕ₀cn0 12437  ...cfz 13461  mzPolycmzp 43154  Diophcdioph 43187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-hash 14293  df-mzpcl 43155  df-mzp 43156  df-dioph 43188

This theorem is referenced by:  diophrex  43207