Theorem 2rexfrabdioph 42791

Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypotheses

Ref	Expression
rexfrabdioph.1	⊢ 𝑀 = (𝑁 + 1)
rexfrabdioph.2	⊢ 𝐿 = (𝑀 + 1)

Assertion

Ref	Expression
2rexfrabdioph	⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Distinct variable groups:   𝑢,𝑡,𝑣,𝑤,𝐿   𝑡,𝑀,𝑢,𝑣,𝑤   𝑡,𝑁,𝑢,𝑣,𝑤   𝜑,𝑡

Allowed substitution hints:   𝜑(𝑤,𝑣,𝑢)

Proof of Theorem 2rexfrabdioph

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sbcrex 42779	. . . 4 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
2	1	rabbii 3414	. . 3 ⊢ {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 𝜑} = {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑}
3		rexfrabdioph.1	. . . . . 6 ⊢ 𝑀 = (𝑁 + 1)
4		peano2nn0 12489	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
5	3, 4	eqeltrid 2833	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ0)
6	5	adantr 480	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → 𝑀 ∈ ℕ0)
7		sbcrot3 42786	. . . . . . . . 9 ⊢ ([(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑 ↔ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
8	7	sbcbii 3813	. . . . . . . 8 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑 ↔ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
9		reseq1 5947	. . . . . . . . . 10 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎 ↾ (1...𝑁)) = ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)))
10	9	sbccomieg 42788	. . . . . . . . 9 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
11		fzssp1 13535	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
12	3	oveq2i 7401	. . . . . . . . . . . 12 ⊢ (1...𝑀) = (1...(𝑁 + 1))
13	11, 12	sseqtrri 3999	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (1...𝑀)
14		resabs1 5980	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ (1...𝑀) → ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)))
15		dfsbcq 3758	. . . . . . . . . . 11 ⊢ (((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)) → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
16	13, 14, 15	mp2b 10	. . . . . . . . . 10 ⊢ ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
17		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
18	17	resex 6003	. . . . . . . . . . . . 13 ⊢ (𝑡 ↾ (1...𝑀)) ∈ V
19		fveq1 6860	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎‘𝑀) = ((𝑡 ↾ (1...𝑀))‘𝑀))
20	19	sbcco3gw 4391	. . . . . . . . . . . . 13 ⊢ ((𝑡 ↾ (1...𝑀)) ∈ V → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
21	18, 20	ax-mp 5	. . . . . . . . . . . 12 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
22		nn0p1nn 12488	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
23	3, 22	eqeltrid 2833	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ)
24		elfz1end 13522	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
25	23, 24	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ (1...𝑀))
26		fvres 6880	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝑀) → ((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀))
27		dfsbcq 3758	. . . . . . . . . . . . 13 ⊢ (((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀) → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
28	25, 26, 27	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
29	21, 28	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
30	29	sbcbidv 3812	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
31	16, 30	bitrid 283	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
32	10, 31	bitrid 283	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
33	8, 32	bitr2id 284	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑))
34	33	rabbidv 3416	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} = {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑})
35	34	eleq1d 2814	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ({𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿) ↔ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐿)))
36	35	biimpa 476	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐿))
37		rexfrabdioph.2	. . . . 5 ⊢ 𝐿 = (𝑀 + 1)
38	37	rexfrabdioph 42790	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐿)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
39	6, 36, 38	syl2anc 584	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
40	2, 39	eqeltrid 2833	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑀))
41	3	rexfrabdioph 42790	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
42	40, 41	syldan 591	1 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  {crab 3408  Vcvv 3450  [wsbc 3756   ⊆ wss 3917   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  1c1 11076   + caddc 11078  ℕcn 12193  ℕ₀cn0 12449  ...cfz 13475  Diophcdioph 42750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-hash 14303  df-mzpcl 42718  df-mzp 42719  df-dioph 42751

This theorem is referenced by:  3rexfrabdioph  42792  4rexfrabdioph  42793  6rexfrabdioph  42794