Theorem 2rexfrabdioph 37360

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 37348	. . . . 5 ⊢ ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑)
2	1	a1i 11	. . . 4 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑀)) → ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]∃𝑤 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑))
3	2	rabbiia 3185	. . 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}
4		rexfrabdioph.1	. . . . . 6 ⊢ 𝑀 = (𝑁 + 1)
5		peano2nn0 11333	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
6	4, 5	syl5eqel 2705	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ0)
7	6	adantr 481	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑𝑚 (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → 𝑀 ∈ ℕ0)
8		sbcrot3 37355	. . . . . . . . 9 ⊢ ([(𝑡‘𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣]𝜑 ↔ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
9	8	sbcbii 3491	. . . . . . . 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)
10		reseq1 5390	. . . . . . . . . 10 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎 ↾ (1...𝑁)) = ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)))
11	10	sbccomieg 37357	. . . . . . . . 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)
12		fzssp1 12384	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
13	4	oveq2i 6661	. . . . . . . . . . . 12 ⊢ (1...𝑀) = (1...(𝑁 + 1))
14	12, 13	sseqtr4i 3638	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (1...𝑀)
15		resabs1 5427	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ (1...𝑀) → ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)))
16		dfsbcq 3437	. . . . . . . . . . 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))
17	14, 15, 16	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)
18		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑡 ∈ V
19	18	resex 5443	. . . . . . . . . . . . 13 ⊢ (𝑡 ↾ (1...𝑀)) ∈ V
20		fveq1 6190	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎‘𝑀) = ((𝑡 ↾ (1...𝑀))‘𝑀))
21	20	sbcco3g 3999	. . . . . . . . . . . . 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))
22	19, 21	ax-mp 5	. . . . . . . . . . . 12 ⊢ ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑)
23		nn0p1nn 11332	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
24	4, 23	syl5eqel 2705	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ)
25		elfz1end 12371	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
26	24, 25	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ (1...𝑀))
27		fvres 6207	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝑀) → ((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀))
28		dfsbcq 3437	. . . . . . . . . . . . 13 ⊢ (((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡‘𝑀) → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
29	26, 27, 28	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
30	22, 29	syl5bb 272	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑 ↔ [(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑))
31	30	sbcbidv 3490	. . . . . . . . . 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))
32	17, 31	syl5bb 272	. . . . . . . . 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))
33	11, 32	syl5bb 272	. . . . . . . 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))
34	9, 33	syl5rbb 273	. . . . . . 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))
35	34	rabbidv 3189	. . . . . 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})
36	35	eleq1d 2686	. . . . 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)))
37	36	biimpa 501	. . . 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))
38		rexfrabdioph.2	. . . . 5 ⊢ 𝐿 = (𝑀 + 1)
39	38	rexfrabdioph 37359	. . . 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))
40	7, 37, 39	syl2anc 693	. . 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))
41	3, 40	syl5eqel 2705	. 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))
42	4	rexfrabdioph 37359	. 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))
43	41, 42	syldan 487	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916  Vcvv 3200  [wsbc 3435   ⊆ wss 3574   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  1c1 9937   + caddc 9939  ℕcn 11020  ℕ₀cn0 11292  ...cfz 12326  Diophcdioph 37318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-mzpcl 37286  df-mzp 37287  df-dioph 37319

This theorem is referenced by:  3rexfrabdioph  37361  4rexfrabdioph  37362  6rexfrabdioph  37363