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Theorem seqcl2 14071
Description: Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqcl2.1 (𝜑 → (𝐹𝑀) ∈ 𝐶)
seqcl2.2 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
seqcl2.3 (𝜑𝑁 ∈ (ℤ𝑀))
seqcl2.4 ((𝜑𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹𝑥) ∈ 𝐷)
Assertion
Ref Expression
seqcl2 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁   𝑥, + ,𝑦   𝜑,𝑥,𝑦
Allowed substitution hint:   𝑁(𝑦)

Proof of Theorem seqcl2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seqcl2.3 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 13592 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2832 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 6920 . . . . . . 7 (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
65eleq1d 2829 . . . . . 6 (𝑥 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
74, 6imbi12d 344 . . . . 5 (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
87imbi2d 340 . . . 4 (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))))
9 eleq1 2832 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10 fveq2 6920 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1110eleq1d 2829 . . . . . 6 (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶))
129, 11imbi12d 344 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)))
1312imbi2d 340 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶))))
14 eleq1 2832 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15 fveq2 6920 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
1615eleq1d 2829 . . . . . 6 (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
1714, 16imbi12d 344 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶)))
1817imbi2d 340 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))))
19 eleq1 2832 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20 fveq2 6920 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2120eleq1d 2829 . . . . . 6 (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))
2219, 21imbi12d 344 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)))
2322imbi2d 340 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))))
24 seqcl2.1 . . . . . 6 (𝜑 → (𝐹𝑀) ∈ 𝐶)
25 seq1 14065 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
2625eleq1d 2829 . . . . . 6 (𝑀 ∈ ℤ → ((seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶 ↔ (𝐹𝑀) ∈ 𝐶))
2724, 26imbitrrid 246 . . . . 5 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
2827a1dd 50 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
29 peano2fzr 13597 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
3029adantl 481 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
3130expr 456 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
3231imim1d 82 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)))
33 fveq2 6920 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
3433eleq1d 2829 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝐷 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝐷))
35 seqcl2.4 . . . . . . . . . 10 ((𝜑𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹𝑥) ∈ 𝐷)
3635ralrimiva 3152 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹𝑥) ∈ 𝐷)
3736adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹𝑥) ∈ 𝐷)
38 eluzp1p1 12931 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ‘(𝑀 + 1)))
3938ad2antrl 727 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (ℤ‘(𝑀 + 1)))
40 elfzuz3 13581 . . . . . . . . . 10 ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
4140ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
42 elfzuzb 13578 . . . . . . . . 9 ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))))
4339, 41, 42sylanbrc 582 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
4434, 37, 43rspcdva 3636 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝐷)
45 seqcl2.2 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
4645caovclg 7642 . . . . . . . . 9 ((𝜑 ∧ ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶)
4746ex 412 . . . . . . . 8 (𝜑 → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
4847adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
4944, 48mpan2d 693 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
50 seqp1 14067 . . . . . . . 8 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5150ad2antrl 727 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5251eleq1d 2829 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶 ↔ ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
5349, 52sylibrd 259 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
5432, 53animpimp2impd 845 . . . 4 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))))
558, 13, 18, 23, 28, 54uzind4 12971 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)))
561, 55mpcom 38 . 2 (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))
573, 56mpd 15 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  wral 3067  cfv 6573  (class class class)co 7448  1c1 11185   + caddc 11187  cz 12639  cuz 12903  ...cfz 13567  seqcseq 14052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053
This theorem is referenced by:  seqf2  14072  seqcl  14073  seqz  14101
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