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Theorem seqcl2 13926
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 13449 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2825 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 6842 . . . . . . 7 (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
65eleq1d 2822 . . . . . 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 2825 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10 fveq2 6842 . . . . . . 7 (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
1110eleq1d 2822 . . . . . 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 2825 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15 fveq2 6842 . . . . . . 7 (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
1615eleq1d 2822 . . . . . 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 2825 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20 fveq2 6842 . . . . . . 7 (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
2120eleq1d 2822 . . . . . 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 13919 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
2625eleq1d 2822 . . . . . 6 (𝑀 ∈ ℤ → ((seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶 ↔ (𝐹𝑀) ∈ 𝐶))
2724, 26syl5ibr 245 . . . . 5 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
2827a1dd 50 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
29 peano2fzr 13454 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
3029adantl 482 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
3130expr 457 . . . . . 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 6842 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
3433eleq1d 2822 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝐹𝑥) ∈ 𝐷 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝐷))
35 seqcl2.4 . . . . . . . . . 10 ((𝜑𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹𝑥) ∈ 𝐷)
3635ralrimiva 3143 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹𝑥) ∈ 𝐷)
3736adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹𝑥) ∈ 𝐷)
38 eluzp1p1 12791 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ‘(𝑀 + 1)))
3938ad2antrl 726 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (ℤ‘(𝑀 + 1)))
40 elfzuz3 13438 . . . . . . . . . 10 ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
4140ad2antll 727 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
42 elfzuzb 13435 . . . . . . . . 9 ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))))
4339, 41, 42sylanbrc 583 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
4434, 37, 43rspcdva 3582 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝐷)
45 seqcl2.2 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
4645caovclg 7546 . . . . . . . . 9 ((𝜑 ∧ ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶)
4746ex 413 . . . . . . . 8 (𝜑 → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
4847adantr 481 . . . . . . 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 692 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
50 seqp1 13921 . . . . . . . 8 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5150ad2antrl 726 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5251eleq1d 2822 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶 ↔ ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
5349, 52sylibrd 258 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
5432, 53animpimp2impd 844 . . . 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 12831 . . 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 396   = wceq 1541  wcel 2106  wral 3064  cfv 6496  (class class class)co 7357  1c1 11052   + caddc 11054  cz 12499  cuz 12763  ...cfz 13424  seqcseq 13906
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907
This theorem is referenced by:  seqf2  13927  seqcl  13928  seqz  13956
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