MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqexw Structured version   Visualization version   GIF version

Theorem seqexw 14059
Description: Weak version of seqex 14045 that holds without ax-rep 5278. A sequence builder exists when its binary operation input exists and its starting index is an integer. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Hypotheses
Ref Expression
seqexw.1 + ∈ V
seqexw.2 𝑀 ∈ ℤ
Assertion
Ref Expression
seqexw seq𝑀( + , 𝐹) ∈ V

Proof of Theorem seqexw
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqexw.2 . . . 4 𝑀 ∈ ℤ
2 seqfn 14055 . . . 4 (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
31, 2ax-mp 5 . . 3 seq𝑀( + , 𝐹) Fn (ℤ𝑀)
4 fnfun 6667 . . 3 (seq𝑀( + , 𝐹) Fn (ℤ𝑀) → Fun seq𝑀( + , 𝐹))
53, 4ax-mp 5 . 2 Fun seq𝑀( + , 𝐹)
63fndmi 6671 . . 3 dom seq𝑀( + , 𝐹) = (ℤ𝑀)
7 fvex 6918 . . 3 (ℤ𝑀) ∈ V
86, 7eqeltri 2836 . 2 dom seq𝑀( + , 𝐹) ∈ V
9 seqexw.1 . . . . 5 + ∈ V
109rnex 7933 . . . 4 ran + ∈ V
11 prex 5436 . . . 4 {∅, (𝐹𝑀)} ∈ V
1210, 11unex 7765 . . 3 (ran + ∪ {∅, (𝐹𝑀)}) ∈ V
13 fveq2 6905 . . . . . . 7 (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
1413eleq1d 2825 . . . . . 6 (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
15 fveq2 6905 . . . . . . 7 (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
1615eleq1d 2825 . . . . . 6 (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
17 fveq2 6905 . . . . . . 7 (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
1817eleq1d 2825 . . . . . 6 (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
19 fveq2 6905 . . . . . . 7 (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
2019eleq1d 2825 . . . . . 6 (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
21 seq1 14056 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
22 ssun2 4178 . . . . . . . 8 {∅, (𝐹𝑀)} ⊆ (ran + ∪ {∅, (𝐹𝑀)})
23 fvex 6918 . . . . . . . . 9 (𝐹𝑀) ∈ V
2423prid2 4762 . . . . . . . 8 (𝐹𝑀) ∈ {∅, (𝐹𝑀)}
2522, 24sselii 3979 . . . . . . 7 (𝐹𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})
2621, 25eqeltrdi 2848 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
27 seqp1 14058 . . . . . . . . 9 (𝑧 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
2827adantr 480 . . . . . . . 8 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29 df-ov 7435 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30 snsspr1 4813 . . . . . . . . . . . 12 {∅} ⊆ {∅, (𝐹𝑀)}
31 unss2 4186 . . . . . . . . . . . 12 ({∅} ⊆ {∅, (𝐹𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
3230, 31ax-mp 5 . . . . . . . . . . 11 (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
33 fvrn0 6935 . . . . . . . . . . 11 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
3432, 33sselii 3979 . . . . . . . . . 10 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3529, 34eqeltri 2836 . . . . . . . . 9 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3635a1i 11 . . . . . . . 8 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
3728, 36eqeltrd 2840 . . . . . . 7 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
3837ex 412 . . . . . 6 (𝑧 ∈ (ℤ𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
3914, 16, 18, 20, 26, 38uzind4 12949 . . . . 5 (𝑥 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
4039rgen 3062 . . . 4 𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})
41 fnfvrnss 7140 . . . 4 ((seq𝑀( + , 𝐹) Fn (ℤ𝑀) ∧ ∀𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
423, 40, 41mp2an 692 . . 3 ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
4312, 42ssexi 5321 . 2 ran seq𝑀( + , 𝐹) ∈ V
44 funexw 7977 . 2 ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
455, 8, 43, 44mp3an 1462 1 seq𝑀( + , 𝐹) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cun 3948  wss 3950  c0 4332  {csn 4625  {cpr 4627  cop 4631  dom cdm 5684  ran crn 5685  Fun wfun 6554   Fn wfn 6555  cfv 6560  (class class class)co 7432  1c1 11157   + caddc 11159  cz 12615  cuz 12879  seqcseq 14043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-seq 14044
This theorem is referenced by:  mulgfval  19088
  Copyright terms: Public domain W3C validator