Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimsuplem1 Structured version   Visualization version   GIF version

Theorem smflimsuplem1 43466
 Description: If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem1.z 𝑍 = (ℤ𝑀)
smflimsuplem1.e 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem1.h 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem1.k (𝜑𝐾𝑍)
Assertion
Ref Expression
smflimsuplem1 (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))
Distinct variable groups:   𝑛,𝐸,𝑥   𝑚,𝐹,𝑛,𝑥   𝑛,𝐾,𝑥   𝑛,𝑍
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝐸(𝑚)   𝐻(𝑥,𝑚,𝑛)   𝐾(𝑚)   𝑀(𝑥,𝑚,𝑛)   𝑍(𝑥,𝑚)

Proof of Theorem smflimsuplem1
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 smflimsuplem1.h . . . . 5 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
2 fveq2 6645 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
32fveq1d 6647 . . . . . . . . . . 11 (𝑚 = 𝑗 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑗)‘𝑥))
43cbvmptv 5133 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
54rneqi 5771 . . . . . . . . 9 ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
65supeq1i 8897 . . . . . . . 8 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )
76mpteq2i 5122 . . . . . . 7 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
87a1i 11 . . . . . 6 (𝑛 = 𝐾 → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
9 fveq2 6645 . . . . . . 7 (𝑛 = 𝐾 → (𝐸𝑛) = (𝐸𝐾))
10 fveq2 6645 . . . . . . . . . 10 (𝑛 = 𝐾 → (ℤ𝑛) = (ℤ𝐾))
1110mpteq1d 5119 . . . . . . . . 9 (𝑛 = 𝐾 → (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1211rneqd 5772 . . . . . . . 8 (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1312supeq1d 8896 . . . . . . 7 (𝑛 = 𝐾 → sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
149, 13mpteq12dv 5115 . . . . . 6 (𝑛 = 𝐾 → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
158, 14eqtrd 2833 . . . . 5 (𝑛 = 𝐾 → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
16 smflimsuplem1.k . . . . 5 (𝜑𝐾𝑍)
17 fvex 6658 . . . . . . 7 (𝐸𝐾) ∈ V
1817mptex 6963 . . . . . 6 (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) ∈ V
1918a1i 11 . . . . 5 (𝜑 → (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) ∈ V)
201, 15, 16, 19fvmptd3 6768 . . . 4 (𝜑 → (𝐻𝐾) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
2120dmeqd 5738 . . 3 (𝜑 → dom (𝐻𝐾) = dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
22 xrltso 12524 . . . . . 6 < Or ℝ*
2322supex 8913 . . . . 5 sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ V
24 eqid 2798 . . . . 5 (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
2523, 24dmmpti 6464 . . . 4 dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝐸𝐾)
2625a1i 11 . . 3 (𝜑 → dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝐸𝐾))
27 smflimsuplem1.e . . . 4 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
282dmeqd 5738 . . . . . . . . . 10 (𝑚 = 𝑗 → dom (𝐹𝑚) = dom (𝐹𝑗))
2928cbviinv 4928 . . . . . . . . 9 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗)
3029eleq2i 2881 . . . . . . . 8 (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗))
316eleq1i 2880 . . . . . . . 8 (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)
3230, 31anbi12i 629 . . . . . . 7 ((𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ∧ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
3332rabbia2 3424 . . . . . 6 {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ}
3433a1i 11 . . . . 5 (𝑛 = 𝐾 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
3510iineq1d 41741 . . . . . . . 8 (𝑛 = 𝐾 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) = 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗))
3635eleq2d 2875 . . . . . . 7 (𝑛 = 𝐾 → (𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ↔ 𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗)))
3713eleq1d 2874 . . . . . . 7 (𝑛 = 𝐾 → (sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
3836, 37anbi12d 633 . . . . . 6 (𝑛 = 𝐾 → ((𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ∧ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∧ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)))
3938rabbidva2 3423 . . . . 5 (𝑛 = 𝐾 → {𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
4034, 39eqtrd 2833 . . . 4 (𝑛 = 𝐾 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
41 eqid 2798 . . . . 5 {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ}
42 smflimsuplem1.z . . . . . . . 8 𝑍 = (ℤ𝑀)
4342, 16eluzelz2d 42065 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
44 uzid 12248 . . . . . . 7 (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ𝐾))
45 ne0i 4250 . . . . . . 7 (𝐾 ∈ (ℤ𝐾) → (ℤ𝐾) ≠ ∅)
4643, 44, 453syl 18 . . . . . 6 (𝜑 → (ℤ𝐾) ≠ ∅)
47 fvex 6658 . . . . . . . . 9 (𝐹𝑗) ∈ V
4847dmex 7600 . . . . . . . 8 dom (𝐹𝑗) ∈ V
4948rgenw 3118 . . . . . . 7 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V
5049a1i 11 . . . . . 6 (𝜑 → ∀𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5146, 50iinexd 41784 . . . . 5 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5241, 51rabexd 5200 . . . 4 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
5327, 40, 16, 52fvmptd3 6768 . . 3 (𝜑 → (𝐸𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
5421, 26, 533eqtrd 2837 . 2 (𝜑 → dom (𝐻𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55 ssrab2 4007 . . . 4 {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗)
5655a1i 11 . . 3 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗))
5743, 44syl 17 . . . 4 (𝜑𝐾 ∈ (ℤ𝐾))
58 fveq2 6645 . . . . 5 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
5958dmeqd 5738 . . . 4 (𝑗 = 𝐾 → dom (𝐹𝑗) = dom (𝐹𝐾))
60 ssid 3937 . . . . 5 dom (𝐹𝐾) ⊆ dom (𝐹𝐾)
6160a1i 11 . . . 4 (𝜑 → dom (𝐹𝐾) ⊆ dom (𝐹𝐾))
6257, 59, 61iinssd 41781 . . 3 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ⊆ dom (𝐹𝐾))
6356, 62sstrd 3925 . 2 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ dom (𝐹𝐾))
6454, 63eqsstrd 3953 1 (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∩ ciin 4882   ↦ cmpt 5110  dom cdm 5519  ran crn 5520  ‘cfv 6324  supcsup 8890  ℝcr 10527  ℝ*cxr 10665   < clt 10666  ℤcz 11971  ℤ≥cuz 12233 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-pre-lttri 10602  ax-pre-lttrn 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-sup 8892  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-neg 10864  df-z 11972  df-uz 12234 This theorem is referenced by:  smflimsuplem4  43469
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