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Theorem smflimsuplem1 41449
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 6304 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
32fveq1d 6306 . . . . . . . . . . 11 (𝑚 = 𝑗 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑗)‘𝑥))
43cbvmptv 4858 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
54rneqi 5459 . . . . . . . . 9 ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
65supeq1i 8469 . . . . . . . 8 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )
76mpteq2i 4849 . . . . . . 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 6304 . . . . . . 7 (𝑛 = 𝐾 → (𝐸𝑛) = (𝐸𝐾))
10 fveq2 6304 . . . . . . . . . 10 (𝑛 = 𝐾 → (ℤ𝑛) = (ℤ𝐾))
1110mpteq1d 4846 . . . . . . . . 9 (𝑛 = 𝐾 → (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1211rneqd 5460 . . . . . . . 8 (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1312supeq1d 8468 . . . . . . 7 (𝑛 = 𝐾 → sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
149, 13mpteq12dv 4841 . . . . . 6 (𝑛 = 𝐾 → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
158, 14eqtrd 2758 . . . . 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 6314 . . . . . . 7 (𝐸𝐾) ∈ V
1817mptex 6602 . . . . . 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 39863 . . . 4 (𝜑 → (𝐻𝐾) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
2120dmeqd 5433 . . 3 (𝜑 → dom (𝐻𝐾) = dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
22 xrltso 12088 . . . . . 6 < Or ℝ*
2322supex 8485 . . . . 5 sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ V
24 eqid 2724 . . . . 5 (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
2523, 24dmmpti 6136 . . . 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 5433 . . . . . . . . . 10 (𝑚 = 𝑗 → dom (𝐹𝑚) = dom (𝐹𝑗))
2928cbviinv 4668 . . . . . . . . 9 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗)
3029eleq2i 2795 . . . . . . . 8 (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗))
316eleq1i 2794 . . . . . . . 8 (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)
3230, 31anbi12i 735 . . . . . . 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 3291 . . . . . 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 39683 . . . . . . . 8 (𝑛 = 𝐾 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) = 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗))
3635eleq2d 2789 . . . . . . 7 (𝑛 = 𝐾 → (𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ↔ 𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗)))
3713eleq1d 2788 . . . . . . 7 (𝑛 = 𝐾 → (sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
3836, 37anbi12d 749 . . . . . 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 3290 . . . . 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 2758 . . . 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 2724 . . . . 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 40055 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
44 uzid 11815 . . . . . . 7 (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ𝐾))
45 ne0i 4029 . . . . . . 7 (𝐾 ∈ (ℤ𝐾) → (ℤ𝐾) ≠ ∅)
4643, 44, 453syl 18 . . . . . 6 (𝜑 → (ℤ𝐾) ≠ ∅)
47 fvex 6314 . . . . . . . . 9 (𝐹𝑗) ∈ V
4847dmex 7216 . . . . . . . 8 dom (𝐹𝑗) ∈ V
4948rgenw 3026 . . . . . . 7 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V
5049a1i 11 . . . . . 6 (𝜑 → ∀𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5146, 50iinexd 39734 . . . . 5 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5241, 51rabexd 4921 . . . 4 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
5327, 40, 16, 52fvmptd3 39863 . . 3 (𝜑 → (𝐸𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
5421, 26, 533eqtrd 2762 . 2 (𝜑 → dom (𝐻𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55 ssrab2 3793 . . . 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 6304 . . . . 5 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
5958dmeqd 5433 . . . 4 (𝑗 = 𝐾 → dom (𝐹𝑗) = dom (𝐹𝐾))
60 ssid 3730 . . . . 5 dom (𝐹𝐾) ⊆ dom (𝐹𝐾)
6160a1i 11 . . . 4 (𝜑 → dom (𝐹𝐾) ⊆ dom (𝐹𝐾))
6257, 59, 61iinssd 39730 . . 3 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ⊆ dom (𝐹𝐾))
6356, 62sstrd 3719 . 2 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ dom (𝐹𝐾))
6454, 63eqsstrd 3745 1 (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  wne 2896  wral 3014  {crab 3018  Vcvv 3304  wss 3680  c0 4023   ciin 4629  cmpt 4837  dom cdm 5218  ran crn 5219  cfv 6001  supcsup 8462  cr 10048  *cxr 10186   < clt 10187  cz 11490  cuz 11800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-pre-lttri 10123  ax-pre-lttrn 10124
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-po 5139  df-so 5140  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-sup 8464  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-neg 10382  df-z 11491  df-uz 11801
This theorem is referenced by:  smflimsuplem4  41452
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