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Theorem smflimsuplem1 45836
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 6892 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
32fveq1d 6894 . . . . . . . . . . 11 (𝑚 = 𝑗 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑗)‘𝑥))
43cbvmptv 5262 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
54rneqi 5937 . . . . . . . . 9 ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
65supeq1i 9445 . . . . . . . 8 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )
76mpteq2i 5254 . . . . . . 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 6892 . . . . . . 7 (𝑛 = 𝐾 → (𝐸𝑛) = (𝐸𝐾))
10 fveq2 6892 . . . . . . . . . 10 (𝑛 = 𝐾 → (ℤ𝑛) = (ℤ𝐾))
1110mpteq1d 5244 . . . . . . . . 9 (𝑛 = 𝐾 → (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1211rneqd 5938 . . . . . . . 8 (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1312supeq1d 9444 . . . . . . 7 (𝑛 = 𝐾 → sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
149, 13mpteq12dv 5240 . . . . . 6 (𝑛 = 𝐾 → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
158, 14eqtrd 2771 . . . . 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 6905 . . . . . . 7 (𝐸𝐾) ∈ V
1817mptex 7228 . . . . . 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 7022 . . . 4 (𝜑 → (𝐻𝐾) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
2120dmeqd 5906 . . 3 (𝜑 → dom (𝐻𝐾) = dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
22 xrltso 13125 . . . . . 6 < Or ℝ*
2322supex 9461 . . . . 5 sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ V
24 eqid 2731 . . . . 5 (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
2523, 24dmmpti 6695 . . . 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 5906 . . . . . . . . . 10 (𝑚 = 𝑗 → dom (𝐹𝑚) = dom (𝐹𝑗))
2928cbviinv 5045 . . . . . . . . 9 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗)
3029eleq2i 2824 . . . . . . . 8 (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗))
316eleq1i 2823 . . . . . . . 8 (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)
3230, 31anbi12i 626 . . . . . . 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 3434 . . . . . 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 44082 . . . . . . . 8 (𝑛 = 𝐾 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) = 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗))
3635eleq2d 2818 . . . . . . 7 (𝑛 = 𝐾 → (𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ↔ 𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗)))
3713eleq1d 2817 . . . . . . 7 (𝑛 = 𝐾 → (sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
3836, 37anbi12d 630 . . . . . 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 3433 . . . . 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 2771 . . . 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 2731 . . . . 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 44423 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
44 uzid 12842 . . . . . . 7 (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ𝐾))
45 ne0i 4335 . . . . . . 7 (𝐾 ∈ (ℤ𝐾) → (ℤ𝐾) ≠ ∅)
4643, 44, 453syl 18 . . . . . 6 (𝜑 → (ℤ𝐾) ≠ ∅)
47 fvex 6905 . . . . . . . . 9 (𝐹𝑗) ∈ V
4847dmex 7905 . . . . . . . 8 dom (𝐹𝑗) ∈ V
4948rgenw 3064 . . . . . . 7 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V
5049a1i 11 . . . . . 6 (𝜑 → ∀𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5146, 50iinexd 44125 . . . . 5 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5241, 51rabexd 5334 . . . 4 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
5327, 40, 16, 52fvmptd3 7022 . . 3 (𝜑 → (𝐸𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
5421, 26, 533eqtrd 2775 . 2 (𝜑 → dom (𝐻𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55 ssrab2 4078 . . . 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 6892 . . . . 5 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
5958dmeqd 5906 . . . 4 (𝑗 = 𝐾 → dom (𝐹𝑗) = dom (𝐹𝐾))
60 ssid 4005 . . . . 5 dom (𝐹𝐾) ⊆ dom (𝐹𝐾)
6160a1i 11 . . . 4 (𝜑 → dom (𝐹𝐾) ⊆ dom (𝐹𝐾))
6257, 59, 61iinssd 44123 . . 3 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ⊆ dom (𝐹𝐾))
6356, 62sstrd 3993 . 2 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ dom (𝐹𝐾))
6454, 63eqsstrd 4021 1 (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wne 2939  wral 3060  {crab 3431  Vcvv 3473  wss 3949  c0 4323   ciin 4999  cmpt 5232  dom cdm 5677  ran crn 5678  cfv 6544  supcsup 9438  cr 11112  *cxr 11252   < clt 11253  cz 12563  cuz 12827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-pre-lttri 11187  ax-pre-lttrn 11188
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-neg 11452  df-z 12564  df-uz 12828
This theorem is referenced by:  smflimsuplem4  45839
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