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Theorem smflimsuplem1 46835
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 6906 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
32fveq1d 6908 . . . . . . . . . . 11 (𝑚 = 𝑗 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑗)‘𝑥))
43cbvmptv 5255 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
54rneqi 5948 . . . . . . . . 9 ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
65supeq1i 9487 . . . . . . . 8 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )
76mpteq2i 5247 . . . . . . 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 6906 . . . . . . 7 (𝑛 = 𝐾 → (𝐸𝑛) = (𝐸𝐾))
10 fveq2 6906 . . . . . . . . . 10 (𝑛 = 𝐾 → (ℤ𝑛) = (ℤ𝐾))
1110mpteq1d 5237 . . . . . . . . 9 (𝑛 = 𝐾 → (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1211rneqd 5949 . . . . . . . 8 (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1312supeq1d 9486 . . . . . . 7 (𝑛 = 𝐾 → sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
149, 13mpteq12dv 5233 . . . . . 6 (𝑛 = 𝐾 → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
158, 14eqtrd 2777 . . . . 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 6919 . . . . . . 7 (𝐸𝐾) ∈ V
1817mptex 7243 . . . . . 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 7039 . . . 4 (𝜑 → (𝐻𝐾) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
2120dmeqd 5916 . . 3 (𝜑 → dom (𝐻𝐾) = dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
22 xrltso 13183 . . . . . 6 < Or ℝ*
2322supex 9503 . . . . 5 sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ V
24 eqid 2737 . . . . 5 (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
2523, 24dmmpti 6712 . . . 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 5916 . . . . . . . . . 10 (𝑚 = 𝑗 → dom (𝐹𝑚) = dom (𝐹𝑗))
2928cbviinv 5041 . . . . . . . . 9 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗)
3029eleq2i 2833 . . . . . . . 8 (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗))
316eleq1i 2832 . . . . . . . 8 (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)
3230, 31anbi12i 628 . . . . . . 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 3439 . . . . . 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 45095 . . . . . . . 8 (𝑛 = 𝐾 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) = 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗))
3635eleq2d 2827 . . . . . . 7 (𝑛 = 𝐾 → (𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ↔ 𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗)))
3713eleq1d 2826 . . . . . . 7 (𝑛 = 𝐾 → (sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
3836, 37anbi12d 632 . . . . . 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 3438 . . . . 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 2777 . . . 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 2737 . . . . 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 45424 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
44 uzid 12893 . . . . . . 7 (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ𝐾))
45 ne0i 4341 . . . . . . 7 (𝐾 ∈ (ℤ𝐾) → (ℤ𝐾) ≠ ∅)
4643, 44, 453syl 18 . . . . . 6 (𝜑 → (ℤ𝐾) ≠ ∅)
47 fvex 6919 . . . . . . . . 9 (𝐹𝑗) ∈ V
4847dmex 7931 . . . . . . . 8 dom (𝐹𝑗) ∈ V
4948rgenw 3065 . . . . . . 7 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V
5049a1i 11 . . . . . 6 (𝜑 → ∀𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5146, 50iinexd 45138 . . . . 5 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5241, 51rabexd 5340 . . . 4 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
5327, 40, 16, 52fvmptd3 7039 . . 3 (𝜑 → (𝐸𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
5421, 26, 533eqtrd 2781 . 2 (𝜑 → dom (𝐻𝐾) = {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55 ssrab2 4080 . . . 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 6906 . . . . 5 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
5958dmeqd 5916 . . . 4 (𝑗 = 𝐾 → dom (𝐹𝑗) = dom (𝐹𝐾))
60 ssid 4006 . . . . 5 dom (𝐹𝐾) ⊆ dom (𝐹𝐾)
6160a1i 11 . . . 4 (𝜑 → dom (𝐹𝐾) ⊆ dom (𝐹𝐾))
6257, 59, 61iinssd 45136 . . 3 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ⊆ dom (𝐹𝐾))
6356, 62sstrd 3994 . 2 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ dom (𝐹𝐾))
6454, 63eqsstrd 4018 1 (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333   ciin 4992  cmpt 5225  dom cdm 5685  ran crn 5686  cfv 6561  supcsup 9480  cr 11154  *cxr 11294   < clt 11295  cz 12613  cuz 12878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879
This theorem is referenced by:  smflimsuplem4  46838
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