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Theorem smflimsuplem1 42658
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 6545 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
32fveq1d 6547 . . . . . . . . . . 11 (𝑚 = 𝑗 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑗)‘𝑥))
43cbvmptv 5068 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
54rneqi 5696 . . . . . . . . 9 ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥))
65supeq1i 8764 . . . . . . . 8 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )
76mpteq2i 5059 . . . . . . 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 6545 . . . . . . 7 (𝑛 = 𝐾 → (𝐸𝑛) = (𝐸𝐾))
10 fveq2 6545 . . . . . . . . . 10 (𝑛 = 𝐾 → (ℤ𝑛) = (ℤ𝐾))
1110mpteq1d 5056 . . . . . . . . 9 (𝑛 = 𝐾 → (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1211rneqd 5697 . . . . . . . 8 (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)))
1312supeq1d 8763 . . . . . . 7 (𝑛 = 𝐾 → sup(ran (𝑗 ∈ (ℤ𝑛) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
149, 13mpteq12dv 5052 . . . . . 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 6558 . . . . . . 7 (𝐸𝐾) ∈ V
1817mptex 6859 . . . . . 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 6664 . . . 4 (𝜑 → (𝐻𝐾) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
2120dmeqd 5667 . . 3 (𝜑 → dom (𝐻𝐾) = dom (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )))
22 xrltso 12388 . . . . . 6 < Or ℝ*
2322supex 8780 . . . . 5 sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ V
24 eqid 2797 . . . . 5 (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝐾) ↦ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ))
2523, 24dmmpti 6367 . . . 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 5667 . . . . . . . . . 10 (𝑚 = 𝑗 → dom (𝐹𝑚) = dom (𝐹𝑗))
2928cbviinv 4873 . . . . . . . . 9 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗)
3029eleq2i 2876 . . . . . . . 8 (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗))
316eleq1i 2875 . . . . . . . 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 3425 . . . . . 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 40916 . . . . . . . 8 (𝑛 = 𝐾 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) = 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗))
3635eleq2d 2870 . . . . . . 7 (𝑛 = 𝐾 → (𝑥 𝑗 ∈ (ℤ𝑛)dom (𝐹𝑗) ↔ 𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗)))
3713eleq1d 2869 . . . . . . 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 3424 . . . . 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 2797 . . . . 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 41250 . . . . . . 7 (𝜑𝐾 ∈ ℤ)
44 uzid 12112 . . . . . . 7 (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ𝐾))
45 ne0i 4226 . . . . . . 7 (𝐾 ∈ (ℤ𝐾) → (ℤ𝐾) ≠ ∅)
4643, 44, 453syl 18 . . . . . 6 (𝜑 → (ℤ𝐾) ≠ ∅)
47 fvex 6558 . . . . . . . . 9 (𝐹𝑗) ∈ V
4847dmex 7479 . . . . . . . 8 dom (𝐹𝑗) ∈ V
4948rgenw 3119 . . . . . . 7 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V
5049a1i 11 . . . . . 6 (𝜑 → ∀𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5146, 50iinexd 40960 . . . . 5 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∈ V)
5241, 51rabexd 5134 . . . 4 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
5327, 40, 16, 52fvmptd3 6664 . . 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 3983 . . . 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 6545 . . . . 5 (𝑗 = 𝐾 → (𝐹𝑗) = (𝐹𝐾))
5958dmeqd 5667 . . . 4 (𝑗 = 𝐾 → dom (𝐹𝑗) = dom (𝐹𝐾))
60 ssid 3916 . . . . 5 dom (𝐹𝐾) ⊆ dom (𝐹𝐾)
6160a1i 11 . . . 4 (𝜑 → dom (𝐹𝐾) ⊆ dom (𝐹𝐾))
6257, 59, 61iinssd 40957 . . 3 (𝜑 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ⊆ dom (𝐹𝐾))
6356, 62sstrd 3905 . 2 (𝜑 → {𝑥 𝑗 ∈ (ℤ𝐾)dom (𝐹𝑗) ∣ sup(ran (𝑗 ∈ (ℤ𝐾) ↦ ((𝐹𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ dom (𝐹𝐾))
6454, 63eqsstrd 3932 1 (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1525  wcel 2083  wne 2986  wral 3107  {crab 3111  Vcvv 3440  wss 3865  c0 4217   ciin 4832  cmpt 5047  dom cdm 5450  ran crn 5451  cfv 6232  supcsup 8757  cr 10389  *cxr 10527   < clt 10528  cz 11835  cuz 12097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-pre-lttri 10464  ax-pre-lttrn 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-po 5369  df-so 5370  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-sup 8759  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-neg 10726  df-z 11836  df-uz 12098
This theorem is referenced by:  smflimsuplem4  42661
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