Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimlem3 Structured version   Visualization version   GIF version

Theorem smflimlem3 43393
Description: The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
smflimlem3.z 𝑍 = (ℤ𝑀)
smflimlem3.s (𝜑𝑆 ∈ SAlg)
smflimlem3.m ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
smflimlem3.d 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem3.a (𝜑𝐴 ∈ ℝ)
smflimlem3.p 𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
smflimlem3.h 𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
smflimlem3.i 𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)
smflimlem3.c ((𝜑𝑦 ∈ ran 𝑃) → (𝐶𝑦) ∈ 𝑦)
smflimlem3.x (𝜑𝑋 ∈ (𝐷𝐼))
smflimlem3.k (𝜑𝐾 ∈ ℕ)
smflimlem3.y (𝜑𝑌 ∈ ℝ+)
smflimlem3.l (𝜑 → (1 / 𝐾) < 𝑌)
Assertion
Ref Expression
smflimlem3 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))
Distinct variable groups:   𝐴,𝑘,𝑚,𝑠,𝑥   𝐶,𝑘,𝑚,𝑠   𝑦,𝐶   𝑖,𝐹,𝑘,𝑚,𝑛,𝑥   𝐹,𝑠,𝑖   𝑖,𝐻,𝑘,𝑚,𝑛   𝑖,𝐾,𝑘,𝑚,𝑠,𝑥   𝑦,𝐾,𝑖   𝑚,𝑀   𝑃,𝑘,𝑚,𝑠   𝑦,𝑃   𝑆,𝑘,𝑚,𝑠   𝑖,𝑋,𝑘,𝑚,𝑥   𝑖,𝑍,𝑘,𝑚,𝑛,𝑥   𝜑,𝑖,𝑘,𝑚   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑛,𝑠)   𝐴(𝑦,𝑖,𝑛)   𝐶(𝑥,𝑖,𝑛)   𝐷(𝑥,𝑦,𝑖,𝑘,𝑚,𝑛,𝑠)   𝑃(𝑥,𝑖,𝑛)   𝑆(𝑥,𝑦,𝑖,𝑛)   𝐹(𝑦)   𝐻(𝑥,𝑦,𝑠)   𝐼(𝑥,𝑦,𝑖,𝑘,𝑚,𝑛,𝑠)   𝐾(𝑛)   𝑀(𝑥,𝑦,𝑖,𝑘,𝑛,𝑠)   𝑋(𝑦,𝑛,𝑠)   𝑌(𝑥,𝑦,𝑖,𝑘,𝑚,𝑛,𝑠)   𝑍(𝑦,𝑠)

Proof of Theorem smflimlem3
StepHypRef Expression
1 smflimlem3.d . . . . . . . . 9 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
2 ssrab2 4010 . . . . . . . . 9 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
31, 2eqsstri 3952 . . . . . . . 8 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
4 inss1 4158 . . . . . . . . 9 (𝐷𝐼) ⊆ 𝐷
5 smflimlem3.x . . . . . . . . 9 (𝜑𝑋 ∈ (𝐷𝐼))
64, 5sseldi 3916 . . . . . . . 8 (𝜑𝑋𝐷)
73, 6sseldi 3916 . . . . . . 7 (𝜑𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
8 fveq2 6649 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝐹𝑖) = (𝐹𝑚))
98dmeqd 5742 . . . . . . . . . . . 12 (𝑖 = 𝑚 → dom (𝐹𝑖) = dom (𝐹𝑚))
10 eqcom 2808 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑚 = 𝑖)
1110imbi1i 353 . . . . . . . . . . . . 13 ((𝑖 = 𝑚 → dom (𝐹𝑖) = dom (𝐹𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹𝑖) = dom (𝐹𝑚)))
12 eqcom 2808 . . . . . . . . . . . . . 14 (dom (𝐹𝑖) = dom (𝐹𝑚) ↔ dom (𝐹𝑚) = dom (𝐹𝑖))
1312imbi2i 339 . . . . . . . . . . . . 13 ((𝑚 = 𝑖 → dom (𝐹𝑖) = dom (𝐹𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹𝑚) = dom (𝐹𝑖)))
1411, 13bitri 278 . . . . . . . . . . . 12 ((𝑖 = 𝑚 → dom (𝐹𝑖) = dom (𝐹𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹𝑚) = dom (𝐹𝑖)))
159, 14mpbi 233 . . . . . . . . . . 11 (𝑚 = 𝑖 → dom (𝐹𝑚) = dom (𝐹𝑖))
1615cbviinv 4931 . . . . . . . . . 10 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖)
1716a1i 11 . . . . . . . . 9 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖))
1817iuneq2i 4905 . . . . . . . 8 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑛𝑍 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖)
19 fveq2 6649 . . . . . . . . . 10 (𝑛 = 𝑚 → (ℤ𝑛) = (ℤ𝑚))
2019iineq1d 41713 . . . . . . . . 9 (𝑛 = 𝑚 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖) = 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖))
2120cbviunv 4930 . . . . . . . 8 𝑛𝑍 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖)
2218, 21eqtri 2824 . . . . . . 7 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖)
237, 22eleqtrdi 2903 . . . . . 6 (𝜑𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖))
24 smflimlem3.z . . . . . . . 8 𝑍 = (ℤ𝑀)
25 eqid 2801 . . . . . . . 8 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖)
2624, 25allbutfi 42016 . . . . . . 7 (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖) ↔ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖))
2726biimpi 219 . . . . . 6 (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖) → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖))
2823, 27syl 17 . . . . 5 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖))
295elin2d 4129 . . . . . . . 8 (𝜑𝑋𝐼)
30 smflimlem3.i . . . . . . . . 9 𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)
31 oveq1 7146 . . . . . . . . . . . . . . 15 (𝑚 = 𝑖 → (𝑚𝐻𝑘) = (𝑖𝐻𝑘))
3231cbviinv 4931 . . . . . . . . . . . . . 14 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘)
3332a1i 11 . . . . . . . . . . . . 13 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘))
3433iuneq2i 4905 . . . . . . . . . . . 12 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘)
3519iineq1d 41713 . . . . . . . . . . . . 13 (𝑛 = 𝑚 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘) = 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘))
3635cbviunv 4930 . . . . . . . . . . . 12 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
3734, 36eqtri 2824 . . . . . . . . . . 11 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
3837a1i 11 . . . . . . . . . 10 (𝑘 ∈ ℕ → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘))
3938iineq2i 4906 . . . . . . . . 9 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑘 ∈ ℕ 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
4030, 39eqtri 2824 . . . . . . . 8 𝐼 = 𝑘 ∈ ℕ 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
4129, 40eleqtrdi 2903 . . . . . . 7 (𝜑𝑋 𝑘 ∈ ℕ 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘))
42 smflimlem3.k . . . . . . 7 (𝜑𝐾 ∈ ℕ)
43 oveq2 7147 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑖𝐻𝑘) = (𝑖𝐻𝐾))
4443adantr 484 . . . . . . . . . 10 ((𝑘 = 𝐾𝑖 ∈ (ℤ𝑚)) → (𝑖𝐻𝑘) = (𝑖𝐻𝐾))
4544iineq2dv 4909 . . . . . . . . 9 (𝑘 = 𝐾 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘) = 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾))
4645iuneq2d 4913 . . . . . . . 8 (𝑘 = 𝐾 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾))
4746eleq2d 2878 . . . . . . 7 (𝑘 = 𝐾 → (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘) ↔ 𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾)))
4841, 42, 47eliind 41692 . . . . . 6 (𝜑𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾))
49 eqid 2801 . . . . . . 7 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾)
5024, 49allbutfi 42016 . . . . . 6 (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾) ↔ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾))
5148, 50sylib 221 . . . . 5 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾))
5228, 51jca 515 . . . 4 (𝜑 → (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖) ∧ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾)))
5324rexanuz2 14704 . . . 4 (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) ↔ (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖) ∧ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾)))
5452, 53sylibr 237 . . 3 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)))
55 simpll 766 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → 𝜑)
56 simpr 488 . . . . . . 7 ((𝜑𝑚𝑍) → 𝑚𝑍)
5724uztrn2 12254 . . . . . . 7 ((𝑚𝑍𝑖 ∈ (ℤ𝑚)) → 𝑖𝑍)
5856, 57sylan 583 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → 𝑖𝑍)
59 simprl 770 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ dom (𝐹𝑖))
60 simp3 1135 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝑖𝐻𝐾))
61 smflimlem3.h . . . . . . . . . . . . . . . . . 18 𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
6261a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))))
63 oveq12 7148 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝑚𝑃𝑘) = (𝑖𝑃𝐾))
6463fveq2d 6653 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑖𝑃𝐾)))
6564adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ (𝑚 = 𝑖𝑘 = 𝐾)) → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑖𝑃𝐾)))
66 simpr 488 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝑖𝑍)
6742adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝐾 ∈ ℕ)
68 fvexd 6664 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ V)
6962, 65, 66, 67, 68ovmpod 7285 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (𝑖𝐻𝐾) = (𝐶‘(𝑖𝑃𝐾)))
70693adant3 1129 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑋 ∈ (𝑖𝐻𝐾)) → (𝑖𝐻𝐾) = (𝐶‘(𝑖𝑃𝐾)))
7160, 70eleqtrd 2895 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
72713expa 1115 . . . . . . . . . . . . 13 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
7372adantrl 715 . . . . . . . . . . . 12 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
7473, 59elind 4124 . . . . . . . . . . 11 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)))
75 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
76 smflimlem3.s . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑆 ∈ SAlg)
7775, 76rabexd 5203 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
7877ralrimivw 3153 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
7978a1d 25 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑚𝑍 → ∀𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V))
8079imp 410 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚𝑍) → ∀𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
8180ralrimiva 3152 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑚𝑍𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
82 smflimlem3.p . . . . . . . . . . . . . . . . . . . . 21 𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
8382fnmpo 7753 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚𝑍𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V → 𝑃 Fn (𝑍 × ℕ))
8481, 83syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃 Fn (𝑍 × ℕ))
8584adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → 𝑃 Fn (𝑍 × ℕ))
86 fnovrn 7307 . . . . . . . . . . . . . . . . . 18 ((𝑃 Fn (𝑍 × ℕ) ∧ 𝑖𝑍𝐾 ∈ ℕ) → (𝑖𝑃𝐾) ∈ ran 𝑃)
8785, 66, 67, 86syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (𝑖𝑃𝐾) ∈ ran 𝑃)
88 ovex 7172 . . . . . . . . . . . . . . . . . 18 (𝑖𝑃𝐾) ∈ V
89 eleq1 2880 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑖𝑃𝐾) → (𝑦 ∈ ran 𝑃 ↔ (𝑖𝑃𝐾) ∈ ran 𝑃))
9089anbi2d 631 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑖𝑃𝐾) → ((𝜑𝑦 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃)))
91 fveq2 6649 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑖𝑃𝐾) → (𝐶𝑦) = (𝐶‘(𝑖𝑃𝐾)))
92 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑖𝑃𝐾) → 𝑦 = (𝑖𝑃𝐾))
9391, 92eleq12d 2887 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑖𝑃𝐾) → ((𝐶𝑦) ∈ 𝑦 ↔ (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾)))
9490, 93imbi12d 348 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑖𝑃𝐾) → (((𝜑𝑦 ∈ ran 𝑃) → (𝐶𝑦) ∈ 𝑦) ↔ ((𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))))
95 smflimlem3.c . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ ran 𝑃) → (𝐶𝑦) ∈ 𝑦)
9688, 94, 95vtocl 3510 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))
9787, 96syldan 594 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))
9882a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}))
9915adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 = 𝑖𝑘 = 𝐾) → dom (𝐹𝑚) = dom (𝐹𝑖))
1008fveq1d 6651 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑚 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥))
10110imbi1i 353 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 = 𝑚 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)))
102 eqcom 2808 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥) ↔ ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥))
103102imbi2i 339 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 = 𝑖 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥)))
104101, 103bitri 278 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥)))
105100, 104mpbi 233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑖 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥))
106105adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑖𝑘 = 𝐾) → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥))
107 oveq2 7147 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝐾 → (1 / 𝑘) = (1 / 𝐾))
108107oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝐾 → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝐾)))
109108adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝐾)))
110106, 109breq12d 5046 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 = 𝑖𝑘 = 𝐾) → (((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))))
11199, 110rabeqbidv 3436 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖𝑘 = 𝐾) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
11215ineq2d 4142 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑖 → (𝑠 ∩ dom (𝐹𝑚)) = (𝑠 ∩ dom (𝐹𝑖)))
113112adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝑠 ∩ dom (𝐹𝑚)) = (𝑠 ∩ dom (𝐹𝑖)))
114111, 113eqeq12d 2817 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑖𝑘 = 𝐾) → ({𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)) ↔ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))))
115114rabbidv 3430 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑖𝑘 = 𝐾) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))})
116115adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ (𝑚 = 𝑖𝑘 = 𝐾)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))})
117 eqid 2801 . . . . . . . . . . . . . . . . . . 19 {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))}
118117, 76rabexd 5203 . . . . . . . . . . . . . . . . . 18 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))} ∈ V)
119118adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))} ∈ V)
12098, 116, 66, 67, 119ovmpod 7285 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (𝑖𝑃𝐾) = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))})
12197, 120eleqtrd 2895 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))})
122 ineq1 4134 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐶‘(𝑖𝑃𝐾)) → (𝑠 ∩ dom (𝐹𝑖)) = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)))
123122eqeq2d 2812 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐶‘(𝑖𝑃𝐾)) → ({𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖)) ↔ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖))))
124123elrab 3631 . . . . . . . . . . . . . . 15 ((𝐶‘(𝑖𝑃𝐾)) ∈ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))} ↔ ((𝐶‘(𝑖𝑃𝐾)) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖))))
125121, 124sylib 221 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ((𝐶‘(𝑖𝑃𝐾)) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖))))
126125simprd 499 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)))
127126eqcomd 2807 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)) = {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
128127adantr 484 . . . . . . . . . . 11 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)) = {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
12974, 128eleqtrd 2895 . . . . . . . . . 10 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
130 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑖)‘𝑋))
131 eqidd 2802 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐴 + (1 / 𝐾)) = (𝐴 + (1 / 𝐾)))
132130, 131breq12d 5046 . . . . . . . . . . 11 (𝑥 = 𝑋 → (((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾)) ↔ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
133132elrab 3631 . . . . . . . . . 10 (𝑋 ∈ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} ↔ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
134129, 133sylib 221 . . . . . . . . 9 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
135134simprd 499 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))
13659, 135jca 515 . . . . . . 7 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
137136ex 416 . . . . . 6 ((𝜑𝑖𝑍) → ((𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
13855, 58, 137syl2anc 587 . . . . 5 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → ((𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
139138ralimdva 3147 . . . 4 ((𝜑𝑚𝑍) → (∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → ∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
140139reximdva 3236 . . 3 (𝜑 → (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
14154, 140mpd 15 . 2 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
142 simprl 770 . . . . . . 7 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → 𝑋 ∈ dom (𝐹𝑖))
143 eleq1 2880 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝑚𝑍𝑖𝑍))
144143anbi2d 631 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝜑𝑚𝑍) ↔ (𝜑𝑖𝑍)))
145 fveq2 6649 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝐹𝑚) = (𝐹𝑖))
146145, 15feq12d 6479 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝐹𝑚):dom (𝐹𝑚)⟶ℝ ↔ (𝐹𝑖):dom (𝐹𝑖)⟶ℝ))
147144, 146imbi12d 348 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ) ↔ ((𝜑𝑖𝑍) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)))
14876adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
149 smflimlem3.m . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
150 eqid 2801 . . . . . . . . . . . . 13 dom (𝐹𝑚) = dom (𝐹𝑚)
151148, 149, 150smff 43353 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
152147, 151chvarvv 2005 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)
153152adantr 484 . . . . . . . . . 10 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ dom (𝐹𝑖)) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)
154 simpr 488 . . . . . . . . . 10 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ dom (𝐹𝑖)) → 𝑋 ∈ dom (𝐹𝑖))
155153, 154ffvelrnd 6833 . . . . . . . . 9 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ dom (𝐹𝑖)) → ((𝐹𝑖)‘𝑋) ∈ ℝ)
156155adantrr 716 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹𝑖)‘𝑋) ∈ ℝ)
157 smflimlem3.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ)
15842nnrecred 11680 . . . . . . . . . 10 (𝜑 → (1 / 𝐾) ∈ ℝ)
159157, 158readdcld 10663 . . . . . . . . 9 (𝜑 → (𝐴 + (1 / 𝐾)) ∈ ℝ)
160159ad2antrr 725 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + (1 / 𝐾)) ∈ ℝ)
161 smflimlem3.y . . . . . . . . . . 11 (𝜑𝑌 ∈ ℝ+)
162161rpred 12423 . . . . . . . . . 10 (𝜑𝑌 ∈ ℝ)
163157, 162readdcld 10663 . . . . . . . . 9 (𝜑 → (𝐴 + 𝑌) ∈ ℝ)
164163ad2antrr 725 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + 𝑌) ∈ ℝ)
165 simprr 772 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))
166 smflimlem3.l . . . . . . . . . 10 (𝜑 → (1 / 𝐾) < 𝑌)
167158, 162, 157, 166ltadd2dd 10792 . . . . . . . . 9 (𝜑 → (𝐴 + (1 / 𝐾)) < (𝐴 + 𝑌))
168167ad2antrr 725 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + (1 / 𝐾)) < (𝐴 + 𝑌))
169156, 160, 164, 165, 168lttrd 10794 . . . . . . 7 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))
170142, 169jca 515 . . . . . 6 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))
171170ex 416 . . . . 5 ((𝜑𝑖𝑍) → ((𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
17255, 58, 171syl2anc 587 . . . 4 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → ((𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
173172ralimdva 3147 . . 3 ((𝜑𝑚𝑍) → (∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → ∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
174173reximdva 3236 . 2 (𝜑 → (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
175141, 174mpd 15 1 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cin 3883   ciun 4884   ciin 4885   class class class wbr 5033  cmpt 5113   × cxp 5521  dom cdm 5523  ran crn 5524   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  cr 10529  1c1 10531   + caddc 10533   < clt 10668   / cdiv 11290  cn 11629  cuz 12235  +crp 12381  cli 14836  SAlgcsalg 42937  SMblFncsmblfn 43321
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-z 11974  df-uz 12236  df-rp 12382  df-ioo 12734  df-ico 12736  df-smblfn 43322
This theorem is referenced by:  smflimlem4  43394
  Copyright terms: Public domain W3C validator