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Theorem smflimlem3 42618
 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 3981 . . . . . . . . 9 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
31, 2eqsstri 3926 . . . . . . . 8 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
4 inss1 4129 . . . . . . . . 9 (𝐷𝐼) ⊆ 𝐷
5 smflimlem3.x . . . . . . . . 9 (𝜑𝑋 ∈ (𝐷𝐼))
64, 5sseldi 3891 . . . . . . . 8 (𝜑𝑋𝐷)
73, 6sseldi 3891 . . . . . . 7 (𝜑𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
8 fveq2 6543 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝐹𝑖) = (𝐹𝑚))
98dmeqd 5665 . . . . . . . . . . . 12 (𝑖 = 𝑚 → dom (𝐹𝑖) = dom (𝐹𝑚))
10 eqcom 2802 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑚 = 𝑖)
1110imbi1i 351 . . . . . . . . . . . . 13 ((𝑖 = 𝑚 → dom (𝐹𝑖) = dom (𝐹𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹𝑖) = dom (𝐹𝑚)))
12 eqcom 2802 . . . . . . . . . . . . . 14 (dom (𝐹𝑖) = dom (𝐹𝑚) ↔ dom (𝐹𝑚) = dom (𝐹𝑖))
1312imbi2i 337 . . . . . . . . . . . . 13 ((𝑚 = 𝑖 → dom (𝐹𝑖) = dom (𝐹𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹𝑚) = dom (𝐹𝑖)))
1411, 13bitri 276 . . . . . . . . . . . 12 ((𝑖 = 𝑚 → dom (𝐹𝑖) = dom (𝐹𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹𝑚) = dom (𝐹𝑖)))
159, 14mpbi 231 . . . . . . . . . . 11 (𝑚 = 𝑖 → dom (𝐹𝑚) = dom (𝐹𝑖))
1615cbviinv 4871 . . . . . . . . . 10 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖)
1716a1i 11 . . . . . . . . 9 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖))
1817iuneq2i 4849 . . . . . . . 8 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑛𝑍 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖)
19 fveq2 6543 . . . . . . . . . 10 (𝑛 = 𝑚 → (ℤ𝑛) = (ℤ𝑚))
2019iineq1d 40921 . . . . . . . . 9 (𝑛 = 𝑚 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖) = 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖))
2120cbviunv 4870 . . . . . . . 8 𝑛𝑍 𝑖 ∈ (ℤ𝑛)dom (𝐹𝑖) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖)
2218, 21eqtri 2819 . . . . . . 7 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖)
237, 22syl6eleq 2893 . . . . . 6 (𝜑𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖))
24 smflimlem3.z . . . . . . . 8 𝑍 = (ℤ𝑀)
25 eqid 2795 . . . . . . . 8 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖)
2624, 25allbutfi 41232 . . . . . . 7 (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖) ↔ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖))
2726biimpi 217 . . . . . 6 (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)dom (𝐹𝑖) → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖))
2823, 27syl 17 . . . . 5 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖))
295elin2d 4101 . . . . . . . 8 (𝜑𝑋𝐼)
30 smflimlem3.i . . . . . . . . 9 𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)
31 oveq1 7028 . . . . . . . . . . . . . . 15 (𝑚 = 𝑖 → (𝑚𝐻𝑘) = (𝑖𝐻𝑘))
3231cbviinv 4871 . . . . . . . . . . . . . 14 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘)
3332a1i 11 . . . . . . . . . . . . 13 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘))
3433iuneq2i 4849 . . . . . . . . . . . 12 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘)
3519iineq1d 40921 . . . . . . . . . . . . 13 (𝑛 = 𝑚 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘) = 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘))
3635cbviunv 4870 . . . . . . . . . . . 12 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
3734, 36eqtri 2819 . . . . . . . . . . 11 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
3837a1i 11 . . . . . . . . . 10 (𝑘 ∈ ℕ → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘))
3938iineq2i 4850 . . . . . . . . 9 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘) = 𝑘 ∈ ℕ 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
4030, 39eqtri 2819 . . . . . . . 8 𝐼 = 𝑘 ∈ ℕ 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘)
4129, 40syl6eleq 2893 . . . . . . 7 (𝜑𝑋 𝑘 ∈ ℕ 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘))
42 smflimlem3.k . . . . . . 7 (𝜑𝐾 ∈ ℕ)
43 oveq2 7029 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑖𝐻𝑘) = (𝑖𝐻𝐾))
4443adantr 481 . . . . . . . . . 10 ((𝑘 = 𝐾𝑖 ∈ (ℤ𝑚)) → (𝑖𝐻𝑘) = (𝑖𝐻𝐾))
4544iineq2dv 4853 . . . . . . . . 9 (𝑘 = 𝐾 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘) = 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾))
4645iuneq2d 4857 . . . . . . . 8 (𝑘 = 𝐾 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾))
4746eleq2d 2868 . . . . . . 7 (𝑘 = 𝐾 → (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝑘) ↔ 𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾)))
4841, 42, 47eliind 40898 . . . . . 6 (𝜑𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾))
49 eqid 2795 . . . . . . 7 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾) = 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾)
5024, 49allbutfi 41232 . . . . . 6 (𝑋 𝑚𝑍 𝑖 ∈ (ℤ𝑚)(𝑖𝐻𝐾) ↔ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾))
5148, 50sylib 219 . . . . 5 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾))
5228, 51jca 512 . . . 4 (𝜑 → (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖) ∧ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾)))
5324rexanuz2 14548 . . . 4 (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) ↔ (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ dom (𝐹𝑖) ∧ ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)𝑋 ∈ (𝑖𝐻𝐾)))
5452, 53sylibr 235 . . 3 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)))
55 simpll 763 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → 𝜑)
56 simpr 485 . . . . . . 7 ((𝜑𝑚𝑍) → 𝑚𝑍)
5724uztrn2 12116 . . . . . . 7 ((𝑚𝑍𝑖 ∈ (ℤ𝑚)) → 𝑖𝑍)
5856, 57sylan 580 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → 𝑖𝑍)
59 simprl 767 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ dom (𝐹𝑖))
60 simp3 1131 . . . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝑚𝑃𝑘) = (𝑖𝑃𝐾))
6463fveq2d 6547 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑖𝑃𝐾)))
6564adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ (𝑚 = 𝑖𝑘 = 𝐾)) → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑖𝑃𝐾)))
66 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝑖𝑍)
6742adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝐾 ∈ ℕ)
68 fvexd 6558 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ V)
6962, 65, 66, 67, 68ovmpod 7163 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (𝑖𝐻𝐾) = (𝐶‘(𝑖𝑃𝐾)))
70693adant3 1125 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍𝑋 ∈ (𝑖𝐻𝐾)) → (𝑖𝐻𝐾) = (𝐶‘(𝑖𝑃𝐾)))
7160, 70eleqtrd 2885 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
72713expa 1111 . . . . . . . . . . . . 13 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
7372adantrl 712 . . . . . . . . . . . 12 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
7473, 59elind 4096 . . . . . . . . . . 11 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)))
75 eqid 2795 . . . . . . . . . . . . . . . . . . . . . . . . 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 5132 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
7877ralrimivw 3150 . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚𝑍) → ∀𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
8180ralrimiva 3149 . . . . . . . . . . . . . . . . . . . 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 7628 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → 𝑃 Fn (𝑍 × ℕ))
86 fnovrn 7184 . . . . . . . . . . . . . . . . . 18 ((𝑃 Fn (𝑍 × ℕ) ∧ 𝑖𝑍𝐾 ∈ ℕ) → (𝑖𝑃𝐾) ∈ ran 𝑃)
8785, 66, 67, 86syl3anc 1364 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → (𝑖𝑃𝐾) ∈ ran 𝑃)
88 ovex 7053 . . . . . . . . . . . . . . . . . 18 (𝑖𝑃𝐾) ∈ V
89 eleq1 2870 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑖𝑃𝐾) → (𝑦 ∈ ran 𝑃 ↔ (𝑖𝑃𝐾) ∈ ran 𝑃))
9089anbi2d 628 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑖𝑃𝐾) → ((𝜑𝑦 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃)))
91 fveq2 6543 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑖𝑃𝐾) → (𝐶𝑦) = (𝐶‘(𝑖𝑃𝐾)))
92 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑖𝑃𝐾) → 𝑦 = (𝑖𝑃𝐾))
9391, 92eleq12d 2877 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑖𝑃𝐾) → ((𝐶𝑦) ∈ 𝑦 ↔ (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾)))
9490, 93imbi12d 346 . . . . . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))
9787, 96syldan 591 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 = 𝑖𝑘 = 𝐾) → dom (𝐹𝑚) = dom (𝐹𝑖))
1008fveq1d 6545 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑚 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥))
10110imbi1i 351 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 = 𝑚 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)))
102 eqcom 2802 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥) ↔ ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥))
103102imbi2i 337 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 = 𝑖 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥)))
104101, 103bitri 276 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥)))
105100, 104mpbi 231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑖 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥))
106105adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑖𝑘 = 𝐾) → ((𝐹𝑚)‘𝑥) = ((𝐹𝑖)‘𝑥))
107 oveq2 7029 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝐾 → (1 / 𝑘) = (1 / 𝐾))
108107oveq2d 7037 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝐾 → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝐾)))
109108adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝐾)))
110106, 109breq12d 4979 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 = 𝑖𝑘 = 𝐾) → (((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))))
11199, 110rabeqbidv 3430 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖𝑘 = 𝐾) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
11215ineq2d 4113 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑖 → (𝑠 ∩ dom (𝐹𝑚)) = (𝑠 ∩ dom (𝐹𝑖)))
113112adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖𝑘 = 𝐾) → (𝑠 ∩ dom (𝐹𝑚)) = (𝑠 ∩ dom (𝐹𝑖)))
114111, 113eqeq12d 2810 . . . . . . . . . . . . . . . . . . 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 3425 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . . 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 5132 . . . . . . . . . . . . . . . . . 18 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))} ∈ V)
119118adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))} ∈ V)
12098, 116, 66, 67, 119ovmpod 7163 . . . . . . . . . . . . . . . 16 ((𝜑𝑖𝑍) → (𝑖𝑃𝐾) = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))})
12197, 120eleqtrd 2885 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹𝑖))})
122 ineq1 4105 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐶‘(𝑖𝑃𝐾)) → (𝑠 ∩ dom (𝐹𝑖)) = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)))
123122eqeq2d 2805 . . . . . . . . . . . . . . . 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 3619 . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → ((𝐶‘(𝑖𝑃𝐾)) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖))))
126125simprd 496 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)))
127126eqcomd 2801 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)) = {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
128127adantr 481 . . . . . . . . . . 11 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹𝑖)) = {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
12974, 128eleqtrd 2885 . . . . . . . . . 10 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
130 fveq2 6543 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝐹𝑖)‘𝑥) = ((𝐹𝑖)‘𝑋))
131 eqidd 2796 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐴 + (1 / 𝐾)) = (𝐴 + (1 / 𝐾)))
132130, 131breq12d 4979 . . . . . . . . . . 11 (𝑥 = 𝑋 → (((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾)) ↔ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
133132elrab 3619 . . . . . . . . . 10 (𝑋 ∈ {𝑥 ∈ dom (𝐹𝑖) ∣ ((𝐹𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} ↔ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
134129, 133sylib 219 . . . . . . . . 9 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
135134simprd 496 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))
13659, 135jca 512 . . . . . . 7 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
137136ex 413 . . . . . 6 ((𝜑𝑖𝑍) → ((𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
13855, 58, 137syl2anc 584 . . . . 5 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → ((𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
139138ralimdva 3144 . . . 4 ((𝜑𝑚𝑍) → (∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → ∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
140139reximdva 3237 . . 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 767 . . . . . . 7 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → 𝑋 ∈ dom (𝐹𝑖))
143 nfv 1892 . . . . . . . . . . . 12 𝑚((𝜑𝑖𝑍) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)
144 eleq1 2870 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝑚𝑍𝑖𝑍))
145144anbi2d 628 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝜑𝑚𝑍) ↔ (𝜑𝑖𝑍)))
146 fveq2 6543 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝐹𝑚) = (𝐹𝑖))
147146, 15feq12d 6375 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝐹𝑚):dom (𝐹𝑚)⟶ℝ ↔ (𝐹𝑖):dom (𝐹𝑖)⟶ℝ))
148145, 147imbi12d 346 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ) ↔ ((𝜑𝑖𝑍) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)))
14976adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
150 smflimlem3.m . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
151 eqid 2795 . . . . . . . . . . . . 13 dom (𝐹𝑚) = dom (𝐹𝑚)
152149, 150, 151smff 42578 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
153143, 148, 152chvar 2369 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)
154153adantr 481 . . . . . . . . . 10 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ dom (𝐹𝑖)) → (𝐹𝑖):dom (𝐹𝑖)⟶ℝ)
155 simpr 485 . . . . . . . . . 10 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ dom (𝐹𝑖)) → 𝑋 ∈ dom (𝐹𝑖))
156154, 155ffvelrnd 6722 . . . . . . . . 9 (((𝜑𝑖𝑍) ∧ 𝑋 ∈ dom (𝐹𝑖)) → ((𝐹𝑖)‘𝑋) ∈ ℝ)
157156adantrr 713 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹𝑖)‘𝑋) ∈ ℝ)
158 smflimlem3.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ)
15942nnrecred 11541 . . . . . . . . . 10 (𝜑 → (1 / 𝐾) ∈ ℝ)
160158, 159readdcld 10521 . . . . . . . . 9 (𝜑 → (𝐴 + (1 / 𝐾)) ∈ ℝ)
161160ad2antrr 722 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + (1 / 𝐾)) ∈ ℝ)
162 smflimlem3.y . . . . . . . . . . 11 (𝜑𝑌 ∈ ℝ+)
163162rpred 12286 . . . . . . . . . 10 (𝜑𝑌 ∈ ℝ)
164158, 163readdcld 10521 . . . . . . . . 9 (𝜑 → (𝐴 + 𝑌) ∈ ℝ)
165164ad2antrr 722 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + 𝑌) ∈ ℝ)
166 simprr 769 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))
167 smflimlem3.l . . . . . . . . . 10 (𝜑 → (1 / 𝐾) < 𝑌)
168159, 163, 158, 167ltadd2dd 10651 . . . . . . . . 9 (𝜑 → (𝐴 + (1 / 𝐾)) < (𝐴 + 𝑌))
169168ad2antrr 722 . . . . . . . 8 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + (1 / 𝐾)) < (𝐴 + 𝑌))
170157, 161, 165, 166, 169lttrd 10653 . . . . . . 7 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))
171142, 170jca 512 . . . . . 6 (((𝜑𝑖𝑍) ∧ (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))
172171ex 413 . . . . 5 ((𝜑𝑖𝑍) → ((𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
17355, 58, 172syl2anc 584 . . . 4 (((𝜑𝑚𝑍) ∧ 𝑖 ∈ (ℤ𝑚)) → ((𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → (𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
174173ralimdva 3144 . . 3 ((𝜑𝑚𝑍) → (∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → ∀𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
175174reximdva 3237 . 2 (𝜑 → (∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌))))
176141, 175mpd 15 1 (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  {crab 3109  Vcvv 3437   ∩ cin 3862  ∪ ciun 4829  ∩ ciin 4830   class class class wbr 4966   ↦ cmpt 5045   × cxp 5446  dom cdm 5448  ran crn 5449   Fn wfn 6225  ⟶wf 6226  ‘cfv 6230  (class class class)co 7021   ∈ cmpo 7023  ℝcr 10387  1c1 10389   + caddc 10391   < clt 10526   / cdiv 11150  ℕcn 11491  ℤ≥cuz 12098  ℝ+crp 12244   ⇝ cli 14680  SAlgcsalg 42162  SMblFncsmblfn 42546 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-iin 4832  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-pm 8264  df-en 8363  df-dom 8364  df-sdom 8365  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-z 11835  df-uz 12099  df-rp 12245  df-ioo 12597  df-ico 12599  df-smblfn 42547 This theorem is referenced by:  smflimlem4  42619
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