Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimlem6 Structured version   Visualization version   GIF version

Theorem smflimlem6 43803
 Description: Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
smflimlem6.1 (𝜑𝑀 ∈ ℤ)
smflimlem6.2 𝑍 = (ℤ𝑀)
smflimlem6.3 (𝜑𝑆 ∈ SAlg)
smflimlem6.4 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflimlem6.5 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem6.6 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
smflimlem6.7 (𝜑𝐴 ∈ ℝ)
smflimlem6.8 𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
Assertion
Ref Expression
smflimlem6 (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
Distinct variable groups:   𝐴,𝑘,𝑚,𝑛,𝑥   𝐴,𝑠,𝑘,𝑚,𝑥   𝐷,𝑘,𝑚,𝑛,𝑥   𝑘,𝐹,𝑚,𝑛,𝑥   𝐹,𝑠   𝑘,𝐺,𝑚,𝑛   𝑚,𝑀   𝑃,𝑘,𝑚,𝑛,𝑥   𝑃,𝑠   𝑆,𝑘,𝑚,𝑛   𝑆,𝑠   𝑘,𝑍,𝑚,𝑛,𝑥   𝑍,𝑠   𝜑,𝑘,𝑚,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑠)   𝐷(𝑠)   𝑆(𝑥)   𝐺(𝑥,𝑠)   𝑀(𝑥,𝑘,𝑛,𝑠)

Proof of Theorem smflimlem6
Dummy variables 𝑐 𝑟 𝑖 𝑗 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smflimlem6.2 . . . . . . . 8 𝑍 = (ℤ𝑀)
21fvexi 6676 . . . . . . 7 𝑍 ∈ V
3 nnex 11685 . . . . . . 7 ℕ ∈ V
42, 3xpex 7479 . . . . . 6 (𝑍 × ℕ) ∈ V
54a1i 11 . . . . 5 (𝜑 → (𝑍 × ℕ) ∈ V)
6 eqid 2758 . . . . . . . . 9 {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
7 smflimlem6.3 . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
86, 7rabexd 5206 . . . . . . . 8 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
98adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
109ralrimivva 3120 . . . . . 6 (𝜑 → ∀𝑚𝑍𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
11 smflimlem6.8 . . . . . . 7 𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
1211fnmpo 7776 . . . . . 6 (∀𝑚𝑍𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V → 𝑃 Fn (𝑍 × ℕ))
1310, 12syl 17 . . . . 5 (𝜑𝑃 Fn (𝑍 × ℕ))
14 fnrndomg 10001 . . . . 5 ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
155, 13, 14sylc 65 . . . 4 (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
161uzct 42098 . . . . . . 7 𝑍 ≼ ω
17 nnct 13403 . . . . . . 7 ℕ ≼ ω
1816, 17pm3.2i 474 . . . . . 6 (𝑍 ≼ ω ∧ ℕ ≼ ω)
19 xpct 9481 . . . . . 6 ((𝑍 ≼ ω ∧ ℕ ≼ ω) → (𝑍 × ℕ) ≼ ω)
2018, 19ax-mp 5 . . . . 5 (𝑍 × ℕ) ≼ ω
2120a1i 11 . . . 4 (𝜑 → (𝑍 × ℕ) ≼ ω)
22 domtr 8585 . . . 4 ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
2315, 21, 22syl2anc 587 . . 3 (𝜑 → ran 𝑃 ≼ ω)
24 vex 3413 . . . . . . 7 𝑦 ∈ V
2511elrnmpog 7286 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran 𝑃 ↔ ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}))
2624, 25ax-mp 5 . . . . . 6 (𝑦 ∈ ran 𝑃 ↔ ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2726biimpi 219 . . . . 5 (𝑦 ∈ ran 𝑃 → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2827adantl 485 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
29 simp3 1135 . . . . . . . 8 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
307adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
31 smflimlem6.4 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
3231ffvelrnda 6847 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
3332adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
34 eqid 2758 . . . . . . . . . . . 12 dom (𝐹𝑚) = dom (𝐹𝑚)
35 smflimlem6.7 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
3635adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
37 nnrecre 11721 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
3837adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
3936, 38readdcld 10713 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4039adantrl 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4130, 33, 34, 40smfpreimalt 43759 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)))
42 fvex 6675 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
4342dmex 7626 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
4443a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝐹𝑚) ∈ V)
45 elrest 16764 . . . . . . . . . . . . 13 ((𝑆 ∈ SAlg ∧ dom (𝐹𝑚) ∈ V) → ({𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))))
467, 44, 45syl2anc 587 . . . . . . . . . . . 12 (𝜑 → ({𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))))
4746adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → ({𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))))
4841, 47mpbid 235 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)))
49 rabn0 4284 . . . . . . . . . 10 ({𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅ ↔ ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)))
5048, 49sylibr 237 . . . . . . . . 9 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
51503adant3 1129 . . . . . . . 8 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
5229, 51eqnetrd 3018 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
53523exp 1116 . . . . . 6 (𝜑 → ((𝑚𝑍𝑘 ∈ ℕ) → (𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
5453rexlimdvv 3217 . . . . 5 (𝜑 → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5554adantr 484 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5628, 55mpd 15 . . 3 ((𝜑𝑦 ∈ ran 𝑃) → 𝑦 ≠ ∅)
5723, 56axccd2 42258 . 2 (𝜑 → ∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦)
58 smflimlem6.1 . . . . . 6 (𝜑𝑀 ∈ ℤ)
5958adantr 484 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
607adantr 484 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
6131adantr 484 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐹:𝑍⟶(SMblFn‘𝑆))
62 smflimlem6.5 . . . . 5 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
63 smflimlem6.6 . . . . 5 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
6435adantr 484 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
65 fvoveq1 7178 . . . . . 6 (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
66 oveq2 7163 . . . . . . 7 (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
6766fveq2d 6666 . . . . . 6 (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
6865, 67cbvmpov 7248 . . . . 5 (𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
69 nfcv 2919 . . . . . 6 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
70 nfcv 2919 . . . . . . 7 𝑗𝑍
71 nfcv 2919 . . . . . . . 8 𝑗(ℤ𝑛)
72 nfcv 2919 . . . . . . . . 9 𝑗𝑚
73 nfmpo2 7234 . . . . . . . . 9 𝑗(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
74 nfcv 2919 . . . . . . . . 9 𝑗𝑘
7572, 73, 74nfov 7185 . . . . . . . 8 𝑗(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7671, 75nfiin 4917 . . . . . . 7 𝑗 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7770, 76nfiun 4916 . . . . . 6 𝑗 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78 oveq2 7163 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
7978adantr 484 . . . . . . . . . 10 ((𝑗 = 𝑘𝑖 ∈ (ℤ𝑛)) → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8079iineq2dv 4911 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
81 oveq1 7162 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8281cbviinv 4933 . . . . . . . . . 10 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
8382a1i 11 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8480, 83eqtrd 2793 . . . . . . . 8 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8584adantr 484 . . . . . . 7 ((𝑗 = 𝑘𝑛𝑍) → 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8685iuneq2dv 4910 . . . . . 6 (𝑗 = 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8769, 77, 86cbviin 4929 . . . . 5 𝑗 ∈ ℕ 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
88 fveq2 6662 . . . . . . . 8 (𝑦 = 𝑟 → (𝑐𝑦) = (𝑐𝑟))
89 id 22 . . . . . . . 8 (𝑦 = 𝑟𝑦 = 𝑟)
9088, 89eleq12d 2846 . . . . . . 7 (𝑦 = 𝑟 → ((𝑐𝑦) ∈ 𝑦 ↔ (𝑐𝑟) ∈ 𝑟))
9190rspccva 3542 . . . . . 6 ((∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9291adantll 713 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9359, 1, 60, 61, 62, 63, 64, 11, 68, 87, 92smflimlem5 43802 . . . 4 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
9493ex 416 . . 3 (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9594exlimdv 1934 . 2 (𝜑 → (∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9657, 95mpd 15 1 (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ∩ cin 3859  ∅c0 4227  ∪ ciun 4886  ∩ ciin 4887   class class class wbr 5035   ↦ cmpt 5115   × cxp 5525  dom cdm 5527  ran crn 5528   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157  ωcom 7584   ≼ cdom 8530  ℝcr 10579  1c1 10581   + caddc 10583   < clt 10718   ≤ cle 10719   / cdiv 11340  ℕcn 11679  ℤcz 12025  ℤ≥cuz 12287   ⇝ cli 14894   ↾t crest 16757  SAlgcsalg 43344  SMblFncsmblfn 43728 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-inf2 9142  ax-cc 9900  ax-ac2 9928  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-oadd 8121  df-omul 8122  df-er 8304  df-map 8423  df-pm 8424  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-sup 8944  df-inf 8945  df-oi 9012  df-card 9406  df-acn 9409  df-ac 9581  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-3 11743  df-n0 11940  df-z 12026  df-uz 12288  df-q 12394  df-rp 12436  df-ioo 12788  df-ico 12790  df-fl 13216  df-seq 13424  df-exp 13485  df-cj 14511  df-re 14512  df-im 14513  df-sqrt 14647  df-abs 14648  df-clim 14898  df-rlim 14899  df-rest 16759  df-salg 43345  df-smblfn 43729 This theorem is referenced by:  smflim  43804
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