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Theorem smflimlem6 42614
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 6552 . . . . . . 7 𝑍 ∈ V
3 nnex 11492 . . . . . . 7 ℕ ∈ V
42, 3xpex 7333 . . . . . 6 (𝑍 × ℕ) ∈ V
54a1i 11 . . . . 5 (𝜑 → (𝑍 × ℕ) ∈ V)
6 eqid 2795 . . . . . . . . 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 5127 . . . . . . . 8 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
98adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
109ralrimivva 3158 . . . . . 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 7623 . . . . . 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 9804 . . . . 5 ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
155, 13, 14sylc 65 . . . 4 (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
161uzct 40883 . . . . . . 7 𝑍 ≼ ω
17 nnct 13199 . . . . . . 7 ℕ ≼ ω
1816, 17pm3.2i 471 . . . . . 6 (𝑍 ≼ ω ∧ ℕ ≼ ω)
19 xpct 9288 . . . . . 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 8410 . . . 4 ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
2315, 21, 22syl2anc 584 . . 3 (𝜑 → ran 𝑃 ≼ ω)
24 vex 3440 . . . . . . 7 𝑦 ∈ V
2511elrnmpog 7142 . . . . . . 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 217 . . . . 5 (𝑦 ∈ ran 𝑃 → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2827adantl 482 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
29 simp3 1131 . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
31 smflimlem6.4 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
3231ffvelrnda 6716 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
3332adantrr 713 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
34 eqid 2795 . . . . . . . . . . . 12 dom (𝐹𝑚) = dom (𝐹𝑚)
35 smflimlem6.7 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
3635adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
37 nnrecre 11527 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
3837adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
3936, 38readdcld 10516 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4039adantrl 712 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4130, 33, 34, 40smfpreimalt 42570 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)))
42 fvex 6551 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
4342dmex 7472 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
4443a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝐹𝑚) ∈ V)
45 elrest 16530 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 233 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)))
49 rabn0 4259 . . . . . . . . . 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 235 . . . . . . . . 9 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
51503adant3 1125 . . . . . . . 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 3051 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
53523exp 1112 . . . . . 6 (𝜑 → ((𝑚𝑍𝑘 ∈ ℕ) → (𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
5453rexlimdvv 3256 . . . . 5 (𝜑 → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5554adantr 481 . . . 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 41055 . 2 (𝜑 → ∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦)
58 smflimlem6.1 . . . . . 6 (𝜑𝑀 ∈ ℤ)
5958adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
607adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
6131adantr 481 . . . . 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 481 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
65 fvoveq1 7039 . . . . . 6 (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
66 oveq2 7024 . . . . . . 7 (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
6766fveq2d 6542 . . . . . 6 (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
6865, 67cbvmpov 7105 . . . . 5 (𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
69 nfcv 2949 . . . . . 6 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
70 nfcv 2949 . . . . . . 7 𝑗𝑍
71 nfcv 2949 . . . . . . . 8 𝑗(ℤ𝑛)
72 nfcv 2949 . . . . . . . . 9 𝑗𝑚
73 nfmpo2 7093 . . . . . . . . 9 𝑗(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
74 nfcv 2949 . . . . . . . . 9 𝑗𝑘
7572, 73, 74nfov 7046 . . . . . . . 8 𝑗(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7671, 75nfiin 4855 . . . . . . 7 𝑗 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7770, 76nfiun 4854 . . . . . 6 𝑗 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78 oveq2 7024 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
7978adantr 481 . . . . . . . . . 10 ((𝑗 = 𝑘𝑖 ∈ (ℤ𝑛)) → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8079iineq2dv 4849 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
81 oveq1 7023 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8281cbviinv 4867 . . . . . . . . . 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 2831 . . . . . . . 8 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8584adantr 481 . . . . . . 7 ((𝑗 = 𝑘𝑛𝑍) → 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8685iuneq2dv 4848 . . . . . 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 4865 . . . . 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 6538 . . . . . . . 8 (𝑦 = 𝑟 → (𝑐𝑦) = (𝑐𝑟))
89 id 22 . . . . . . . 8 (𝑦 = 𝑟𝑦 = 𝑟)
9088, 89eleq12d 2877 . . . . . . 7 (𝑦 = 𝑟 → ((𝑐𝑦) ∈ 𝑦 ↔ (𝑐𝑟) ∈ 𝑟))
9190rspccva 3558 . . . . . 6 ((∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9291adantll 710 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9359, 1, 60, 61, 62, 63, 64, 11, 68, 87, 92smflimlem5 42613 . . . 4 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
9493ex 413 . . 3 (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9594exlimdv 1911 . 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 207  wa 396  w3a 1080   = wceq 1522  wex 1761  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  cin 3858  c0 4211   ciun 4825   ciin 4826   class class class wbr 4962  cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  ωcom 7436  cdom 8355  cr 10382  1c1 10384   + caddc 10386   < clt 10521  cle 10522   / cdiv 11145  cn 11486  cz 11829  cuz 12093  cli 14675  t crest 16523  SAlgcsalg 42155  SMblFncsmblfn 42539
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-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cc 9703  ax-ac2 9731  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
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 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-omul 7958  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-acn 9217  df-ac 9388  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-ioo 12592  df-ico 12594  df-fl 13012  df-seq 13220  df-exp 13280  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-rest 16525  df-salg 42156  df-smblfn 42540
This theorem is referenced by:  smflim  42615
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