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Theorem smflimlem6 46791
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 6920 . . . . . . 7 𝑍 ∈ V
3 nnex 12272 . . . . . . 7 ℕ ∈ V
42, 3xpex 7773 . . . . . 6 (𝑍 × ℕ) ∈ V
54a1i 11 . . . . 5 (𝜑 → (𝑍 × ℕ) ∈ V)
6 eqid 2737 . . . . . . . . 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 5340 . . . . . . . 8 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
98adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
109ralrimivva 3202 . . . . . 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 8094 . . . . . 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 10576 . . . . 5 ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
155, 13, 14sylc 65 . . . 4 (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
161uzct 45068 . . . . . . 7 𝑍 ≼ ω
17 nnct 14022 . . . . . . 7 ℕ ≼ ω
1816, 17pm3.2i 470 . . . . . 6 (𝑍 ≼ ω ∧ ℕ ≼ ω)
19 xpct 10056 . . . . . 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 9047 . . . 4 ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
2315, 21, 22syl2anc 584 . . 3 (𝜑 → ran 𝑃 ≼ ω)
24 vex 3484 . . . . . . 7 𝑦 ∈ V
2511elrnmpog 7568 . . . . . . 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 216 . . . . 5 (𝑦 ∈ ran 𝑃 → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2827adantl 481 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
29 simp3 1139 . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
31 smflimlem6.4 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
3231ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
3332adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
34 eqid 2737 . . . . . . . . . . . 12 dom (𝐹𝑚) = dom (𝐹𝑚)
35 smflimlem6.7 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
3635adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
37 nnrecre 12308 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
3837adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
3936, 38readdcld 11290 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4039adantrl 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4130, 33, 34, 40smfpreimalt 46746 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)))
42 fvex 6919 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
4342dmex 7931 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
4443a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝐹𝑚) ∈ V)
45 elrest 17472 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 232 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)))
49 rabn0 4389 . . . . . . . . . 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 234 . . . . . . . . 9 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
51503adant3 1133 . . . . . . . 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 3008 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
53523exp 1120 . . . . . 6 (𝜑 → ((𝑚𝑍𝑘 ∈ ℕ) → (𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
5453rexlimdvv 3212 . . . . 5 (𝜑 → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5554adantr 480 . . . 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 45235 . 2 (𝜑 → ∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦)
58 smflimlem6.1 . . . . . 6 (𝜑𝑀 ∈ ℤ)
5958adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
607adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
6131adantr 480 . . . . 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 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
65 fvoveq1 7454 . . . . . 6 (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
66 oveq2 7439 . . . . . . 7 (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
6766fveq2d 6910 . . . . . 6 (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
6865, 67cbvmpov 7528 . . . . 5 (𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
69 nfcv 2905 . . . . . 6 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
70 nfcv 2905 . . . . . . 7 𝑗𝑍
71 nfcv 2905 . . . . . . . 8 𝑗(ℤ𝑛)
72 nfcv 2905 . . . . . . . . 9 𝑗𝑚
73 nfmpo2 7514 . . . . . . . . 9 𝑗(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
74 nfcv 2905 . . . . . . . . 9 𝑗𝑘
7572, 73, 74nfov 7461 . . . . . . . 8 𝑗(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7671, 75nfiin 5024 . . . . . . 7 𝑗 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7770, 76nfiun 5023 . . . . . 6 𝑗 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78 oveq2 7439 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
7978adantr 480 . . . . . . . . . 10 ((𝑗 = 𝑘𝑖 ∈ (ℤ𝑛)) → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8079iineq2dv 5017 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
81 oveq1 7438 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8281cbviinv 5041 . . . . . . . . . 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 2777 . . . . . . . 8 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8584adantr 480 . . . . . . 7 ((𝑗 = 𝑘𝑛𝑍) → 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8685iuneq2dv 5016 . . . . . 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 5037 . . . . 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 6906 . . . . . . . 8 (𝑦 = 𝑟 → (𝑐𝑦) = (𝑐𝑟))
89 id 22 . . . . . . . 8 (𝑦 = 𝑟𝑦 = 𝑟)
9088, 89eleq12d 2835 . . . . . . 7 (𝑦 = 𝑟 → ((𝑐𝑦) ∈ 𝑦 ↔ (𝑐𝑟) ∈ 𝑟))
9190rspccva 3621 . . . . . 6 ((∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9291adantll 714 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9359, 1, 60, 61, 62, 63, 64, 11, 68, 87, 92smflimlem5 46790 . . . 4 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
9493ex 412 . . 3 (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9594exlimdv 1933 . 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  c0 4333   ciun 4991   ciin 4992   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  cdom 8983  cr 11154  1c1 11156   + caddc 11158   < clt 11295  cle 11296   / cdiv 11920  cn 12266  cz 12613  cuz 12878  cli 15520  t crest 17465  SAlgcsalg 46323  SMblFncsmblfn 46710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-rest 17467  df-salg 46324  df-smblfn 46711
This theorem is referenced by:  smflim  46792
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