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Theorem smflimlem6 41305
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)
2 fvex 6239 . . . . . . . 8 (ℤ𝑀) ∈ V
31, 2eqeltri 2726 . . . . . . 7 𝑍 ∈ V
4 nnex 11064 . . . . . . 7 ℕ ∈ V
53, 4xpex 7004 . . . . . 6 (𝑍 × ℕ) ∈ V
65a1i 11 . . . . 5 (𝜑 → (𝑍 × ℕ) ∈ V)
7 eqid 2651 . . . . . . . . 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))}
8 smflimlem6.3 . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
97, 8rabexd 4846 . . . . . . . 8 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
109adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
1110ralrimivva 3000 . . . . . 6 (𝜑 → ∀𝑚𝑍𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
12 smflimlem6.8 . . . . . . 7 𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
1312fnmpt2 7283 . . . . . 6 (∀𝑚𝑍𝑘 ∈ ℕ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V → 𝑃 Fn (𝑍 × ℕ))
1411, 13syl 17 . . . . 5 (𝜑𝑃 Fn (𝑍 × ℕ))
15 fnrndomg 9396 . . . . 5 ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
166, 14, 15sylc 65 . . . 4 (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
171uzct 39546 . . . . . . 7 𝑍 ≼ ω
18 nnct 12820 . . . . . . 7 ℕ ≼ ω
1917, 18pm3.2i 470 . . . . . 6 (𝑍 ≼ ω ∧ ℕ ≼ ω)
20 xpct 8877 . . . . . 6 ((𝑍 ≼ ω ∧ ℕ ≼ ω) → (𝑍 × ℕ) ≼ ω)
2119, 20ax-mp 5 . . . . 5 (𝑍 × ℕ) ≼ ω
2221a1i 11 . . . 4 (𝜑 → (𝑍 × ℕ) ≼ ω)
23 domtr 8050 . . . 4 ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
2416, 22, 23syl2anc 694 . . 3 (𝜑 → ran 𝑃 ≼ ω)
25 vex 3234 . . . . . . 7 𝑦 ∈ V
2612elrnmpt2g 6814 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran 𝑃 ↔ ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}))
2725, 26ax-mp 5 . . . . . 6 (𝑦 ∈ ran 𝑃 ↔ ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2827biimpi 206 . . . . 5 (𝑦 ∈ ran 𝑃 → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2928adantl 481 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
30 simp3 1083 . . . . . . . 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))})
318adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
32 smflimlem6.4 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
3332ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
3433adantrr 753 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
35 eqid 2651 . . . . . . . . . . . 12 dom (𝐹𝑚) = dom (𝐹𝑚)
36 smflimlem6.7 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
3736adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38 nnrecre 11095 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
3938adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
4037, 39readdcld 10107 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4140adantrl 752 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4231, 34, 35, 41smfpreimalt 41261 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)))
43 fvex 6239 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
4443dmex 7141 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
4544a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝐹𝑚) ∈ V)
46 elrest 16135 . . . . . . . . . . . . 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))))
478, 45, 46syl2anc 694 . . . . . . . . . . . 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))))
4847adantr 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))))
4942, 48mpbid 222 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)))
50 rabn0 3991 . . . . . . . . . 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)))
5149, 50sylibr 224 . . . . . . . . 9 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
52513adant3 1101 . . . . . . . 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))} ≠ ∅)
5330, 52eqnetrd 2890 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
54533exp 1283 . . . . . 6 (𝜑 → ((𝑚𝑍𝑘 ∈ ℕ) → (𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
5554rexlimdvv 3066 . . . . 5 (𝜑 → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5655adantr 480 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5729, 56mpd 15 . . 3 ((𝜑𝑦 ∈ ran 𝑃) → 𝑦 ≠ ∅)
5824, 57axccd2 39744 . 2 (𝜑 → ∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦)
59 smflimlem6.1 . . . . . 6 (𝜑𝑀 ∈ ℤ)
6059adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
618adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
6232adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐹:𝑍⟶(SMblFn‘𝑆))
63 smflimlem6.5 . . . . 5 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
64 smflimlem6.6 . . . . 5 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
6536adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
66 oveq1 6697 . . . . . . 7 (𝑙 = 𝑚 → (𝑙𝑃𝑗) = (𝑚𝑃𝑗))
6766fveq2d 6233 . . . . . 6 (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
68 oveq2 6698 . . . . . . 7 (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
6968fveq2d 6233 . . . . . 6 (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
7067, 69cbvmpt2v 6777 . . . . 5 (𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
71 nfcv 2793 . . . . . 6 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
72 nfcv 2793 . . . . . . 7 𝑗𝑍
73 nfcv 2793 . . . . . . . 8 𝑗(ℤ𝑛)
74 nfcv 2793 . . . . . . . . 9 𝑗𝑚
75 nfmpt22 6765 . . . . . . . . 9 𝑗(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
76 nfcv 2793 . . . . . . . . 9 𝑗𝑘
7774, 75, 76nfov 6716 . . . . . . . 8 𝑗(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7873, 77nfiin 4581 . . . . . . 7 𝑗 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7972, 78nfiun 4580 . . . . . 6 𝑗 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
80 oveq2 6698 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8180adantr 480 . . . . . . . . . 10 ((𝑗 = 𝑘𝑖 ∈ (ℤ𝑛)) → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8281iineq2dv 4575 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
83 oveq1 6697 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8483cbviinv 4592 . . . . . . . . . 10 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
8584a1i 11 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8682, 85eqtrd 2685 . . . . . . . 8 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8786adantr 480 . . . . . . 7 ((𝑗 = 𝑘𝑛𝑍) → 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8887iuneq2dv 4574 . . . . . 6 (𝑗 = 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8971, 79, 88cbviin 4590 . . . . 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)
90 fveq2 6229 . . . . . . . 8 (𝑦 = 𝑟 → (𝑐𝑦) = (𝑐𝑟))
91 id 22 . . . . . . . 8 (𝑦 = 𝑟𝑦 = 𝑟)
9290, 91eleq12d 2724 . . . . . . 7 (𝑦 = 𝑟 → ((𝑐𝑦) ∈ 𝑦 ↔ (𝑐𝑟) ∈ 𝑟))
9392rspccva 3339 . . . . . 6 ((∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9493adantll 750 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9560, 1, 61, 62, 63, 64, 65, 12, 70, 89, 94smflimlem5 41304 . . . 4 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
9695ex 449 . . 3 (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9796exlimdv 1901 . 2 (𝜑 → (∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9858, 97mpd 15 1 (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  c0 3948   ciun 4552   ciin 4553   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  cdom 7995  cr 9973  1c1 9975   + caddc 9977   < clt 10112  cle 10113   / cdiv 10722  cn 11058  cz 11415  cuz 11725  cli 14259  t crest 16128  SAlgcsalg 40846  SMblFncsmblfn 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ico 12219  df-fl 12633  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-rest 16130  df-salg 40847  df-smblfn 41231
This theorem is referenced by:  smflim  41306
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