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Theorem smflimlem6 41624
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 6389 . . . . . . 7 𝑍 ∈ V
3 nnex 11281 . . . . . . 7 ℕ ∈ V
42, 3xpex 7160 . . . . . 6 (𝑍 × ℕ) ∈ V
54a1i 11 . . . . 5 (𝜑 → (𝑍 × ℕ) ∈ V)
6 eqid 2765 . . . . . . . . 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 4974 . . . . . . . 8 (𝜑 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
98adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ∈ V)
109ralrimivva 3118 . . . . . 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))})
1211fnmpt2 7439 . . . . . 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 9611 . . . . 5 ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
155, 13, 14sylc 65 . . . 4 (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
161uzct 39883 . . . . . . 7 𝑍 ≼ ω
17 nnct 12988 . . . . . . 7 ℕ ≼ ω
1816, 17pm3.2i 462 . . . . . 6 (𝑍 ≼ ω ∧ ℕ ≼ ω)
19 xpct 9090 . . . . . 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 8213 . . . 4 ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
2315, 21, 22syl2anc 579 . . 3 (𝜑 → ran 𝑃 ≼ ω)
24 vex 3353 . . . . . . 7 𝑦 ∈ V
2511elrnmpt2g 6970 . . . . . . 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 207 . . . . 5 (𝑦 ∈ ran 𝑃 → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
2827adantl 473 . . . 4 ((𝜑𝑦 ∈ ran 𝑃) → ∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
29 simp3 1168 . . . . . . . 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 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
31 smflimlem6.4 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
3231ffvelrnda 6549 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
3332adantrr 708 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
34 eqid 2765 . . . . . . . . . . . 12 dom (𝐹𝑚) = dom (𝐹𝑚)
35 smflimlem6.7 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
3635adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
37 nnrecre 11314 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
3837adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
3936, 38readdcld 10323 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4039adantrl 707 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
4130, 33, 34, 40smfpreimalt 41580 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆t dom (𝐹𝑚)))
42 fvex 6388 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
4342dmex 7297 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
4443a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝐹𝑚) ∈ V)
45 elrest 16354 . . . . . . . . . . . . 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 579 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 223 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → ∃𝑠𝑆 {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚)))
49 rabn0 4122 . . . . . . . . . 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 225 . . . . . . . . 9 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ)) → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
51503adant3 1162 . . . . . . . 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 3004 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
53523exp 1148 . . . . . 6 (𝜑 → ((𝑚𝑍𝑘 ∈ ℕ) → (𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
5453rexlimdvv 3184 . . . . 5 (𝜑 → (∃𝑚𝑍𝑘 ∈ ℕ 𝑦 = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
5554adantr 472 . . . 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 40072 . 2 (𝜑 → ∃𝑐𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦)
58 smflimlem6.1 . . . . . 6 (𝜑𝑀 ∈ ℤ)
5958adantr 472 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
607adantr 472 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
6131adantr 472 . . . . 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 472 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
65 fvoveq1 6865 . . . . . 6 (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
66 oveq2 6850 . . . . . . 7 (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
6766fveq2d 6379 . . . . . 6 (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
6865, 67cbvmpt2v 6933 . . . . 5 (𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
69 nfcv 2907 . . . . . 6 𝑘 𝑛𝑍 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
70 nfcv 2907 . . . . . . 7 𝑗𝑍
71 nfcv 2907 . . . . . . . 8 𝑗(ℤ𝑛)
72 nfcv 2907 . . . . . . . . 9 𝑗𝑚
73 nfmpt22 6921 . . . . . . . . 9 𝑗(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
74 nfcv 2907 . . . . . . . . 9 𝑗𝑘
7572, 73, 74nfov 6872 . . . . . . . 8 𝑗(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7671, 75nfiin 4705 . . . . . . 7 𝑗 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
7770, 76nfiun 4704 . . . . . 6 𝑗 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78 oveq2 6850 . . . . . . . . . . 11 (𝑗 = 𝑘 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
7978adantr 472 . . . . . . . . . 10 ((𝑗 = 𝑘𝑖 ∈ (ℤ𝑛)) → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8079iineq2dv 4699 . . . . . . . . 9 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
81 oveq1 6849 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8281cbviinv 4716 . . . . . . . . . 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 2799 . . . . . . . 8 (𝑗 = 𝑘 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8584adantr 472 . . . . . . 7 ((𝑗 = 𝑘𝑛𝑍) → 𝑖 ∈ (ℤ𝑛)(𝑖(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = 𝑚 ∈ (ℤ𝑛)(𝑚(𝑙𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
8685iuneq2dv 4698 . . . . . 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 4714 . . . . 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 6375 . . . . . . . 8 (𝑦 = 𝑟 → (𝑐𝑦) = (𝑐𝑟))
89 id 22 . . . . . . . 8 (𝑦 = 𝑟𝑦 = 𝑟)
9088, 89eleq12d 2838 . . . . . . 7 (𝑦 = 𝑟 → ((𝑐𝑦) ∈ 𝑦 ↔ (𝑐𝑟) ∈ 𝑟))
9190rspccva 3460 . . . . . 6 ((∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9291adantll 705 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐𝑟) ∈ 𝑟)
9359, 1, 60, 61, 62, 63, 64, 11, 68, 87, 92smflimlem5 41623 . . . 4 ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦) → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))
9493ex 401 . . 3 (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐𝑦) ∈ 𝑦 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷)))
9594exlimdv 2028 . 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cin 3731  c0 4079   ciun 4676   ciin 4677   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  ωcom 7263  cdom 8158  cr 10188  1c1 10190   + caddc 10192   < clt 10328  cle 10329   / cdiv 10938  cn 11274  cz 11624  cuz 11886  cli 14500  t crest 16347  SAlgcsalg 41165  SMblFncsmblfn 41549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cc 9510  ax-ac2 9538  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-omul 7769  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-acn 9019  df-ac 9190  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-ioo 12381  df-ico 12383  df-fl 12801  df-seq 13009  df-exp 13068  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-rlim 14505  df-rest 16349  df-salg 41166  df-smblfn 41550
This theorem is referenced by:  smflim  41625
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