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Theorem smfsupmpt 42554
Description: The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smfsupmpt.n 𝑛𝜑
smfsupmpt.x 𝑥𝜑
smfsupmpt.y 𝑦𝜑
smfsupmpt.m (𝜑𝑀 ∈ ℤ)
smfsupmpt.z 𝑍 = (ℤ𝑀)
smfsupmpt.s (𝜑𝑆 ∈ SAlg)
smfsupmpt.b ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
smfsupmpt.f ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
smfsupmpt.d 𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦}
smfsupmpt.g 𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < ))
Assertion
Ref Expression
smfsupmpt (𝜑𝐺 ∈ (SMblFn‘𝑆))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑆,𝑛   𝑛,𝑍,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑛)   𝐴(𝑛)   𝐵(𝑥,𝑛)   𝐷(𝑥,𝑦,𝑛)   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑛)   𝑀(𝑥,𝑦,𝑛)   𝑉(𝑥,𝑦,𝑛)

Proof of Theorem smfsupmpt
StepHypRef Expression
1 smfsupmpt.g . . . 4 𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < ))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < )))
3 smfsupmpt.x . . . . 5 𝑥𝜑
4 smfsupmpt.d . . . . . . 7 𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦}
54a1i 11 . . . . . 6 (𝜑𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦})
6 smfsupmpt.n . . . . . . . . 9 𝑛𝜑
7 eqidd 2772 . . . . . . . . . . . 12 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵)))
8 smfsupmpt.f . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
97, 8fvmpt2d 6605 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = (𝑥𝐴𝐵))
109dmeqd 5620 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = dom (𝑥𝐴𝐵))
11 nfcv 2925 . . . . . . . . . . . . 13 𝑥𝑛
12 nfcv 2925 . . . . . . . . . . . . 13 𝑥𝑍
1311, 12nfel 2937 . . . . . . . . . . . 12 𝑥 𝑛𝑍
143, 13nfan 1863 . . . . . . . . . . 11 𝑥(𝜑𝑛𝑍)
15 eqid 2771 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
16 smfsupmpt.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1716adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
18 smfsupmpt.b . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
19183expa 1099 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2014, 17, 19, 8smffmpt 42544 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2120fvmptelrn 6698 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵 ∈ ℝ)
2214, 15, 21dmmptdf 40948 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
23 eqidd 2772 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝐴 = 𝐴)
2410, 22, 233eqtrrd 2812 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐴 = dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
256, 24iineq2d 4810 . . . . . . . 8 (𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
26 nfcv 2925 . . . . . . . . 9 𝑥 𝑛𝑍 𝐴
27 nfmpt1 5021 . . . . . . . . . . . . 13 𝑥(𝑥𝐴𝐵)
2812, 27nfmpt 5020 . . . . . . . . . . . 12 𝑥(𝑛𝑍 ↦ (𝑥𝐴𝐵))
2928, 11nffv 6506 . . . . . . . . . . 11 𝑥((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3029nfdm 5663 . . . . . . . . . 10 𝑥dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3112, 30nfiin 4818 . . . . . . . . 9 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3226, 31rabeqf 3397 . . . . . . . 8 ( 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) → {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦})
3325, 32syl 17 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦})
34 smfsupmpt.y . . . . . . . . . 10 𝑦𝜑
35 nfv 1874 . . . . . . . . . 10 𝑦 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3634, 35nfan 1863 . . . . . . . . 9 𝑦(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
37 nfcv 2925 . . . . . . . . . . . 12 𝑛𝑥
38 nfii1 4820 . . . . . . . . . . . 12 𝑛 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3937, 38nfel 2937 . . . . . . . . . . 11 𝑛 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
406, 39nfan 1863 . . . . . . . . . 10 𝑛(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
41 simpll 755 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝜑)
42 simpr 477 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
43 eliinid 40836 . . . . . . . . . . . . 13 ((𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4443adantll 702 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4524eqcomd 2777 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4645adantlr 703 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4744, 46eleqtrd 2861 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥𝐴)
489fveq1d 6498 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
49483adant3 1113 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
50 simp3 1119 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝑥𝐴)
5115fvmpt2 6603 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5250, 18, 51syl2anc 576 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5349, 52eqtr2d 2808 . . . . . . . . . . . 12 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
5453breq1d 4935 . . . . . . . . . . 11 ((𝜑𝑛𝑍𝑥𝐴) → (𝐵𝑦 ↔ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
5541, 42, 47, 54syl3anc 1352 . . . . . . . . . 10 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → (𝐵𝑦 ↔ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
5640, 55ralbida 3170 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∀𝑛𝑍 𝐵𝑦 ↔ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
5736, 56rexbid 3256 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦))
583, 57rabbida 40819 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
5933, 58eqtrd 2807 . . . . . 6 (𝜑 → {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
605, 59eqtrd 2807 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
613, 60alrimi 2144 . . . 4 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦})
62 nfcv 2925 . . . . . . . . . . . . . 14 𝑛
63 nfra1 3162 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝐵𝑦
6462, 63nfrex 3246 . . . . . . . . . . . . 13 𝑛𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦
65 nfii1 4820 . . . . . . . . . . . . 13 𝑛 𝑛𝑍 𝐴
6664, 65nfrab 3318 . . . . . . . . . . . 12 𝑛{𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦}
674, 66nfcxfr 2923 . . . . . . . . . . 11 𝑛𝐷
6837, 67nfel 2937 . . . . . . . . . 10 𝑛 𝑥𝐷
696, 68nfan 1863 . . . . . . . . 9 𝑛(𝜑𝑥𝐷)
70 simpll 755 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝜑)
71 simpr 477 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑛𝑍)
724eleq2i 2850 . . . . . . . . . . . . . . 15 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦})
7372biimpi 208 . . . . . . . . . . . . . 14 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦})
74 rabidim1 3312 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦} → 𝑥 𝑛𝑍 𝐴)
7573, 74syl 17 . . . . . . . . . . . . 13 (𝑥𝐷𝑥 𝑛𝑍 𝐴)
7675adantr 473 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑥 𝑛𝑍 𝐴)
77 simpr 477 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
78 eliinid 40836 . . . . . . . . . . . 12 ((𝑥 𝑛𝑍 𝐴𝑛𝑍) → 𝑥𝐴)
7976, 77, 78syl2anc 576 . . . . . . . . . . 11 ((𝑥𝐷𝑛𝑍) → 𝑥𝐴)
8079adantll 702 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥𝐴)
8153idi 1 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8270, 71, 80, 81syl3anc 1352 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8369, 82mpteq2da 5017 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝑛𝑍𝐵) = (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8483rneqd 5648 . . . . . . 7 ((𝜑𝑥𝐷) → ran (𝑛𝑍𝐵) = ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8584supeq1d 8703 . . . . . 6 ((𝜑𝑥𝐷) → sup(ran (𝑛𝑍𝐵), ℝ, < ) = sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
8685ex 405 . . . . 5 (𝜑 → (𝑥𝐷 → sup(ran (𝑛𝑍𝐵), ℝ, < ) = sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
873, 86ralrimi 3159 . . . 4 (𝜑 → ∀𝑥𝐷 sup(ran (𝑛𝑍𝐵), ℝ, < ) = sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
88 mpteq12f 5006 . . . 4 ((∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ∧ ∀𝑥𝐷 sup(ran (𝑛𝑍𝐵), ℝ, < ) = sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )) → (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
8961, 87, 88syl2anc 576 . . 3 (𝜑 → (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
902, 89eqtrd 2807 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
91 nfmpt1 5021 . . 3 𝑛(𝑛𝑍 ↦ (𝑥𝐴𝐵))
92 smfsupmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
93 smfsupmpt.z . . 3 𝑍 = (ℤ𝑀)
94 eqid 2771 . . . 4 (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵))
956, 8, 94fmptdf 6702 . . 3 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
96 eqid 2771 . . 3 {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦}
97 eqid 2771 . . 3 (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
9891, 28, 92, 93, 16, 95, 96, 97smfsup 42553 . 2 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
9990, 98eqeltrd 2859 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069  wal 1506   = wceq 1508  wnf 1747  wcel 2051  wral 3081  wrex 3082  {crab 3085   ciin 4789   class class class wbr 4925  cmpt 5004  dom cdm 5403  ran crn 5404  cfv 6185  supcsup 8697  cr 10332   < clt 10472  cle 10473  cz 11791  cuz 12056  SAlgcsalg 42058  SMblFncsmblfn 42442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896  ax-cc 9653  ax-ac2 9681  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-omul 7908  df-er 8087  df-map 8206  df-pm 8207  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-sup 8699  df-inf 8700  df-oi 8767  df-card 9160  df-acn 9163  df-ac 9334  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-n0 11706  df-z 11792  df-uz 12057  df-q 12161  df-rp 12203  df-ioo 12556  df-ioc 12557  df-ico 12558  df-fl 12975  df-rest 16550  df-topgen 16571  df-top 21221  df-bases 21273  df-salg 42059  df-salgen 42063  df-smblfn 42443
This theorem is referenced by:  smfinflem  42556
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