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Theorem smfinfmpt 42955
Description: The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smfinfmpt.n 𝑛𝜑
smfinfmpt.x 𝑥𝜑
smfinfmpt.y 𝑦𝜑
smfinfmpt.m (𝜑𝑀 ∈ ℤ)
smfinfmpt.z 𝑍 = (ℤ𝑀)
smfinfmpt.s (𝜑𝑆 ∈ SAlg)
smfinfmpt.b ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
smfinfmpt.f ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
smfinfmpt.d 𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}
smfinfmpt.g 𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < ))
Assertion
Ref Expression
smfinfmpt (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 smfinfmpt
StepHypRef Expression
1 smfinfmpt.g . . . 4 𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < ))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < )))
3 smfinfmpt.x . . . . 5 𝑥𝜑
4 smfinfmpt.d . . . . . . 7 𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}
54a1i 11 . . . . . 6 (𝜑𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
6 smfinfmpt.n . . . . . . . . 9 𝑛𝜑
7 eqidd 2826 . . . . . . . . . . . 12 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵)))
8 smfinfmpt.f . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
97, 8fvmpt2d 6776 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = (𝑥𝐴𝐵))
109dmeqd 5772 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = dom (𝑥𝐴𝐵))
11 nfcv 2981 . . . . . . . . . . . . 13 𝑥𝑛
12 nfcv 2981 . . . . . . . . . . . . 13 𝑥𝑍
1311, 12nfel 2996 . . . . . . . . . . . 12 𝑥 𝑛𝑍
143, 13nfan 1893 . . . . . . . . . . 11 𝑥(𝜑𝑛𝑍)
15 eqid 2825 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
16 smfinfmpt.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1716adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
18 smfinfmpt.b . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
19183expa 1112 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2014, 17, 19, 8smffmpt 42941 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2120fvmptelrn 6872 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵 ∈ ℝ)
2214, 15, 21dmmptdf 41349 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
23 eqidd 2826 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝐴 = 𝐴)
2410, 22, 233eqtrrd 2865 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐴 = dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
256, 24iineq2d 4938 . . . . . . . 8 (𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
26 nfcv 2981 . . . . . . . . 9 𝑥 𝑛𝑍 𝐴
27 nfmpt1 5160 . . . . . . . . . . . . 13 𝑥(𝑥𝐴𝐵)
2812, 27nfmpt 5159 . . . . . . . . . . . 12 𝑥(𝑛𝑍 ↦ (𝑥𝐴𝐵))
2928, 11nffv 6676 . . . . . . . . . . 11 𝑥((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3029nfdm 5821 . . . . . . . . . 10 𝑥dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3112, 30nfiin 4946 . . . . . . . . 9 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3226, 31rabeqf 3486 . . . . . . . 8 ( 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) → {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
3325, 32syl 17 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
34 smfinfmpt.y . . . . . . . . . 10 𝑦𝜑
35 nfv 1908 . . . . . . . . . 10 𝑦 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3634, 35nfan 1893 . . . . . . . . 9 𝑦(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
37 nfcv 2981 . . . . . . . . . . . 12 𝑛𝑥
38 nfii1 4950 . . . . . . . . . . . 12 𝑛 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3937, 38nfel 2996 . . . . . . . . . . 11 𝑛 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
406, 39nfan 1893 . . . . . . . . . 10 𝑛(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
41 simpll 763 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝜑)
42 simpr 485 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
43 eliinid 41239 . . . . . . . . . . . . 13 ((𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4443adantll 710 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4524eqcomd 2831 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4645adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4744, 46eleqtrd 2919 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥𝐴)
489fveq1d 6668 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
49483adant3 1126 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
50 simp3 1132 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝑥𝐴)
5115fvmpt2 6774 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5250, 18, 51syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5349, 52eqtr2d 2861 . . . . . . . . . . . 12 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
5453breq2d 5074 . . . . . . . . . . 11 ((𝜑𝑛𝑍𝑥𝐴) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5541, 42, 47, 54syl3anc 1365 . . . . . . . . . 10 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5640, 55ralbida 3234 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∀𝑛𝑍 𝑦𝐵 ↔ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5736, 56rexbid 3324 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
583, 57rabbida 3479 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
5933, 58eqtrd 2860 . . . . . 6 (𝜑 → {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
605, 59eqtrd 2860 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
613, 60alrimi 2206 . . . 4 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
62 nfcv 2981 . . . . . . . . . . . . . 14 𝑛
63 nfra1 3223 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦𝐵
6462, 63nfrex 3313 . . . . . . . . . . . . 13 𝑛𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵
65 nfii1 4950 . . . . . . . . . . . . 13 𝑛 𝑛𝑍 𝐴
6664, 65nfrab 3391 . . . . . . . . . . . 12 𝑛{𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}
674, 66nfcxfr 2979 . . . . . . . . . . 11 𝑛𝐷
6837, 67nfel 2996 . . . . . . . . . 10 𝑛 𝑥𝐷
696, 68nfan 1893 . . . . . . . . 9 𝑛(𝜑𝑥𝐷)
70 simpll 763 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝜑)
71 simpr 485 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑛𝑍)
724eleq2i 2908 . . . . . . . . . . . . . . 15 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
7372biimpi 217 . . . . . . . . . . . . . 14 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
74 rabidim1 3385 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵} → 𝑥 𝑛𝑍 𝐴)
7573, 74syl 17 . . . . . . . . . . . . 13 (𝑥𝐷𝑥 𝑛𝑍 𝐴)
7675adantr 481 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑥 𝑛𝑍 𝐴)
77 simpr 485 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
78 eliinid 41239 . . . . . . . . . . . 12 ((𝑥 𝑛𝑍 𝐴𝑛𝑍) → 𝑥𝐴)
7976, 77, 78syl2anc 584 . . . . . . . . . . 11 ((𝑥𝐷𝑛𝑍) → 𝑥𝐴)
8079adantll 710 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥𝐴)
8153idi 1 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8270, 71, 80, 81syl3anc 1365 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8369, 82mpteq2da 5156 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝑛𝑍𝐵) = (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8483rneqd 5806 . . . . . . 7 ((𝜑𝑥𝐷) → ran (𝑛𝑍𝐵) = ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8584infeq1d 8933 . . . . . 6 ((𝜑𝑥𝐷) → inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
8685ex 413 . . . . 5 (𝜑 → (𝑥𝐷 → inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
873, 86ralrimi 3220 . . . 4 (𝜑 → ∀𝑥𝐷 inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
88 mpteq12f 5145 . . . 4 ((∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ∧ ∀𝑥𝐷 inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )) → (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
8961, 87, 88syl2anc 584 . . 3 (𝜑 → (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
902, 89eqtrd 2860 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
91 nfmpt1 5160 . . 3 𝑛(𝑛𝑍 ↦ (𝑥𝐴𝐵))
92 smfinfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
93 smfinfmpt.z . . 3 𝑍 = (ℤ𝑀)
94 eqid 2825 . . . 4 (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵))
956, 8, 94fmptdf 6876 . . 3 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
96 eqid 2825 . . 3 {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)}
97 eqid 2825 . . 3 (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
9891, 28, 92, 93, 16, 95, 96, 97smfinf 42954 . 2 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)} ↦ inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
9990, 98eqeltrd 2917 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081  wal 1528   = wceq 1530  wnf 1777  wcel 2107  wral 3142  wrex 3143  {crab 3146   ciin 4917   class class class wbr 5062  cmpt 5142  dom cdm 5553  ran crn 5554  cfv 6351  infcinf 8897  cr 10528   < clt 10667  cle 10668  cz 11973  cuz 12235  SAlgcsalg 42455  SMblFncsmblfn 42839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cc 9849  ax-ac2 9877  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-omul 8101  df-er 8282  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-acn 9363  df-ac 9534  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-ioo 12735  df-ioc 12736  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-seq 13363  df-exp 13423  df-hash 13684  df-word 13855  df-concat 13916  df-s1 13943  df-s2 14203  df-s3 14204  df-s4 14205  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-rest 16688  df-topgen 16709  df-top 21418  df-bases 21470  df-salg 42456  df-salgen 42460  df-smblfn 42840
This theorem is referenced by: (None)
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