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Theorem smfinfmpt 41346
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 2652 . . . . . . . . . . . 12 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵)))
8 smfinfmpt.f . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
97, 8fvmpt2d 6332 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = (𝑥𝐴𝐵))
109dmeqd 5358 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = dom (𝑥𝐴𝐵))
11 nfcv 2793 . . . . . . . . . . . . 13 𝑥𝑛
12 nfcv 2793 . . . . . . . . . . . . 13 𝑥𝑍
1311, 12nfel 2806 . . . . . . . . . . . 12 𝑥 𝑛𝑍
143, 13nfan 1868 . . . . . . . . . . 11 𝑥(𝜑𝑛𝑍)
15 eqid 2651 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
16 smfinfmpt.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1716adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
18 smfinfmpt.b . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
19183expa 1284 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2014, 17, 19, 8smffmpt 41332 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2120fvmptelrn 39742 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵 ∈ ℝ)
2214, 15, 21dmmptdf 39731 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
23 eqidd 2652 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝐴 = 𝐴)
2410, 22, 233eqtrrd 2690 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐴 = dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
256, 24iineq2d 4573 . . . . . . . 8 (𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
26 nfcv 2793 . . . . . . . . 9 𝑥 𝑛𝑍 𝐴
27 nfmpt1 4780 . . . . . . . . . . . . 13 𝑥(𝑥𝐴𝐵)
2812, 27nfmpt 4779 . . . . . . . . . . . 12 𝑥(𝑛𝑍 ↦ (𝑥𝐴𝐵))
2928, 11nffv 6236 . . . . . . . . . . 11 𝑥((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3029nfdm 5399 . . . . . . . . . 10 𝑥dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3112, 30nfiin 4581 . . . . . . . . 9 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3226, 31rabeqf 3221 . . . . . . . 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 1883 . . . . . . . . . 10 𝑦 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3634, 35nfan 1868 . . . . . . . . 9 𝑦(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
37 nfcv 2793 . . . . . . . . . . . 12 𝑛𝑥
38 nfii1 4583 . . . . . . . . . . . 12 𝑛 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3937, 38nfel 2806 . . . . . . . . . . 11 𝑛 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
406, 39nfan 1868 . . . . . . . . . 10 𝑛(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
41 simpll 805 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝜑)
42 simpr 476 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
43 eliinid 39608 . . . . . . . . . . . . 13 ((𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4443adantll 750 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4524eqcomd 2657 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4645adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4744, 46eleqtrd 2732 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥𝐴)
489fveq1d 6231 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
49483adant3 1101 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
50 simp3 1083 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝑥𝐴)
5115fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5250, 18, 51syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5349, 52eqtr2d 2686 . . . . . . . . . . . 12 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
5453breq2d 4697 . . . . . . . . . . 11 ((𝜑𝑛𝑍𝑥𝐴) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5541, 42, 47, 54syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5640, 55ralbida 3011 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∀𝑛𝑍 𝑦𝐵 ↔ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5736, 56rexbid 3080 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
583, 57rabbida 39588 . . . . . . 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 2685 . . . . . 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 2685 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
613, 60alrimi 2120 . . . 4 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
62 nfcv 2793 . . . . . . . . . . . . . 14 𝑛
63 nfra1 2970 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦𝐵
6462, 63nfrex 3036 . . . . . . . . . . . . 13 𝑛𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵
65 nfii1 4583 . . . . . . . . . . . . 13 𝑛 𝑛𝑍 𝐴
6664, 65nfrab 3153 . . . . . . . . . . . 12 𝑛{𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}
674, 66nfcxfr 2791 . . . . . . . . . . 11 𝑛𝐷
6837, 67nfel 2806 . . . . . . . . . 10 𝑛 𝑥𝐷
696, 68nfan 1868 . . . . . . . . 9 𝑛(𝜑𝑥𝐷)
70 simpll 805 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝜑)
71 simpr 476 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑛𝑍)
724eleq2i 2722 . . . . . . . . . . . . . . 15 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
7372biimpi 206 . . . . . . . . . . . . . 14 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
74 rabidim1 3147 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵} → 𝑥 𝑛𝑍 𝐴)
7573, 74syl 17 . . . . . . . . . . . . 13 (𝑥𝐷𝑥 𝑛𝑍 𝐴)
7675adantr 480 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑥 𝑛𝑍 𝐴)
77 simpr 476 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
78 eliinid 39608 . . . . . . . . . . . 12 ((𝑥 𝑛𝑍 𝐴𝑛𝑍) → 𝑥𝐴)
7976, 77, 78syl2anc 694 . . . . . . . . . . 11 ((𝑥𝐷𝑛𝑍) → 𝑥𝐴)
8079adantll 750 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥𝐴)
8153idi 2 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8270, 71, 80, 81syl3anc 1366 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8369, 82mpteq2da 4776 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝑛𝑍𝐵) = (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8483rneqd 5385 . . . . . . 7 ((𝜑𝑥𝐷) → ran (𝑛𝑍𝐵) = ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8584infeq1d 8424 . . . . . 6 ((𝜑𝑥𝐷) → inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
8685ex 449 . . . . 5 (𝜑 → (𝑥𝐷 → inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
873, 86ralrimi 2986 . . . 4 (𝜑 → ∀𝑥𝐷 inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
88 mpteq12f 4764 . . . 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 694 . . 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 2685 . 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 4780 . . 3 𝑛(𝑛𝑍 ↦ (𝑥𝐴𝐵))
92 smfinfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
93 smfinfmpt.z . . 3 𝑍 = (ℤ𝑀)
94 eqid 2651 . . . 4 (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵))
956, 8, 94fmptdf 6427 . . 3 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
96 eqid 2651 . . 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 2651 . . 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 41345 . 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 2730 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wnf 1748  wcel 2030  wral 2941  wrex 2942  {crab 2945   ciin 4553   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  cfv 5926  infcinf 8388  cr 9973   < clt 10112  cle 10113  cz 11415  cuz 11725  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-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-s4 13641  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-rest 16130  df-topgen 16151  df-top 20747  df-bases 20798  df-salg 40847  df-salgen 40851  df-smblfn 41231
This theorem is referenced by: (None)
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