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Theorem smfinfmpt 42970
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 2819 . . . . . . . . . . . 12 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵)))
8 smfinfmpt.f . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
97, 8fvmpt2d 6773 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = (𝑥𝐴𝐵))
109dmeqd 5767 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = dom (𝑥𝐴𝐵))
11 nfcv 2974 . . . . . . . . . . . . 13 𝑥𝑛
12 nfcv 2974 . . . . . . . . . . . . 13 𝑥𝑍
1311, 12nfel 2989 . . . . . . . . . . . 12 𝑥 𝑛𝑍
143, 13nfan 1891 . . . . . . . . . . 11 𝑥(𝜑𝑛𝑍)
15 eqid 2818 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
16 smfinfmpt.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1716adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
18 smfinfmpt.b . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
19183expa 1110 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2014, 17, 19, 8smffmpt 42956 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2120fvmptelrn 6869 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵 ∈ ℝ)
2214, 15, 21dmmptdf 41364 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
23 eqidd 2819 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝐴 = 𝐴)
2410, 22, 233eqtrrd 2858 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐴 = dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
256, 24iineq2d 4933 . . . . . . . 8 (𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
26 nfcv 2974 . . . . . . . . 9 𝑥 𝑛𝑍 𝐴
27 nfmpt1 5155 . . . . . . . . . . . . 13 𝑥(𝑥𝐴𝐵)
2812, 27nfmpt 5154 . . . . . . . . . . . 12 𝑥(𝑛𝑍 ↦ (𝑥𝐴𝐵))
2928, 11nffv 6673 . . . . . . . . . . 11 𝑥((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3029nfdm 5816 . . . . . . . . . 10 𝑥dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3112, 30nfiin 4941 . . . . . . . . 9 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3226, 31rabeqf 3479 . . . . . . . 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 1906 . . . . . . . . . 10 𝑦 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3634, 35nfan 1891 . . . . . . . . 9 𝑦(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
37 nfcv 2974 . . . . . . . . . . . 12 𝑛𝑥
38 nfii1 4945 . . . . . . . . . . . 12 𝑛 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3937, 38nfel 2989 . . . . . . . . . . 11 𝑛 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
406, 39nfan 1891 . . . . . . . . . 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 41254 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4645adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4744, 46eleqtrd 2912 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥𝐴)
489fveq1d 6665 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
49483adant3 1124 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
50 simp3 1130 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝑥𝐴)
5115fvmpt2 6771 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5250, 18, 51syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5349, 52eqtr2d 2854 . . . . . . . . . . . 12 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
5453breq2d 5069 . . . . . . . . . . 11 ((𝜑𝑛𝑍𝑥𝐴) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5541, 42, 47, 54syl3anc 1363 . . . . . . . . . 10 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5640, 55ralbida 3227 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∀𝑛𝑍 𝑦𝐵 ↔ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5736, 56rexbid 3317 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
583, 57rabbida 3472 . . . . . . 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 2853 . . . . . 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 2853 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
613, 60alrimi 2203 . . . 4 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
62 nfcv 2974 . . . . . . . . . . . . . 14 𝑛
63 nfra1 3216 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦𝐵
6462, 63nfrex 3306 . . . . . . . . . . . . 13 𝑛𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵
65 nfii1 4945 . . . . . . . . . . . . 13 𝑛 𝑛𝑍 𝐴
6664, 65nfrab 3384 . . . . . . . . . . . 12 𝑛{𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}
674, 66nfcxfr 2972 . . . . . . . . . . 11 𝑛𝐷
6837, 67nfel 2989 . . . . . . . . . 10 𝑛 𝑥𝐷
696, 68nfan 1891 . . . . . . . . 9 𝑛(𝜑𝑥𝐷)
70 simpll 763 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝜑)
71 simpr 485 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑛𝑍)
724eleq2i 2901 . . . . . . . . . . . . . . 15 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
7372biimpi 217 . . . . . . . . . . . . . 14 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
74 rabidim1 3378 . . . . . . . . . . . . . 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 41254 . . . . . . . . . . . 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 1363 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8369, 82mpteq2da 5151 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝑛𝑍𝐵) = (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8483rneqd 5801 . . . . . . 7 ((𝜑𝑥𝐷) → ran (𝑛𝑍𝐵) = ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8584infeq1d 8929 . . . . . 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 3213 . . . 4 (𝜑 → ∀𝑥𝐷 inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
88 mpteq12f 5140 . . . 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 2853 . 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 5155 . . 3 𝑛(𝑛𝑍 ↦ (𝑥𝐴𝐵))
92 smfinfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
93 smfinfmpt.z . . 3 𝑍 = (ℤ𝑀)
94 eqid 2818 . . . 4 (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵))
956, 8, 94fmptdf 6873 . . 3 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
96 eqid 2818 . . 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 2818 . . 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 42969 . 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 2910 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wnf 1775  wcel 2105  wral 3135  wrex 3136  {crab 3139   ciin 4911   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  cfv 6348  infcinf 8893  cr 10524   < clt 10663  cle 10664  cz 11969  cuz 12231  SAlgcsalg 42470  SMblFncsmblfn 42854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845  ax-ac2 9873  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-acn 9359  df-ac 9530  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938  df-s2 14198  df-s3 14199  df-s4 14200  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-rest 16684  df-topgen 16705  df-top 21430  df-bases 21482  df-salg 42471  df-salgen 42475  df-smblfn 42855
This theorem is referenced by: (None)
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