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Theorem smfinfmpt 40788
 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 2621 . . . . . . . . . . . 12 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵)))
8 smfinfmpt.f . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
97, 8fvmpt2d 6280 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = (𝑥𝐴𝐵))
109dmeqd 5315 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = dom (𝑥𝐴𝐵))
11 nfcv 2762 . . . . . . . . . . . . 13 𝑥𝑛
12 nfcv 2762 . . . . . . . . . . . . 13 𝑥𝑍
1311, 12nfel 2774 . . . . . . . . . . . 12 𝑥 𝑛𝑍
143, 13nfan 1826 . . . . . . . . . . 11 𝑥(𝜑𝑛𝑍)
15 eqid 2620 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
16 smfinfmpt.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1716adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
18 smfinfmpt.b . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)
19183expa 1263 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2014, 17, 19, 8smffmpt 40774 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2120fvmptelrn 39244 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑥𝐴) → 𝐵 ∈ ℝ)
2214, 15, 21dmmptdf 39233 . . . . . . . . . 10 ((𝜑𝑛𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
23 eqidd 2621 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝐴 = 𝐴)
2410, 22, 233eqtrrd 2659 . . . . . . . . 9 ((𝜑𝑛𝑍) → 𝐴 = dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
256, 24iineq2d 4532 . . . . . . . 8 (𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
26 nfcv 2762 . . . . . . . . 9 𝑥 𝑛𝑍 𝐴
27 nfmpt1 4738 . . . . . . . . . . . . 13 𝑥(𝑥𝐴𝐵)
2812, 27nfmpt 4737 . . . . . . . . . . . 12 𝑥(𝑛𝑍 ↦ (𝑥𝐴𝐵))
2928, 11nffv 6185 . . . . . . . . . . 11 𝑥((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3029nfdm 5356 . . . . . . . . . 10 𝑥dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3112, 30nfiin 4540 . . . . . . . . 9 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3226, 31rabeqf 3185 . . . . . . . 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 1841 . . . . . . . . . 10 𝑦 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3634, 35nfan 1826 . . . . . . . . 9 𝑦(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
37 nfcv 2762 . . . . . . . . . . . 12 𝑛𝑥
38 nfii1 4542 . . . . . . . . . . . 12 𝑛 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
3937, 38nfel 2774 . . . . . . . . . . 11 𝑛 𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)
406, 39nfan 1826 . . . . . . . . . 10 𝑛(𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
41 simpll 789 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝜑)
42 simpr 477 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
43 eliinid 39114 . . . . . . . . . . . . 13 ((𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4443adantll 749 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥 ∈ dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛))
4524eqcomd 2626 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4645adantlr 750 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) = 𝐴)
4744, 46eleqtrd 2701 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → 𝑥𝐴)
489fveq1d 6180 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
49483adant3 1079 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
50 simp3 1061 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥𝐴) → 𝑥𝐴)
5115fvmpt2 6278 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5250, 18, 51syl2anc 692 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5349, 52eqtr2d 2655 . . . . . . . . . . . 12 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
5453breq2d 4656 . . . . . . . . . . 11 ((𝜑𝑛𝑍𝑥𝐴) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5541, 42, 47, 54syl3anc 1324 . . . . . . . . . 10 (((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) ∧ 𝑛𝑍) → (𝑦𝐵𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5640, 55ralbida 2979 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∀𝑛𝑍 𝑦𝐵 ↔ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
5736, 56rexbid 3047 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
583, 57rabbida 39094 . . . . . . 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 2654 . . . . . 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 2654 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
613, 60alrimi 2080 . . . 4 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom ((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)})
62 nfcv 2762 . . . . . . . . . . . . . 14 𝑛
63 nfra1 2938 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦𝐵
6462, 63nfrex 3004 . . . . . . . . . . . . 13 𝑛𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵
65 nfii1 4542 . . . . . . . . . . . . 13 𝑛 𝑛𝑍 𝐴
6664, 65nfrab 3118 . . . . . . . . . . . 12 𝑛{𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}
674, 66nfcxfr 2760 . . . . . . . . . . 11 𝑛𝐷
6837, 67nfel 2774 . . . . . . . . . 10 𝑛 𝑥𝐷
696, 68nfan 1826 . . . . . . . . 9 𝑛(𝜑𝑥𝐷)
70 simpll 789 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝜑)
71 simpr 477 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑛𝑍)
724eleq2i 2691 . . . . . . . . . . . . . . 15 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
7372biimpi 206 . . . . . . . . . . . . . 14 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵})
74 rabidim1 3112 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
78 eliinid 39114 . . . . . . . . . . . 12 ((𝑥 𝑛𝑍 𝐴𝑛𝑍) → 𝑥𝐴)
7976, 77, 78syl2anc 692 . . . . . . . . . . 11 ((𝑥𝐷𝑛𝑍) → 𝑥𝐴)
8079adantll 749 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥𝐴)
8153idi 2 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥𝐴) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8270, 71, 80, 81syl3anc 1324 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝐵 = (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥))
8369, 82mpteq2da 4734 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝑛𝑍𝐵) = (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8483rneqd 5342 . . . . . . 7 ((𝜑𝑥𝐷) → ran (𝑛𝑍𝐵) = ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)))
8584infeq1d 8368 . . . . . 6 ((𝜑𝑥𝐷) → inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
8685ex 450 . . . . 5 (𝜑 → (𝑥𝐷 → inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < )))
873, 86ralrimi 2954 . . . 4 (𝜑 → ∀𝑥𝐷 inf(ran (𝑛𝑍𝐵), ℝ, < ) = inf(ran (𝑛𝑍 ↦ (((𝑛𝑍 ↦ (𝑥𝐴𝐵))‘𝑛)‘𝑥)), ℝ, < ))
88 mpteq12f 4722 . . . 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 692 . . 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 2654 . 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 4738 . . 3 𝑛(𝑛𝑍 ↦ (𝑥𝐴𝐵))
92 smfinfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
93 smfinfmpt.z . . 3 𝑍 = (ℤ𝑀)
94 eqid 2620 . . . 4 (𝑛𝑍 ↦ (𝑥𝐴𝐵)) = (𝑛𝑍 ↦ (𝑥𝐴𝐵))
956, 8, 94fmptdf 6373 . . 3 (𝜑 → (𝑛𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
96 eqid 2620 . . 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 2620 . . 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 40787 . 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 2699 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036  ∀wal 1479   = wceq 1481  Ⅎwnf 1706   ∈ wcel 1988  ∀wral 2909  ∃wrex 2910  {crab 2913  ∩ ciin 4512   class class class wbr 4644   ↦ cmpt 4720  dom cdm 5104  ran crn 5105  ‘cfv 5876  infcinf 8332  ℝcr 9920   < clt 10059   ≤ cle 10060  ℤcz 11362  ℤ≥cuz 11672  SAlgcsalg 40291  SMblFncsmblfn 40672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cc 9242  ax-ac2 9270  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-ioo 12164  df-ioc 12165  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-hash 13101  df-word 13282  df-concat 13284  df-s1 13285  df-s2 13574  df-s3 13575  df-s4 13576  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-rest 16064  df-topgen 16085  df-top 20680  df-bases 20731  df-salg 40292  df-salgen 40296  df-smblfn 40673 This theorem is referenced by: (None)
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