Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smfinflem Structured version   Visualization version   GIF version

Theorem smfinflem 41344
 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
smfinflem.m (𝜑𝑀 ∈ ℤ)
smfinflem.z 𝑍 = (ℤ𝑀)
smfinflem.s (𝜑𝑆 ∈ SAlg)
smfinflem.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smfinflem.d 𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)}
smfinflem.g 𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
Assertion
Ref Expression
smfinflem (𝜑𝐺 ∈ (SMblFn‘𝑆))
Distinct variable groups:   𝐷,𝑛,𝑥,𝑦   𝑛,𝐹,𝑥,𝑦   𝑆,𝑛   𝑛,𝑍,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑛)   𝑀(𝑥,𝑦,𝑛)

Proof of Theorem smfinflem
Dummy variables 𝑚 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smfinflem.g . . . 4 𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < )))
3 nfv 1883 . . . . 5 𝑛(𝜑𝑥𝐷)
4 smfinflem.m . . . . . . 7 (𝜑𝑀 ∈ ℤ)
5 smfinflem.z . . . . . . 7 𝑍 = (ℤ𝑀)
64, 5uzn0d 39965 . . . . . 6 (𝜑𝑍 ≠ ∅)
76adantr 480 . . . . 5 ((𝜑𝑥𝐷) → 𝑍 ≠ ∅)
8 smfinflem.s . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
98adantr 480 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
10 smfinflem.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1110ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
12 eqid 2651 . . . . . . . 8 dom (𝐹𝑛) = dom (𝐹𝑛)
139, 11, 12smff 41262 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
1413adantlr 751 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
15 ssrab2 3720 . . . . . . . . . 10 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} ⊆ 𝑛𝑍 dom (𝐹𝑛)
16 smfinflem.d . . . . . . . . . . . 12 𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)}
1716eleq2i 2722 . . . . . . . . . . 11 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
1817biimpi 206 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
1915, 18sseldi 3634 . . . . . . . . 9 (𝑥𝐷𝑥 𝑛𝑍 dom (𝐹𝑛))
2019adantr 480 . . . . . . . 8 ((𝑥𝐷𝑛𝑍) → 𝑥 𝑛𝑍 dom (𝐹𝑛))
21 simpr 476 . . . . . . . 8 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
22 eliinid 39608 . . . . . . . 8 ((𝑥 𝑛𝑍 dom (𝐹𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2320, 21, 22syl2anc 694 . . . . . . 7 ((𝑥𝐷𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2423adantll 750 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2514, 24ffvelrnd 6400 . . . . 5 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
26 rabidim2 39598 . . . . . . 7 (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
2718, 26syl 17 . . . . . 6 (𝑥𝐷 → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
2827adantl 481 . . . . 5 ((𝜑𝑥𝐷) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
293, 7, 25, 28infnsuprnmpt 39779 . . . 4 ((𝜑𝑥𝐷) → inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ))
3029mpteq2dva 4777 . . 3 (𝜑 → (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < )) = (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
312, 30eqtrd 2685 . 2 (𝜑𝐺 = (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
32 nfv 1883 . . 3 𝑥𝜑
33 fvex 6239 . . . . . . . 8 (𝐹𝑛) ∈ V
3433dmex 7141 . . . . . . 7 dom (𝐹𝑛) ∈ V
3534rgenw 2953 . . . . . 6 𝑛𝑍 dom (𝐹𝑛) ∈ V
3635a1i 11 . . . . 5 (𝜑 → ∀𝑛𝑍 dom (𝐹𝑛) ∈ V)
376, 36iinexd 39632 . . . 4 (𝜑 𝑛𝑍 dom (𝐹𝑛) ∈ V)
3816, 37rabexd 4846 . . 3 (𝜑𝐷 ∈ V)
3925renegcld 10495 . . . 4 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
40 fveq2 6229 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝐹𝑚)‘𝑤) = ((𝐹𝑚)‘𝑥))
4140breq2d 4697 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
4241ralbidv 3015 . . . . . . . . . 10 (𝑤 = 𝑥 → (∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
4342rexbidv 3081 . . . . . . . . 9 (𝑤 = 𝑥 → (∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
44 nfcv 2793 . . . . . . . . . . 11 𝑤 𝑛𝑍 dom (𝐹𝑛)
45 nfcv 2793 . . . . . . . . . . . 12 𝑥𝑍
46 nfcv 2793 . . . . . . . . . . . . 13 𝑥(𝐹𝑚)
4746nfdm 5399 . . . . . . . . . . . 12 𝑥dom (𝐹𝑚)
4845, 47nfiin 4581 . . . . . . . . . . 11 𝑥 𝑚𝑍 dom (𝐹𝑚)
49 nfv 1883 . . . . . . . . . . 11 𝑤𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)
50 nfv 1883 . . . . . . . . . . 11 𝑥𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)
51 nfcv 2793 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑛)
52 nfcv 2793 . . . . . . . . . . . . . 14 𝑛(𝐹𝑚)
5352nfdm 5399 . . . . . . . . . . . . 13 𝑛dom (𝐹𝑚)
54 fveq2 6229 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
5554dmeqd 5358 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → dom (𝐹𝑛) = dom (𝐹𝑚))
5651, 53, 55cbviin 4590 . . . . . . . . . . . 12 𝑛𝑍 dom (𝐹𝑛) = 𝑚𝑍 dom (𝐹𝑚)
5756a1i 11 . . . . . . . . . . 11 (𝑥 = 𝑤 𝑛𝑍 dom (𝐹𝑛) = 𝑚𝑍 dom (𝐹𝑚))
58 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑤))
5958breq2d 4697 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ 𝑦 ≤ ((𝐹𝑛)‘𝑤)))
6059ralbidv 3015 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤)))
61 nfv 1883 . . . . . . . . . . . . . . . 16 𝑚 𝑦 ≤ ((𝐹𝑛)‘𝑤)
62 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑛𝑦
63 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑛
64 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑛𝑤
6552, 64nffv 6236 . . . . . . . . . . . . . . . . 17 𝑛((𝐹𝑚)‘𝑤)
6662, 63, 65nfbr 4732 . . . . . . . . . . . . . . . 16 𝑛 𝑦 ≤ ((𝐹𝑚)‘𝑤)
6754fveq1d 6231 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑤) = ((𝐹𝑚)‘𝑤))
6867breq2d 4697 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
6961, 66, 68cbvral 3197 . . . . . . . . . . . . . . 15 (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤))
7069a1i 11 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
7160, 70bitrd 268 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
7271rexbidv 3081 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
73 breq1 4688 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7473ralbidv 3015 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7574cbvrexv 3202 . . . . . . . . . . . . 13 (∃𝑦 ∈ ℝ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤))
7675a1i 11 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7772, 76bitrd 268 . . . . . . . . . . 11 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7844, 48, 49, 50, 57, 77cbvrabcsf 3601 . . . . . . . . . 10 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} = {𝑤 𝑚𝑍 dom (𝐹𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)}
7916, 78eqtri 2673 . . . . . . . . 9 𝐷 = {𝑤 𝑚𝑍 dom (𝐹𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)}
8043, 79elrab2 3399 . . . . . . . 8 (𝑥𝐷 ↔ (𝑥 𝑚𝑍 dom (𝐹𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
8180biimpi 206 . . . . . . 7 (𝑥𝐷 → (𝑥 𝑚𝑍 dom (𝐹𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
8281simprd 478 . . . . . 6 (𝑥𝐷 → ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥))
8382adantl 481 . . . . 5 ((𝜑𝑥𝐷) → ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥))
84 renegcl 10382 . . . . . . . . 9 (𝑧 ∈ ℝ → -𝑧 ∈ ℝ)
8584ad2antlr 763 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → -𝑧 ∈ ℝ)
86 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
8786fveq1d 6231 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑛)‘𝑥))
8887breq2d 4697 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑧 ≤ ((𝐹𝑚)‘𝑥) ↔ 𝑧 ≤ ((𝐹𝑛)‘𝑥)))
8988rspcva 3338 . . . . . . . . . . . 12 ((𝑛𝑍 ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
9089ancoms 468 . . . . . . . . . . 11 ((∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥) ∧ 𝑛𝑍) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
9190adantll 750 . . . . . . . . . 10 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
92 simpllr 815 . . . . . . . . . . 11 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → 𝑧 ∈ ℝ)
9325ad4ant14 1317 . . . . . . . . . . 11 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
9492, 93lenegd 10644 . . . . . . . . . 10 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → (𝑧 ≤ ((𝐹𝑛)‘𝑥) ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑧))
9591, 94mpbid 222 . . . . . . . . 9 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ -𝑧)
9695ralrimiva 2995 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧)
97 breq2 4689 . . . . . . . . . 10 (𝑦 = -𝑧 → (-((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑧))
9897ralbidv 3015 . . . . . . . . 9 (𝑦 = -𝑧 → (∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧))
9998rspcev 3340 . . . . . . . 8 ((-𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
10085, 96, 99syl2anc 694 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
101100ex 449 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) → (∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦))
102101rexlimdva 3060 . . . . 5 ((𝜑𝑥𝐷) → (∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦))
10383, 102mpd 15 . . . 4 ((𝜑𝑥𝐷) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
1043, 7, 39, 103suprclrnmpt 39780 . . 3 ((𝜑𝑥𝐷) → sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
10516a1i 11 . . . . . . 7 (𝜑𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
106 nfv 1883 . . . . . . . . . 10 𝑦(𝜑𝑥 𝑛𝑍 dom (𝐹𝑛))
107 nfv 1883 . . . . . . . . . 10 𝑦𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧
108 renegcl 10382 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1091083ad2ant2 1103 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → -𝑦 ∈ ℝ)
110 nfv 1883 . . . . . . . . . . . . . . 15 𝑛𝜑
111 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑛𝑥
112 nfii1 4583 . . . . . . . . . . . . . . . 16 𝑛 𝑛𝑍 dom (𝐹𝑛)
113111, 112nfel 2806 . . . . . . . . . . . . . . 15 𝑛 𝑥 𝑛𝑍 dom (𝐹𝑛)
114110, 113nfan 1868 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥 𝑛𝑍 dom (𝐹𝑛))
11562nfel1 2808 . . . . . . . . . . . . . 14 𝑛 𝑦 ∈ ℝ
116 nfra1 2970 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)
117114, 115, 116nf3an 1871 . . . . . . . . . . . . 13 𝑛((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
118 simpl2 1085 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
119 simpll 805 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝜑)
120 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
12122adantll 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
122133adant3 1101 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
123 simp3 1083 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → 𝑥 ∈ dom (𝐹𝑛))
124122, 123ffvelrnd 6400 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
125119, 120, 121, 124syl3anc 1366 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
1261253ad2antl1 1243 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
127 rspa 2959 . . . . . . . . . . . . . . . 16 ((∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ∧ 𝑛𝑍) → 𝑦 ≤ ((𝐹𝑛)‘𝑥))
1281273ad2antl3 1245 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → 𝑦 ≤ ((𝐹𝑛)‘𝑥))
129 leneg 10569 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ ((𝐹𝑛)‘𝑥) ∈ ℝ) → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑦))
130129biimp3a 1472 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ ∧ ((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → -((𝐹𝑛)‘𝑥) ≤ -𝑦)
131118, 126, 128, 130syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ -𝑦)
132131ex 449 . . . . . . . . . . . . 13 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → (𝑛𝑍 → -((𝐹𝑛)‘𝑥) ≤ -𝑦))
133117, 132ralrimi 2986 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦)
134 breq2 4689 . . . . . . . . . . . . . 14 (𝑧 = -𝑦 → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑦))
135134ralbidv 3015 . . . . . . . . . . . . 13 (𝑧 = -𝑦 → (∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦))
136135rspcev 3340 . . . . . . . . . . . 12 ((-𝑦 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
137109, 133, 136syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
1381373exp 1283 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (𝑦 ∈ ℝ → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)))
139106, 107, 138rexlimd 3055 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧))
140843ad2ant2 1103 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ∈ ℝ)
141 nfv 1883 . . . . . . . . . . . . . 14 𝑛 𝑧 ∈ ℝ
142 nfra1 2970 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧
143114, 141, 142nf3an 1871 . . . . . . . . . . . . 13 𝑛((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
1441253ad2antl1 1243 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
145 simpl2 1085 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → 𝑧 ∈ ℝ)
146 rspa 2959 . . . . . . . . . . . . . . . 16 ((∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ 𝑧)
1471463ad2antl3 1245 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ 𝑧)
148 simp3 1083 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -((𝐹𝑛)‘𝑥) ≤ 𝑧)
149 renegcl 10382 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑛)‘𝑥) ∈ ℝ → -((𝐹𝑛)‘𝑥) ∈ ℝ)
150149adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
151 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
152 leneg 10569 . . . . . . . . . . . . . . . . . . 19 ((-((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
153150, 151, 152syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
1541533adant3 1101 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
155148, 154mpbid 222 . . . . . . . . . . . . . . . 16 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ --((𝐹𝑛)‘𝑥))
156 recn 10064 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑛)‘𝑥) ∈ ℝ → ((𝐹𝑛)‘𝑥) ∈ ℂ)
157156negnegd 10421 . . . . . . . . . . . . . . . . 17 (((𝐹𝑛)‘𝑥) ∈ ℝ → --((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
1581573ad2ant1 1102 . . . . . . . . . . . . . . . 16 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → --((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
159155, 158breqtrd 4711 . . . . . . . . . . . . . . 15 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ ((𝐹𝑛)‘𝑥))
160144, 145, 147, 159syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → -𝑧 ≤ ((𝐹𝑛)‘𝑥))
161160ex 449 . . . . . . . . . . . . 13 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → (𝑛𝑍 → -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
162143, 161ralrimi 2986 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥))
163 breq1 4688 . . . . . . . . . . . . . 14 (𝑦 = -𝑧 → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
164163ralbidv 3015 . . . . . . . . . . . . 13 (𝑦 = -𝑧 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
165164rspcev 3340 . . . . . . . . . . . 12 ((-𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
166140, 162, 165syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
1671663exp 1283 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (𝑧 ∈ ℝ → (∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))))
168167rexlimdv 3059 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)))
169139, 168impbid 202 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧))
17032, 169rabbida 39588 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
171105, 170eqtrd 2685 . . . . . 6 (𝜑𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
17232, 171alrimi 2120 . . . . 5 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
173 eqid 2651 . . . . . . 7 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )
174173rgenw 2953 . . . . . 6 𝑥𝐷 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )
175174a1i 11 . . . . 5 (𝜑 → ∀𝑥𝐷 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ))
176 mpteq12f 4764 . . . . 5 ((∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ∧ ∀𝑥𝐷 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) → (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
177172, 175, 176syl2anc 694 . . . 4 (𝜑 → (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
178 nfv 1883 . . . . 5 𝑧𝜑
179124renegcld 10495 . . . . 5 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
180 nfv 1883 . . . . . 6 𝑥(𝜑𝑛𝑍)
18134a1i 11 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ∈ V)
1821243expa 1284 . . . . . 6 (((𝜑𝑛𝑍) ∧ 𝑥 ∈ dom (𝐹𝑛)) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
18313feqmptd 6288 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹𝑛) = (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)))
184183eqcomd 2657 . . . . . . 7 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)) = (𝐹𝑛))
185184, 11eqeltrd 2730 . . . . . 6 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
186180, 9, 181, 182, 185smfneg 41331 . . . . 5 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ -((𝐹𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
187 eqid 2651 . . . . 5 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧}
188 eqid 2651 . . . . 5 (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ))
189110, 32, 178, 4, 5, 8, 179, 186, 187, 188smfsupmpt 41342 . . . 4 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
190177, 189eqeltrd 2730 . . 3 (𝜑 → (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
19132, 8, 38, 104, 190smfneg 41331 . 2 (𝜑 → (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
19231, 191eqeltrd 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   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231  ∅c0 3948  ∩ ciin 4553   class class class wbr 4685   ↦ cmpt 4762  dom cdm 5143  ran crn 5144  ⟶wf 5922  ‘cfv 5926  supcsup 8387  infcinf 8388  ℝcr 9973   < clt 10112   ≤ cle 10113  -cneg 10305  ℤcz 11415  ℤ≥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:  smfinf  41345
 Copyright terms: Public domain W3C validator