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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  cr 9973   < clt 10112  cle 10113  -cneg 10305  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:  smfinf  41345
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