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Theorem smfinflem 43374
 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 1916 . . . . 5 𝑛(𝜑𝑥𝐷)
4 smfinflem.m . . . . . . 7 (𝜑𝑀 ∈ ℤ)
5 smfinflem.z . . . . . . 7 𝑍 = (ℤ𝑀)
64, 5uzn0d 41988 . . . . . 6 (𝜑𝑍 ≠ ∅)
76adantr 484 . . . . 5 ((𝜑𝑥𝐷) → 𝑍 ≠ ∅)
8 smfinflem.s . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
98adantr 484 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
10 smfinflem.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1110ffvelrnda 6842 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
12 eqid 2824 . . . . . . . 8 dom (𝐹𝑛) = dom (𝐹𝑛)
139, 11, 12smff 43292 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
1413adantlr 714 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
15 ssrab2 4042 . . . . . . . . . 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 2907 . . . . . . . . . . 11 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
1817biimpi 219 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
1915, 18sseldi 3951 . . . . . . . . 9 (𝑥𝐷𝑥 𝑛𝑍 dom (𝐹𝑛))
2019adantr 484 . . . . . . . 8 ((𝑥𝐷𝑛𝑍) → 𝑥 𝑛𝑍 dom (𝐹𝑛))
21 simpr 488 . . . . . . . 8 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
22 eliinid 41669 . . . . . . . 8 ((𝑥 𝑛𝑍 dom (𝐹𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2320, 21, 22syl2anc 587 . . . . . . 7 ((𝑥𝐷𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2423adantll 713 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2514, 24ffvelrnd 6843 . . . . 5 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
26 rabidim2 41660 . . . . . . 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 485 . . . . 5 ((𝜑𝑥𝐷) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
293, 7, 25, 28infnsuprnmpt 41813 . . . 4 ((𝜑𝑥𝐷) → inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ))
3029mpteq2dva 5147 . . 3 (𝜑 → (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < )) = (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
312, 30eqtrd 2859 . 2 (𝜑𝐺 = (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
32 nfv 1916 . . 3 𝑥𝜑
33 fvex 6674 . . . . . . . 8 (𝐹𝑛) ∈ V
3433dmex 7611 . . . . . . 7 dom (𝐹𝑛) ∈ V
3534rgenw 3145 . . . . . 6 𝑛𝑍 dom (𝐹𝑛) ∈ V
3635a1i 11 . . . . 5 (𝜑 → ∀𝑛𝑍 dom (𝐹𝑛) ∈ V)
376, 36iinexd 41691 . . . 4 (𝜑 𝑛𝑍 dom (𝐹𝑛) ∈ V)
3816, 37rabexd 5222 . . 3 (𝜑𝐷 ∈ V)
3925renegcld 11065 . . . 4 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
40 fveq2 6661 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝐹𝑚)‘𝑤) = ((𝐹𝑚)‘𝑥))
4140breq2d 5064 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
4241ralbidv 3192 . . . . . . . . . 10 (𝑤 = 𝑥 → (∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
4342rexbidv 3289 . . . . . . . . 9 (𝑤 = 𝑥 → (∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
44 nfcv 2982 . . . . . . . . . . 11 𝑤 𝑛𝑍 dom (𝐹𝑛)
45 nfcv 2982 . . . . . . . . . . . 12 𝑥𝑍
46 nfcv 2982 . . . . . . . . . . . . 13 𝑥(𝐹𝑚)
4746nfdm 5810 . . . . . . . . . . . 12 𝑥dom (𝐹𝑚)
4845, 47nfiin 4936 . . . . . . . . . . 11 𝑥 𝑚𝑍 dom (𝐹𝑚)
49 nfv 1916 . . . . . . . . . . 11 𝑤𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)
50 nfv 1916 . . . . . . . . . . 11 𝑥𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)
51 nfcv 2982 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑛)
52 nfcv 2982 . . . . . . . . . . . . . 14 𝑛(𝐹𝑚)
5352nfdm 5810 . . . . . . . . . . . . 13 𝑛dom (𝐹𝑚)
54 fveq2 6661 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
5554dmeqd 5761 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → dom (𝐹𝑛) = dom (𝐹𝑚))
5651, 53, 55cbviin 4948 . . . . . . . . . . . 12 𝑛𝑍 dom (𝐹𝑛) = 𝑚𝑍 dom (𝐹𝑚)
5756a1i 11 . . . . . . . . . . 11 (𝑥 = 𝑤 𝑛𝑍 dom (𝐹𝑛) = 𝑚𝑍 dom (𝐹𝑚))
58 fveq2 6661 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑤))
5958breq2d 5064 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ 𝑦 ≤ ((𝐹𝑛)‘𝑤)))
6059ralbidv 3192 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤)))
61 nfv 1916 . . . . . . . . . . . . . . . 16 𝑚 𝑦 ≤ ((𝐹𝑛)‘𝑤)
62 nfcv 2982 . . . . . . . . . . . . . . . . 17 𝑛𝑦
63 nfcv 2982 . . . . . . . . . . . . . . . . 17 𝑛
64 nfcv 2982 . . . . . . . . . . . . . . . . . 18 𝑛𝑤
6552, 64nffv 6671 . . . . . . . . . . . . . . . . 17 𝑛((𝐹𝑚)‘𝑤)
6662, 63, 65nfbr 5099 . . . . . . . . . . . . . . . 16 𝑛 𝑦 ≤ ((𝐹𝑚)‘𝑤)
6754fveq1d 6663 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑤) = ((𝐹𝑚)‘𝑤))
6867breq2d 5064 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
6961, 66, 68cbvralw 3425 . . . . . . . . . . . . . . 15 (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤))
7069a1i 11 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
7160, 70bitrd 282 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
7271rexbidv 3289 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
73 breq1 5055 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7473ralbidv 3192 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7574cbvrexvw 3435 . . . . . . . . . . . . 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 282 . . . . . . . . . . 11 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7844, 48, 49, 50, 57, 77cbvrabcsfw 3907 . . . . . . . . . 10 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} = {𝑤 𝑚𝑍 dom (𝐹𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)}
7916, 78eqtri 2847 . . . . . . . . 9 𝐷 = {𝑤 𝑚𝑍 dom (𝐹𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)}
8043, 79elrab2 3669 . . . . . . . 8 (𝑥𝐷 ↔ (𝑥 𝑚𝑍 dom (𝐹𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
8180biimpi 219 . . . . . . 7 (𝑥𝐷 → (𝑥 𝑚𝑍 dom (𝐹𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
8281simprd 499 . . . . . 6 (𝑥𝐷 → ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥))
8382adantl 485 . . . . 5 ((𝜑𝑥𝐷) → ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥))
84 renegcl 10947 . . . . . . . 8 (𝑧 ∈ ℝ → -𝑧 ∈ ℝ)
8584ad2antlr 726 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → -𝑧 ∈ ℝ)
86 fveq2 6661 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
8786fveq1d 6663 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑛)‘𝑥))
8887breq2d 5064 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑧 ≤ ((𝐹𝑚)‘𝑥) ↔ 𝑧 ≤ ((𝐹𝑛)‘𝑥)))
8988rspcva 3607 . . . . . . . . . . 11 ((𝑛𝑍 ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
9089ancoms 462 . . . . . . . . . 10 ((∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥) ∧ 𝑛𝑍) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
9190adantll 713 . . . . . . . . 9 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
92 simpllr 775 . . . . . . . . . 10 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → 𝑧 ∈ ℝ)
9325ad4ant14 751 . . . . . . . . . 10 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
9492, 93lenegd 11217 . . . . . . . . 9 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → (𝑧 ≤ ((𝐹𝑛)‘𝑥) ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑧))
9591, 94mpbid 235 . . . . . . . 8 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ -𝑧)
9695ralrimiva 3177 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧)
97 brralrspcev 5112 . . . . . . 7 ((-𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
9885, 96, 97syl2anc 587 . . . . . 6 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
9998rexlimdva2 3279 . . . . 5 ((𝜑𝑥𝐷) → (∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦))
10083, 99mpd 15 . . . 4 ((𝜑𝑥𝐷) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
1013, 7, 39, 100suprclrnmpt 41814 . . 3 ((𝜑𝑥𝐷) → sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
10216a1i 11 . . . . . . 7 (𝜑𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
103 nfv 1916 . . . . . . . . . 10 𝑦(𝜑𝑥 𝑛𝑍 dom (𝐹𝑛))
104 nfv 1916 . . . . . . . . . 10 𝑦𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧
105 renegcl 10947 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1061053ad2ant2 1131 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → -𝑦 ∈ ℝ)
107 nfv 1916 . . . . . . . . . . . . . . 15 𝑛𝜑
108 nfcv 2982 . . . . . . . . . . . . . . . 16 𝑛𝑥
109 nfii1 4940 . . . . . . . . . . . . . . . 16 𝑛 𝑛𝑍 dom (𝐹𝑛)
110108, 109nfel 2996 . . . . . . . . . . . . . . 15 𝑛 𝑥 𝑛𝑍 dom (𝐹𝑛)
111107, 110nfan 1901 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥 𝑛𝑍 dom (𝐹𝑛))
11262nfel1 2998 . . . . . . . . . . . . . 14 𝑛 𝑦 ∈ ℝ
113 nfra1 3213 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)
114111, 112, 113nf3an 1903 . . . . . . . . . . . . 13 𝑛((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
115 simpl2 1189 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
116 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝜑)
117 simpr 488 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
11822adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
119133adant3 1129 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
120 simp3 1135 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → 𝑥 ∈ dom (𝐹𝑛))
121119, 120ffvelrnd 6843 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
122116, 117, 118, 121syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
1231223ad2antl1 1182 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
124 rspa 3201 . . . . . . . . . . . . . . . 16 ((∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ∧ 𝑛𝑍) → 𝑦 ≤ ((𝐹𝑛)‘𝑥))
1251243ad2antl3 1184 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → 𝑦 ≤ ((𝐹𝑛)‘𝑥))
126 leneg 11141 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ ((𝐹𝑛)‘𝑥) ∈ ℝ) → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑦))
127126biimp3a 1466 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ ∧ ((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → -((𝐹𝑛)‘𝑥) ≤ -𝑦)
128115, 123, 125, 127syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ -𝑦)
129128ex 416 . . . . . . . . . . . . 13 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → (𝑛𝑍 → -((𝐹𝑛)‘𝑥) ≤ -𝑦))
130114, 129ralrimi 3210 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦)
131 brralrspcev 5112 . . . . . . . . . . . 12 ((-𝑦 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
132106, 130, 131syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
1331323exp 1116 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (𝑦 ∈ ℝ → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)))
134103, 104, 133rexlimd 3309 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧))
135843ad2ant2 1131 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ∈ ℝ)
136 nfv 1916 . . . . . . . . . . . . . 14 𝑛 𝑧 ∈ ℝ
137 nfra1 3213 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧
138111, 136, 137nf3an 1903 . . . . . . . . . . . . 13 𝑛((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
1391223ad2antl1 1182 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
140 simpl2 1189 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → 𝑧 ∈ ℝ)
141 rspa 3201 . . . . . . . . . . . . . . . 16 ((∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ 𝑧)
1421413ad2antl3 1184 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ 𝑧)
143 simp3 1135 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -((𝐹𝑛)‘𝑥) ≤ 𝑧)
144 renegcl 10947 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑛)‘𝑥) ∈ ℝ → -((𝐹𝑛)‘𝑥) ∈ ℝ)
145144adantr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
146 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
147 leneg 11141 . . . . . . . . . . . . . . . . . . 19 ((-((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
148145, 146, 147syl2anc 587 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
1491483adant3 1129 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
150143, 149mpbid 235 . . . . . . . . . . . . . . . 16 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ --((𝐹𝑛)‘𝑥))
151 recn 10625 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑛)‘𝑥) ∈ ℝ → ((𝐹𝑛)‘𝑥) ∈ ℂ)
152151negnegd 10986 . . . . . . . . . . . . . . . . 17 (((𝐹𝑛)‘𝑥) ∈ ℝ → --((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
1531523ad2ant1 1130 . . . . . . . . . . . . . . . 16 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → --((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑥))
154150, 153breqtrd 5078 . . . . . . . . . . . . . . 15 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ ((𝐹𝑛)‘𝑥))
155139, 140, 142, 154syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) ∧ 𝑛𝑍) → -𝑧 ≤ ((𝐹𝑛)‘𝑥))
156155ex 416 . . . . . . . . . . . . 13 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → (𝑛𝑍 → -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
157138, 156ralrimi 3210 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥))
158 breq1 5055 . . . . . . . . . . . . . 14 (𝑦 = -𝑧 → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
159158ralbidv 3192 . . . . . . . . . . . . 13 (𝑦 = -𝑧 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
160159rspcev 3609 . . . . . . . . . . . 12 ((-𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
161135, 157, 160syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
1621613exp 1116 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (𝑧 ∈ ℝ → (∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))))
163162rexlimdv 3275 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)))
164134, 163impbid 215 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧))
16532, 164rabbida 3459 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
166102, 165eqtrd 2859 . . . . . 6 (𝜑𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
16732, 166alrimi 2215 . . . . 5 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
168 eqid 2824 . . . . . . 7 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )
169168rgenw 3145 . . . . . 6 𝑥𝐷 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )
170169a1i 11 . . . . 5 (𝜑 → ∀𝑥𝐷 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ))
171 mpteq12f 5135 . . . . 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, < )))
172167, 170, 171syl2anc 587 . . . 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, < )))
173 nfv 1916 . . . . 5 𝑧𝜑
174121renegcld 11065 . . . . 5 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
175 nfv 1916 . . . . . 6 𝑥(𝜑𝑛𝑍)
17634a1i 11 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ∈ V)
1771213expa 1115 . . . . . 6 (((𝜑𝑛𝑍) ∧ 𝑥 ∈ dom (𝐹𝑛)) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
17813feqmptd 6724 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹𝑛) = (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)))
179178eqcomd 2830 . . . . . . 7 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)) = (𝐹𝑛))
180179, 11eqeltrd 2916 . . . . . 6 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
181175, 9, 176, 177, 180smfneg 43361 . . . . 5 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ -((𝐹𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
182 eqid 2824 . . . . 5 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧}
183 eqid 2824 . . . . 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, < ))
184107, 32, 173, 4, 5, 8, 174, 181, 182, 183smfsupmpt 43372 . . . 4 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
185172, 184eqeltrd 2916 . . 3 (𝜑 → (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
18632, 8, 38, 101, 185smfneg 43361 . 2 (𝜑 → (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
18731, 186eqeltrd 2916 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  {crab 3137  Vcvv 3480  ∅c0 4276  ∩ ciin 4906   class class class wbr 5052   ↦ cmpt 5132  dom cdm 5542  ran crn 5543  ⟶wf 6339  ‘cfv 6343  supcsup 8901  infcinf 8902  ℝcr 10534   < clt 10673   ≤ cle 10674  -cneg 10869  ℤcz 11978  ℤ≥cuz 12240  SAlgcsalg 42876  SMblFncsmblfn 43260 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cc 9855  ax-ac2 9883  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-omul 8103  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-acn 9368  df-ac 9540  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-ioo 12739  df-ioc 12740  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-word 13867  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-s4 14212  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-rest 16696  df-topgen 16717  df-top 21502  df-bases 21554  df-salg 42877  df-salgen 42881  df-smblfn 43261 This theorem is referenced by:  smfinf  43375
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