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Theorem smfinflem 46473
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 1910 . . . . 5 𝑛(𝜑𝑥𝐷)
4 smfinflem.m . . . . . . 7 (𝜑𝑀 ∈ ℤ)
5 smfinflem.z . . . . . . 7 𝑍 = (ℤ𝑀)
64, 5uzn0d 45075 . . . . . 6 (𝜑𝑍 ≠ ∅)
76adantr 479 . . . . 5 ((𝜑𝑥𝐷) → 𝑍 ≠ ∅)
8 smfinflem.s . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
98adantr 479 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
10 smfinflem.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1110ffvelcdmda 7089 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
12 eqid 2726 . . . . . . . 8 dom (𝐹𝑛) = dom (𝐹𝑛)
139, 11, 12smff 46388 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
1413adantlr 713 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → (𝐹𝑛):dom (𝐹𝑛)⟶ℝ)
15 ssrab2 4075 . . . . . . . . . 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 2818 . . . . . . . . . . 11 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
1817biimpi 215 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)})
1915, 18sselid 3978 . . . . . . . . 9 (𝑥𝐷𝑥 𝑛𝑍 dom (𝐹𝑛))
2019adantr 479 . . . . . . . 8 ((𝑥𝐷𝑛𝑍) → 𝑥 𝑛𝑍 dom (𝐹𝑛))
21 simpr 483 . . . . . . . 8 ((𝑥𝐷𝑛𝑍) → 𝑛𝑍)
22 eliinid 44748 . . . . . . . 8 ((𝑥 𝑛𝑍 dom (𝐹𝑛) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2320, 21, 22syl2anc 582 . . . . . . 7 ((𝑥𝐷𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2423adantll 712 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → 𝑥 ∈ dom (𝐹𝑛))
2514, 24ffvelcdmd 7090 . . . . 5 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
26 rabidim2 44739 . . . . . . 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 480 . . . . 5 ((𝜑𝑥𝐷) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥))
293, 7, 25, 28infnsuprnmpt 44894 . . . 4 ((𝜑𝑥𝐷) → inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ) = -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ))
3029mpteq2dva 5245 . . 3 (𝜑 → (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < )) = (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
312, 30eqtrd 2766 . 2 (𝜑𝐺 = (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )))
32 nfv 1910 . . 3 𝑥𝜑
33 fvex 6905 . . . . . . . 8 (𝐹𝑛) ∈ V
3433dmex 7913 . . . . . . 7 dom (𝐹𝑛) ∈ V
3534rgenw 3055 . . . . . 6 𝑛𝑍 dom (𝐹𝑛) ∈ V
3635a1i 11 . . . . 5 (𝜑 → ∀𝑛𝑍 dom (𝐹𝑛) ∈ V)
376, 36iinexd 44770 . . . 4 (𝜑 𝑛𝑍 dom (𝐹𝑛) ∈ V)
3816, 37rabexd 5332 . . 3 (𝜑𝐷 ∈ V)
3925renegcld 11681 . . . 4 (((𝜑𝑥𝐷) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
40 fveq2 6892 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ((𝐹𝑚)‘𝑤) = ((𝐹𝑚)‘𝑥))
4140breq2d 5157 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
4241ralbidv 3168 . . . . . . . . . 10 (𝑤 = 𝑥 → (∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
4342rexbidv 3169 . . . . . . . . 9 (𝑤 = 𝑥 → (∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
44 nfcv 2892 . . . . . . . . . . 11 𝑤 𝑛𝑍 dom (𝐹𝑛)
45 nfcv 2892 . . . . . . . . . . . 12 𝑥𝑍
46 nfcv 2892 . . . . . . . . . . . . 13 𝑥(𝐹𝑚)
4746nfdm 5949 . . . . . . . . . . . 12 𝑥dom (𝐹𝑚)
4845, 47nfiin 5026 . . . . . . . . . . 11 𝑥 𝑚𝑍 dom (𝐹𝑚)
49 nfv 1910 . . . . . . . . . . 11 𝑤𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)
50 nfv 1910 . . . . . . . . . . 11 𝑥𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)
51 nfcv 2892 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑛)
52 nfcv 2892 . . . . . . . . . . . . . 14 𝑛(𝐹𝑚)
5352nfdm 5949 . . . . . . . . . . . . 13 𝑛dom (𝐹𝑚)
54 fveq2 6892 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
5554dmeqd 5904 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → dom (𝐹𝑛) = dom (𝐹𝑚))
5651, 53, 55cbviin 5039 . . . . . . . . . . . 12 𝑛𝑍 dom (𝐹𝑛) = 𝑚𝑍 dom (𝐹𝑚)
5756a1i 11 . . . . . . . . . . 11 (𝑥 = 𝑤 𝑛𝑍 dom (𝐹𝑛) = 𝑚𝑍 dom (𝐹𝑚))
58 fveq2 6892 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝐹𝑛)‘𝑥) = ((𝐹𝑛)‘𝑤))
5958breq2d 5157 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ 𝑦 ≤ ((𝐹𝑛)‘𝑤)))
6059ralbidv 3168 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤)))
61 nfv 1910 . . . . . . . . . . . . . . . 16 𝑚 𝑦 ≤ ((𝐹𝑛)‘𝑤)
62 nfcv 2892 . . . . . . . . . . . . . . . . 17 𝑛𝑦
63 nfcv 2892 . . . . . . . . . . . . . . . . 17 𝑛
64 nfcv 2892 . . . . . . . . . . . . . . . . . 18 𝑛𝑤
6552, 64nffv 6902 . . . . . . . . . . . . . . . . 17 𝑛((𝐹𝑚)‘𝑤)
6662, 63, 65nfbr 5192 . . . . . . . . . . . . . . . 16 𝑛 𝑦 ≤ ((𝐹𝑚)‘𝑤)
6754fveq1d 6894 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐹𝑛)‘𝑤) = ((𝐹𝑚)‘𝑤))
6867breq2d 5157 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
6961, 66, 68cbvralw 3294 . . . . . . . . . . . . . . 15 (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤))
7069a1i 11 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑤) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
7160, 70bitrd 278 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
7271rexbidv 3169 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤)))
73 breq1 5148 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7473ralbidv 3168 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (∀𝑚𝑍 𝑦 ≤ ((𝐹𝑚)‘𝑤) ↔ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7574cbvrexvw 3226 . . . . . . . . . . . . 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 278 . . . . . . . . . . 11 (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)))
7844, 48, 49, 50, 57, 77cbvrabcsfw 3937 . . . . . . . . . 10 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} = {𝑤 𝑚𝑍 dom (𝐹𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)}
7916, 78eqtri 2754 . . . . . . . . 9 𝐷 = {𝑤 𝑚𝑍 dom (𝐹𝑚) ∣ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑤)}
8043, 79elrab2 3685 . . . . . . . 8 (𝑥𝐷 ↔ (𝑥 𝑚𝑍 dom (𝐹𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
8180biimpi 215 . . . . . . 7 (𝑥𝐷 → (𝑥 𝑚𝑍 dom (𝐹𝑚) ∧ ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)))
8281simprd 494 . . . . . 6 (𝑥𝐷 → ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥))
8382adantl 480 . . . . 5 ((𝜑𝑥𝐷) → ∃𝑧 ∈ ℝ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥))
84 renegcl 11563 . . . . . . . 8 (𝑧 ∈ ℝ → -𝑧 ∈ ℝ)
8584ad2antlr 725 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → -𝑧 ∈ ℝ)
86 fveq2 6892 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
8786fveq1d 6894 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑛)‘𝑥))
8887breq2d 5157 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑧 ≤ ((𝐹𝑚)‘𝑥) ↔ 𝑧 ≤ ((𝐹𝑛)‘𝑥)))
8988rspcva 3607 . . . . . . . . . . 11 ((𝑛𝑍 ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
9089ancoms 457 . . . . . . . . . 10 ((∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥) ∧ 𝑛𝑍) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
9190adantll 712 . . . . . . . . 9 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → 𝑧 ≤ ((𝐹𝑛)‘𝑥))
92 simpllr 774 . . . . . . . . . 10 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → 𝑧 ∈ ℝ)
9325ad4ant14 750 . . . . . . . . . 10 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
9492, 93lenegd 11833 . . . . . . . . 9 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → (𝑧 ≤ ((𝐹𝑛)‘𝑥) ↔ -((𝐹𝑛)‘𝑥) ≤ -𝑧))
9591, 94mpbid 231 . . . . . . . 8 (((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) ∧ 𝑛𝑍) → -((𝐹𝑛)‘𝑥) ≤ -𝑧)
9695ralrimiva 3136 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧)
97 brralrspcev 5205 . . . . . . 7 ((-𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑧) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
9885, 96, 97syl2anc 582 . . . . . 6 ((((𝜑𝑥𝐷) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑚𝑍 𝑧 ≤ ((𝐹𝑚)‘𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑦)
9998rexlimdva2 3147 . . . . 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 44895 . . 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 1910 . . . . . . . . . 10 𝑦(𝜑𝑥 𝑛𝑍 dom (𝐹𝑛))
104 nfv 1910 . . . . . . . . . 10 𝑦𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧
105 renegcl 11563 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1061053ad2ant2 1131 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → -𝑦 ∈ ℝ)
107 nfv 1910 . . . . . . . . . . . . . . 15 𝑛𝜑
108 nfcv 2892 . . . . . . . . . . . . . . . 16 𝑛𝑥
109 nfii1 5031 . . . . . . . . . . . . . . . 16 𝑛 𝑛𝑍 dom (𝐹𝑛)
110108, 109nfel 2907 . . . . . . . . . . . . . . 15 𝑛 𝑥 𝑛𝑍 dom (𝐹𝑛)
111107, 110nfan 1895 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥 𝑛𝑍 dom (𝐹𝑛))
11262nfel1 2909 . . . . . . . . . . . . . 14 𝑛 𝑦 ∈ ℝ
113 nfra1 3272 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)
114111, 112, 113nf3an 1897 . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝜑)
117 simpr 483 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑛𝑍) → 𝑛𝑍)
11822adantll 712 . . . . . . . . . . . . . . . . 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, 120ffvelcdmd 7090 . . . . . . . . . . . . . . . . 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 3236 . . . . . . . . . . . . . . . 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 11757 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 13 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → (𝑛𝑍 → -((𝐹𝑛)‘𝑥) ≤ -𝑦))
130114, 129ralrimi 3245 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑦 ∈ ℝ ∧ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)) → ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦)
131 brralrspcev 5205 . . . . . . . . . . . 12 ((-𝑦 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧)
132106, 130, 131syl2anc 582 . . . . . . . . . . 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 3254 . . . . . . . . 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 1910 . . . . . . . . . . . . . 14 𝑛 𝑧 ∈ ℝ
137 nfra1 3272 . . . . . . . . . . . . . 14 𝑛𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧
138111, 136, 137nf3an 1897 . . . . . . . . . . . . 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 3236 . . . . . . . . . . . . . . . 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 11563 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑛)‘𝑥) ∈ ℝ → -((𝐹𝑛)‘𝑥) ∈ ℝ)
145144adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
146 simpr 483 . . . . . . . . . . . . . . . . . . 19 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
147 leneg 11757 . . . . . . . . . . . . . . . . . . 19 ((-((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-((𝐹𝑛)‘𝑥) ≤ 𝑧 ↔ -𝑧 ≤ --((𝐹𝑛)‘𝑥)))
148145, 146, 147syl2anc 582 . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . 16 ((((𝐹𝑛)‘𝑥) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ -((𝐹𝑛)‘𝑥) ≤ 𝑧) → -𝑧 ≤ --((𝐹𝑛)‘𝑥))
151 recn 11238 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑛)‘𝑥) ∈ ℝ → ((𝐹𝑛)‘𝑥) ∈ ℂ)
152151negnegd 11602 . . . . . . . . . . . . . . . . 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 5171 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 13 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → (𝑛𝑍 → -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
157138, 156ralrimi 3245 . . . . . . . . . . . 12 (((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) ∧ 𝑧 ∈ ℝ ∧ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧) → ∀𝑛𝑍 -𝑧 ≤ ((𝐹𝑛)‘𝑥))
158 breq1 5148 . . . . . . . . . . . . . 14 (𝑦 = -𝑧 → (𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ -𝑧 ≤ ((𝐹𝑛)‘𝑥)))
159158ralbidv 3168 . . . . . . . . . . . . 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 582 . . . . . . . . . . 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 3143 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)))
164134, 163impbid 211 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 dom (𝐹𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥) ↔ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧))
16532, 164rabbida 3447 . . . . . . 7 (𝜑 → {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)} = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
166102, 165eqtrd 2766 . . . . . 6 (𝜑𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
16732, 166alrimi 2202 . . . . 5 (𝜑 → ∀𝑥 𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧})
168 eqid 2726 . . . . . . 7 sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )
169168rgenw 3055 . . . . . 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 5233 . . . . 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 582 . . . 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 1910 . . . . 5 𝑧𝜑
174121renegcld 11681 . . . . 5 ((𝜑𝑛𝑍𝑥 ∈ dom (𝐹𝑛)) → -((𝐹𝑛)‘𝑥) ∈ ℝ)
175 nfv 1910 . . . . . 6 𝑥(𝜑𝑛𝑍)
17634a1i 11 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ∈ V)
1771213expa 1115 . . . . . 6 (((𝜑𝑛𝑍) ∧ 𝑥 ∈ dom (𝐹𝑛)) → ((𝐹𝑛)‘𝑥) ∈ ℝ)
17813feqmptd 6962 . . . . . . . 8 ((𝜑𝑛𝑍) → (𝐹𝑛) = (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)))
179178eqcomd 2732 . . . . . . 7 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)) = (𝐹𝑛))
180179, 11eqeltrd 2826 . . . . . 6 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ ((𝐹𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
181175, 9, 176, 177, 180smfneg 46459 . . . . 5 ((𝜑𝑛𝑍) → (𝑥 ∈ dom (𝐹𝑛) ↦ -((𝐹𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
182 eqid 2726 . . . . 5 {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧}
183 eqid 2726 . . . . 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 46471 . . . 4 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑧 ∈ ℝ ∀𝑛𝑍 -((𝐹𝑛)‘𝑥) ≤ 𝑧} ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
185172, 184eqeltrd 2826 . . 3 (𝜑 → (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
18632, 8, 38, 101, 185smfneg 46459 . 2 (𝜑 → (𝑥𝐷 ↦ -sup(ran (𝑛𝑍 ↦ -((𝐹𝑛)‘𝑥)), ℝ, < )) ∈ (SMblFn‘𝑆))
18731, 186eqeltrd 2826 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1532   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3420  Vcvv 3464  c0 4324   ciin 4996   class class class wbr 5145  cmpt 5228  dom cdm 5674  ran crn 5675  wf 6541  cfv 6545  supcsup 9475  infcinf 9476  cr 11147   < clt 11288  cle 11289  -cneg 11485  cz 12603  cuz 12867  SAlgcsalg 45964  SMblFncsmblfn 46351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737  ax-inf2 9676  ax-cc 10468  ax-ac2 10496  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225  ax-pre-sup 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-int 4949  df-iun 4997  df-iin 4998  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-isom 6554  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-omul 8492  df-er 8725  df-map 8848  df-pm 8849  df-en 8966  df-dom 8967  df-sdom 8968  df-fin 8969  df-sup 9477  df-inf 9478  df-oi 9545  df-card 9974  df-acn 9977  df-ac 10151  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12258  df-2 12320  df-3 12321  df-4 12322  df-n0 12518  df-z 12604  df-uz 12868  df-q 12978  df-rp 13022  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13532  df-fzo 13675  df-fl 13805  df-seq 14015  df-exp 14075  df-hash 14342  df-word 14517  df-concat 14573  df-s1 14598  df-s2 14851  df-s3 14852  df-s4 14853  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-rest 17431  df-topgen 17452  df-top 22883  df-bases 22936  df-salg 45965  df-salgen 45969  df-smblfn 46352
This theorem is referenced by:  smfinf  46474
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