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Theorem amgmlem 26188
Description: Lemma for amgm 26189. (Contributed by Mario Carneiro, 21-Jun-2015.)
Hypotheses
Ref Expression
amgm.1 𝑀 = (mulGrpβ€˜β„‚fld)
amgm.2 (πœ‘ β†’ 𝐴 ∈ Fin)
amgm.3 (πœ‘ β†’ 𝐴 β‰  βˆ…)
amgm.4 (πœ‘ β†’ 𝐹:π΄βŸΆβ„+)
Assertion
Ref Expression
amgmlem (πœ‘ β†’ ((𝑀 Ξ£g 𝐹)↑𝑐(1 / (β™―β€˜π΄))) ≀ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))

Proof of Theorem amgmlem
Dummy variables π‘Ž 𝑏 π‘˜ 𝑠 𝑒 𝑣 𝑀 π‘₯ 𝑦 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnfld0 20671 . . . . . . . 8 0 = (0gβ€˜β„‚fld)
2 cnring 20669 . . . . . . . . 9 β„‚fld ∈ Ring
3 ringabl 19868 . . . . . . . . 9 (β„‚fld ∈ Ring β†’ β„‚fld ∈ Abel)
42, 3mp1i 13 . . . . . . . 8 (πœ‘ β†’ β„‚fld ∈ Abel)
5 amgm.2 . . . . . . . 8 (πœ‘ β†’ 𝐴 ∈ Fin)
6 resubdrg 20862 . . . . . . . . . 10 (ℝ ∈ (SubRingβ€˜β„‚fld) ∧ ℝfld ∈ DivRing)
76simpli 485 . . . . . . . . 9 ℝ ∈ (SubRingβ€˜β„‚fld)
8 subrgsubg 20079 . . . . . . . . 9 (ℝ ∈ (SubRingβ€˜β„‚fld) β†’ ℝ ∈ (SubGrpβ€˜β„‚fld))
97, 8mp1i 13 . . . . . . . 8 (πœ‘ β†’ ℝ ∈ (SubGrpβ€˜β„‚fld))
10 amgm.4 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹:π΄βŸΆβ„+)
1110ffvelcdmda 6993 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (πΉβ€˜π‘˜) ∈ ℝ+)
1211relogcld 25827 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (logβ€˜(πΉβ€˜π‘˜)) ∈ ℝ)
1312renegcld 11452 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ -(logβ€˜(πΉβ€˜π‘˜)) ∈ ℝ)
1413fmpttd 7021 . . . . . . . 8 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜))):π΄βŸΆβ„)
15 c0ex 11019 . . . . . . . . . 10 0 ∈ V
1615a1i 11 . . . . . . . . 9 (πœ‘ β†’ 0 ∈ V)
1714, 5, 16fdmfifsupp 9186 . . . . . . . 8 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜))) finSupp 0)
181, 4, 5, 9, 14, 17gsumsubgcl 19570 . . . . . . 7 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) ∈ ℝ)
1918recnd 11053 . . . . . 6 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) ∈ β„‚)
20 amgm.3 . . . . . . . 8 (πœ‘ β†’ 𝐴 β‰  βˆ…)
21 hashnncl 14130 . . . . . . . . 9 (𝐴 ∈ Fin β†’ ((β™―β€˜π΄) ∈ β„• ↔ 𝐴 β‰  βˆ…))
225, 21syl 17 . . . . . . . 8 (πœ‘ β†’ ((β™―β€˜π΄) ∈ β„• ↔ 𝐴 β‰  βˆ…))
2320, 22mpbird 257 . . . . . . 7 (πœ‘ β†’ (β™―β€˜π΄) ∈ β„•)
2423nncnd 12039 . . . . . 6 (πœ‘ β†’ (β™―β€˜π΄) ∈ β„‚)
2523nnne0d 12073 . . . . . 6 (πœ‘ β†’ (β™―β€˜π΄) β‰  0)
2619, 24, 25divnegd 11814 . . . . 5 (πœ‘ β†’ -((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) = (-(β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)))
2712recnd 11053 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (logβ€˜(πΉβ€˜π‘˜)) ∈ β„‚)
285, 27gsumfsum 20714 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (logβ€˜(πΉβ€˜π‘˜)))) = Ξ£π‘˜ ∈ 𝐴 (logβ€˜(πΉβ€˜π‘˜)))
2927negnegd 11373 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ --(logβ€˜(πΉβ€˜π‘˜)) = (logβ€˜(πΉβ€˜π‘˜)))
3029sumeq2dv 15464 . . . . . . . . 9 (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 --(logβ€˜(πΉβ€˜π‘˜)) = Ξ£π‘˜ ∈ 𝐴 (logβ€˜(πΉβ€˜π‘˜)))
3113recnd 11053 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ -(logβ€˜(πΉβ€˜π‘˜)) ∈ β„‚)
325, 31fsumneg 15548 . . . . . . . . 9 (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 --(logβ€˜(πΉβ€˜π‘˜)) = -Ξ£π‘˜ ∈ 𝐴 -(logβ€˜(πΉβ€˜π‘˜)))
3328, 30, 323eqtr2rd 2783 . . . . . . . 8 (πœ‘ β†’ -Ξ£π‘˜ ∈ 𝐴 -(logβ€˜(πΉβ€˜π‘˜)) = (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (logβ€˜(πΉβ€˜π‘˜)))))
345, 31gsumfsum 20714 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) = Ξ£π‘˜ ∈ 𝐴 -(logβ€˜(πΉβ€˜π‘˜)))
3534negeqd 11265 . . . . . . . 8 (πœ‘ β†’ -(β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) = -Ξ£π‘˜ ∈ 𝐴 -(logβ€˜(πΉβ€˜π‘˜)))
3610feqmptd 6869 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜)))
37 relogf1o 25771 . . . . . . . . . . . . 13 (log β†Ύ ℝ+):ℝ+–1-1-onto→ℝ
38 f1of 6746 . . . . . . . . . . . . 13 ((log β†Ύ ℝ+):ℝ+–1-1-onto→ℝ β†’ (log β†Ύ ℝ+):ℝ+βŸΆβ„)
3937, 38mp1i 13 . . . . . . . . . . . 12 (πœ‘ β†’ (log β†Ύ ℝ+):ℝ+βŸΆβ„)
4039feqmptd 6869 . . . . . . . . . . 11 (πœ‘ β†’ (log β†Ύ ℝ+) = (π‘₯ ∈ ℝ+ ↦ ((log β†Ύ ℝ+)β€˜π‘₯)))
41 fvres 6823 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ+ β†’ ((log β†Ύ ℝ+)β€˜π‘₯) = (logβ€˜π‘₯))
4241mpteq2ia 5184 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ ↦ ((log β†Ύ ℝ+)β€˜π‘₯)) = (π‘₯ ∈ ℝ+ ↦ (logβ€˜π‘₯))
4340, 42eqtrdi 2792 . . . . . . . . . 10 (πœ‘ β†’ (log β†Ύ ℝ+) = (π‘₯ ∈ ℝ+ ↦ (logβ€˜π‘₯)))
44 fveq2 6804 . . . . . . . . . 10 (π‘₯ = (πΉβ€˜π‘˜) β†’ (logβ€˜π‘₯) = (logβ€˜(πΉβ€˜π‘˜)))
4511, 36, 43, 44fmptco 7033 . . . . . . . . 9 (πœ‘ β†’ ((log β†Ύ ℝ+) ∘ 𝐹) = (π‘˜ ∈ 𝐴 ↦ (logβ€˜(πΉβ€˜π‘˜))))
4645oveq2d 7323 . . . . . . . 8 (πœ‘ β†’ (β„‚fld Ξ£g ((log β†Ύ ℝ+) ∘ 𝐹)) = (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (logβ€˜(πΉβ€˜π‘˜)))))
4733, 35, 463eqtr4d 2786 . . . . . . 7 (πœ‘ β†’ -(β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) = (β„‚fld Ξ£g ((log β†Ύ ℝ+) ∘ 𝐹)))
48 amgm.1 . . . . . . . . . . . . . . 15 𝑀 = (mulGrpβ€˜β„‚fld)
4948oveq1i 7317 . . . . . . . . . . . . . 14 (𝑀 β†Ύs (β„‚ βˆ– {0})) = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))
5049rpmsubg 20711 . . . . . . . . . . . . 13 ℝ+ ∈ (SubGrpβ€˜(𝑀 β†Ύs (β„‚ βˆ– {0})))
51 subgsubm 18826 . . . . . . . . . . . . 13 (ℝ+ ∈ (SubGrpβ€˜(𝑀 β†Ύs (β„‚ βˆ– {0}))) β†’ ℝ+ ∈ (SubMndβ€˜(𝑀 β†Ύs (β„‚ βˆ– {0}))))
5250, 51ax-mp 5 . . . . . . . . . . . 12 ℝ+ ∈ (SubMndβ€˜(𝑀 β†Ύs (β„‚ βˆ– {0})))
53 cnfldbas 20650 . . . . . . . . . . . . . . 15 β„‚ = (Baseβ€˜β„‚fld)
54 cndrng 20676 . . . . . . . . . . . . . . 15 β„‚fld ∈ DivRing
5553, 1, 54drngui 20046 . . . . . . . . . . . . . 14 (β„‚ βˆ– {0}) = (Unitβ€˜β„‚fld)
5655, 48unitsubm 19961 . . . . . . . . . . . . 13 (β„‚fld ∈ Ring β†’ (β„‚ βˆ– {0}) ∈ (SubMndβ€˜π‘€))
57 eqid 2736 . . . . . . . . . . . . . 14 (𝑀 β†Ύs (β„‚ βˆ– {0})) = (𝑀 β†Ύs (β„‚ βˆ– {0}))
5857subsubm 18504 . . . . . . . . . . . . 13 ((β„‚ βˆ– {0}) ∈ (SubMndβ€˜π‘€) β†’ (ℝ+ ∈ (SubMndβ€˜(𝑀 β†Ύs (β„‚ βˆ– {0}))) ↔ (ℝ+ ∈ (SubMndβ€˜π‘€) ∧ ℝ+ βŠ† (β„‚ βˆ– {0}))))
592, 56, 58mp2b 10 . . . . . . . . . . . 12 (ℝ+ ∈ (SubMndβ€˜(𝑀 β†Ύs (β„‚ βˆ– {0}))) ↔ (ℝ+ ∈ (SubMndβ€˜π‘€) ∧ ℝ+ βŠ† (β„‚ βˆ– {0})))
6052, 59mpbi 229 . . . . . . . . . . 11 (ℝ+ ∈ (SubMndβ€˜π‘€) ∧ ℝ+ βŠ† (β„‚ βˆ– {0}))
6160simpli 485 . . . . . . . . . 10 ℝ+ ∈ (SubMndβ€˜π‘€)
62 eqid 2736 . . . . . . . . . . 11 (𝑀 β†Ύs ℝ+) = (𝑀 β†Ύs ℝ+)
6362submbas 18502 . . . . . . . . . 10 (ℝ+ ∈ (SubMndβ€˜π‘€) β†’ ℝ+ = (Baseβ€˜(𝑀 β†Ύs ℝ+)))
6461, 63ax-mp 5 . . . . . . . . 9 ℝ+ = (Baseβ€˜(𝑀 β†Ύs ℝ+))
65 cnfld1 20672 . . . . . . . . . . . 12 1 = (1rβ€˜β„‚fld)
6648, 65ringidval 19788 . . . . . . . . . . 11 1 = (0gβ€˜π‘€)
6762, 66subm0 18503 . . . . . . . . . 10 (ℝ+ ∈ (SubMndβ€˜π‘€) β†’ 1 = (0gβ€˜(𝑀 β†Ύs ℝ+)))
6861, 67ax-mp 5 . . . . . . . . 9 1 = (0gβ€˜(𝑀 β†Ύs ℝ+))
69 cncrng 20668 . . . . . . . . . . 11 β„‚fld ∈ CRing
7048crngmgp 19840 . . . . . . . . . . 11 (β„‚fld ∈ CRing β†’ 𝑀 ∈ CMnd)
7169, 70mp1i 13 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ CMnd)
7262submmnd 18501 . . . . . . . . . . 11 (ℝ+ ∈ (SubMndβ€˜π‘€) β†’ (𝑀 β†Ύs ℝ+) ∈ Mnd)
7361, 72mp1i 13 . . . . . . . . . 10 (πœ‘ β†’ (𝑀 β†Ύs ℝ+) ∈ Mnd)
7462subcmn 19487 . . . . . . . . . 10 ((𝑀 ∈ CMnd ∧ (𝑀 β†Ύs ℝ+) ∈ Mnd) β†’ (𝑀 β†Ύs ℝ+) ∈ CMnd)
7571, 73, 74syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (𝑀 β†Ύs ℝ+) ∈ CMnd)
76 df-refld 20859 . . . . . . . . . . . 12 ℝfld = (β„‚fld β†Ύs ℝ)
7776subrgring 20076 . . . . . . . . . . 11 (ℝ ∈ (SubRingβ€˜β„‚fld) β†’ ℝfld ∈ Ring)
787, 77ax-mp 5 . . . . . . . . . 10 ℝfld ∈ Ring
79 ringmnd 19842 . . . . . . . . . 10 (ℝfld ∈ Ring β†’ ℝfld ∈ Mnd)
8078, 79mp1i 13 . . . . . . . . 9 (πœ‘ β†’ ℝfld ∈ Mnd)
8148oveq1i 7317 . . . . . . . . . . . 12 (𝑀 β†Ύs ℝ+) = ((mulGrpβ€˜β„‚fld) β†Ύs ℝ+)
8281reloggim 25803 . . . . . . . . . . 11 (log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) GrpIso ℝfld)
83 gimghm 18929 . . . . . . . . . . 11 ((log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) GrpIso ℝfld) β†’ (log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) GrpHom ℝfld))
8482, 83ax-mp 5 . . . . . . . . . 10 (log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) GrpHom ℝfld)
85 ghmmhm 18893 . . . . . . . . . 10 ((log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) GrpHom ℝfld) β†’ (log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) MndHom ℝfld))
8684, 85mp1i 13 . . . . . . . . 9 (πœ‘ β†’ (log β†Ύ ℝ+) ∈ ((𝑀 β†Ύs ℝ+) MndHom ℝfld))
87 1ex 11021 . . . . . . . . . . 11 1 ∈ V
8887a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 1 ∈ V)
8910, 5, 88fdmfifsupp 9186 . . . . . . . . 9 (πœ‘ β†’ 𝐹 finSupp 1)
9064, 68, 75, 80, 5, 86, 10, 89gsummhm 19588 . . . . . . . 8 (πœ‘ β†’ (ℝfld Ξ£g ((log β†Ύ ℝ+) ∘ 𝐹)) = ((log β†Ύ ℝ+)β€˜((𝑀 β†Ύs ℝ+) Ξ£g 𝐹)))
91 subgsubm 18826 . . . . . . . . . 10 (ℝ ∈ (SubGrpβ€˜β„‚fld) β†’ ℝ ∈ (SubMndβ€˜β„‚fld))
929, 91syl 17 . . . . . . . . 9 (πœ‘ β†’ ℝ ∈ (SubMndβ€˜β„‚fld))
93 fco 6654 . . . . . . . . . 10 (((log β†Ύ ℝ+):ℝ+βŸΆβ„ ∧ 𝐹:π΄βŸΆβ„+) β†’ ((log β†Ύ ℝ+) ∘ 𝐹):π΄βŸΆβ„)
9439, 10, 93syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ ((log β†Ύ ℝ+) ∘ 𝐹):π΄βŸΆβ„)
955, 92, 94, 76gsumsubm 18522 . . . . . . . 8 (πœ‘ β†’ (β„‚fld Ξ£g ((log β†Ύ ℝ+) ∘ 𝐹)) = (ℝfld Ξ£g ((log β†Ύ ℝ+) ∘ 𝐹)))
9661a1i 11 . . . . . . . . . 10 (πœ‘ β†’ ℝ+ ∈ (SubMndβ€˜π‘€))
975, 96, 10, 62gsumsubm 18522 . . . . . . . . 9 (πœ‘ β†’ (𝑀 Ξ£g 𝐹) = ((𝑀 β†Ύs ℝ+) Ξ£g 𝐹))
9897fveq2d 6808 . . . . . . . 8 (πœ‘ β†’ ((log β†Ύ ℝ+)β€˜(𝑀 Ξ£g 𝐹)) = ((log β†Ύ ℝ+)β€˜((𝑀 β†Ύs ℝ+) Ξ£g 𝐹)))
9990, 95, 983eqtr4d 2786 . . . . . . 7 (πœ‘ β†’ (β„‚fld Ξ£g ((log β†Ύ ℝ+) ∘ 𝐹)) = ((log β†Ύ ℝ+)β€˜(𝑀 Ξ£g 𝐹)))
10066, 71, 5, 96, 10, 89gsumsubmcl 19569 . . . . . . . 8 (πœ‘ β†’ (𝑀 Ξ£g 𝐹) ∈ ℝ+)
101100fvresd 6824 . . . . . . 7 (πœ‘ β†’ ((log β†Ύ ℝ+)β€˜(𝑀 Ξ£g 𝐹)) = (logβ€˜(𝑀 Ξ£g 𝐹)))
10247, 99, 1013eqtrd 2780 . . . . . 6 (πœ‘ β†’ -(β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) = (logβ€˜(𝑀 Ξ£g 𝐹)))
103102oveq1d 7322 . . . . 5 (πœ‘ β†’ (-(β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) = ((logβ€˜(𝑀 Ξ£g 𝐹)) / (β™―β€˜π΄)))
104100relogcld 25827 . . . . . . 7 (πœ‘ β†’ (logβ€˜(𝑀 Ξ£g 𝐹)) ∈ ℝ)
105104recnd 11053 . . . . . 6 (πœ‘ β†’ (logβ€˜(𝑀 Ξ£g 𝐹)) ∈ β„‚)
106105, 24, 25divrec2d 11805 . . . . 5 (πœ‘ β†’ ((logβ€˜(𝑀 Ξ£g 𝐹)) / (β™―β€˜π΄)) = ((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))))
10726, 103, 1063eqtrd 2780 . . . 4 (πœ‘ β†’ -((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) = ((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))))
10836oveq2d 7323 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g 𝐹) = (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜))))
10911rpcnd 12824 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
1105, 109gsumfsum 20714 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜))) = Ξ£π‘˜ ∈ 𝐴 (πΉβ€˜π‘˜))
111108, 110eqtrd 2776 . . . . . . . 8 (πœ‘ β†’ (β„‚fld Ξ£g 𝐹) = Ξ£π‘˜ ∈ 𝐴 (πΉβ€˜π‘˜))
1125, 20, 11fsumrpcl 15498 . . . . . . . 8 (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 (πΉβ€˜π‘˜) ∈ ℝ+)
113111, 112eqeltrd 2837 . . . . . . 7 (πœ‘ β†’ (β„‚fld Ξ£g 𝐹) ∈ ℝ+)
11423nnrpd 12820 . . . . . . 7 (πœ‘ β†’ (β™―β€˜π΄) ∈ ℝ+)
115113, 114rpdivcld 12839 . . . . . 6 (πœ‘ β†’ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) ∈ ℝ+)
116115relogcld 25827 . . . . 5 (πœ‘ β†’ (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) ∈ ℝ)
11718, 23nndivred 12077 . . . . 5 (πœ‘ β†’ ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) ∈ ℝ)
118 rpssre 12787 . . . . . . . . 9 ℝ+ βŠ† ℝ
119118a1i 11 . . . . . . . 8 (πœ‘ β†’ ℝ+ βŠ† ℝ)
120 relogcl 25780 . . . . . . . . . . 11 (𝑀 ∈ ℝ+ β†’ (logβ€˜π‘€) ∈ ℝ)
121120adantl 483 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ ℝ+) β†’ (logβ€˜π‘€) ∈ ℝ)
122121renegcld 11452 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ ℝ+) β†’ -(logβ€˜π‘€) ∈ ℝ)
123122fmpttd 7021 . . . . . . . 8 (πœ‘ β†’ (𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)):ℝ+βŸΆβ„)
124 ioorp 13207 . . . . . . . . . . . 12 (0(,)+∞) = ℝ+
125124eleq2i 2828 . . . . . . . . . . 11 (π‘Ž ∈ (0(,)+∞) ↔ π‘Ž ∈ ℝ+)
126124eleq2i 2828 . . . . . . . . . . 11 (𝑏 ∈ (0(,)+∞) ↔ 𝑏 ∈ ℝ+)
127 iccssioo2 13202 . . . . . . . . . . 11 ((π‘Ž ∈ (0(,)+∞) ∧ 𝑏 ∈ (0(,)+∞)) β†’ (π‘Ž[,]𝑏) βŠ† (0(,)+∞))
128125, 126, 127syl2anbr 600 . . . . . . . . . 10 ((π‘Ž ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) β†’ (π‘Ž[,]𝑏) βŠ† (0(,)+∞))
129128, 124sseqtrdi 3976 . . . . . . . . 9 ((π‘Ž ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) β†’ (π‘Ž[,]𝑏) βŠ† ℝ+)
130129adantl 483 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ ℝ+ ∧ 𝑏 ∈ ℝ+)) β†’ (π‘Ž[,]𝑏) βŠ† ℝ+)
13123nnrecred 12074 . . . . . . . . . 10 (πœ‘ β†’ (1 / (β™―β€˜π΄)) ∈ ℝ)
132114rpreccld 12832 . . . . . . . . . . 11 (πœ‘ β†’ (1 / (β™―β€˜π΄)) ∈ ℝ+)
133132rpge0d 12826 . . . . . . . . . 10 (πœ‘ β†’ 0 ≀ (1 / (β™―β€˜π΄)))
134 elrege0 13236 . . . . . . . . . 10 ((1 / (β™―β€˜π΄)) ∈ (0[,)+∞) ↔ ((1 / (β™―β€˜π΄)) ∈ ℝ ∧ 0 ≀ (1 / (β™―β€˜π΄))))
135131, 133, 134sylanbrc 584 . . . . . . . . 9 (πœ‘ β†’ (1 / (β™―β€˜π΄)) ∈ (0[,)+∞))
136 fconst6g 6693 . . . . . . . . 9 ((1 / (β™―β€˜π΄)) ∈ (0[,)+∞) β†’ (𝐴 Γ— {(1 / (β™―β€˜π΄))}):𝐴⟢(0[,)+∞))
137135, 136syl 17 . . . . . . . 8 (πœ‘ β†’ (𝐴 Γ— {(1 / (β™―β€˜π΄))}):𝐴⟢(0[,)+∞))
138 0lt1 11547 . . . . . . . . 9 0 < 1
139 fconstmpt 5660 . . . . . . . . . . 11 (𝐴 Γ— {(1 / (β™―β€˜π΄))}) = (π‘˜ ∈ 𝐴 ↦ (1 / (β™―β€˜π΄)))
140139oveq2i 7318 . . . . . . . . . 10 (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})) = (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (1 / (β™―β€˜π΄))))
141 ringmnd 19842 . . . . . . . . . . . . 13 (β„‚fld ∈ Ring β†’ β„‚fld ∈ Mnd)
1422, 141mp1i 13 . . . . . . . . . . . 12 (πœ‘ β†’ β„‚fld ∈ Mnd)
143131recnd 11053 . . . . . . . . . . . 12 (πœ‘ β†’ (1 / (β™―β€˜π΄)) ∈ β„‚)
144 eqid 2736 . . . . . . . . . . . . 13 (.gβ€˜β„‚fld) = (.gβ€˜β„‚fld)
14553, 144gsumconst 19584 . . . . . . . . . . . 12 ((β„‚fld ∈ Mnd ∧ 𝐴 ∈ Fin ∧ (1 / (β™―β€˜π΄)) ∈ β„‚) β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (1 / (β™―β€˜π΄)))) = ((β™―β€˜π΄)(.gβ€˜β„‚fld)(1 / (β™―β€˜π΄))))
146142, 5, 143, 145syl3anc 1371 . . . . . . . . . . 11 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (1 / (β™―β€˜π΄)))) = ((β™―β€˜π΄)(.gβ€˜β„‚fld)(1 / (β™―β€˜π΄))))
14723nnzd 12475 . . . . . . . . . . . 12 (πœ‘ β†’ (β™―β€˜π΄) ∈ β„€)
148 cnfldmulg 20679 . . . . . . . . . . . 12 (((β™―β€˜π΄) ∈ β„€ ∧ (1 / (β™―β€˜π΄)) ∈ β„‚) β†’ ((β™―β€˜π΄)(.gβ€˜β„‚fld)(1 / (β™―β€˜π΄))) = ((β™―β€˜π΄) Β· (1 / (β™―β€˜π΄))))
149147, 143, 148syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ ((β™―β€˜π΄)(.gβ€˜β„‚fld)(1 / (β™―β€˜π΄))) = ((β™―β€˜π΄) Β· (1 / (β™―β€˜π΄))))
15024, 25recidd 11796 . . . . . . . . . . 11 (πœ‘ β†’ ((β™―β€˜π΄) Β· (1 / (β™―β€˜π΄))) = 1)
151146, 149, 1503eqtrd 2780 . . . . . . . . . 10 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (1 / (β™―β€˜π΄)))) = 1)
152140, 151eqtrid 2788 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})) = 1)
153138, 152breqtrrid 5119 . . . . . . . 8 (πœ‘ β†’ 0 < (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})))
154 logccv 25867 . . . . . . . . . . . 12 (((π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))) < (logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))))
1551543adant1 1130 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))) < (logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))))
156 ioossre 13190 . . . . . . . . . . . . . . 15 (0(,)1) βŠ† ℝ
157 simp3 1138 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ 𝑑 ∈ (0(,)1))
158156, 157sselid 3924 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ 𝑑 ∈ ℝ)
159 simp21 1206 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ π‘₯ ∈ ℝ+)
160159relogcld 25827 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (logβ€˜π‘₯) ∈ ℝ)
161158, 160remulcld 11055 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (𝑑 Β· (logβ€˜π‘₯)) ∈ ℝ)
162 1re 11025 . . . . . . . . . . . . . . 15 1 ∈ ℝ
163 resubcl 11335 . . . . . . . . . . . . . . 15 ((1 ∈ ℝ ∧ 𝑑 ∈ ℝ) β†’ (1 βˆ’ 𝑑) ∈ ℝ)
164162, 158, 163sylancr 588 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (1 βˆ’ 𝑑) ∈ ℝ)
165 simp22 1207 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ 𝑦 ∈ ℝ+)
166165relogcld 25827 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (logβ€˜π‘¦) ∈ ℝ)
167164, 166remulcld 11055 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦)) ∈ ℝ)
168161, 167readdcld 11054 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))) ∈ ℝ)
169 simp1 1136 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ πœ‘)
170 ioossicc 13215 . . . . . . . . . . . . . . 15 (0(,)1) βŠ† (0[,]1)
171170, 157sselid 3924 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ 𝑑 ∈ (0[,]1))
172119, 130cvxcl 26183 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ 𝑑 ∈ (0[,]1))) β†’ ((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦)) ∈ ℝ+)
173169, 159, 165, 171, 172syl13anc 1372 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦)) ∈ ℝ+)
174173relogcld 25827 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) ∈ ℝ)
175168, 174ltnegd 11603 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))) < (logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) ↔ -(logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) < -((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦)))))
176155, 175mpbid 231 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ -(logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) < -((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))))
177 fveq2 6804 . . . . . . . . . . . . 13 (𝑀 = ((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦)) β†’ (logβ€˜π‘€) = (logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))))
178177negeqd 11265 . . . . . . . . . . . 12 (𝑀 = ((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦)) β†’ -(logβ€˜π‘€) = -(logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))))
179 eqid 2736 . . . . . . . . . . . 12 (𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) = (𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))
180 negex 11269 . . . . . . . . . . . 12 -(logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) ∈ V
181178, 179, 180fvmpt 6907 . . . . . . . . . . 11 (((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦)) ∈ ℝ+ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) = -(logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))))
182173, 181syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) = -(logβ€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))))
183 fveq2 6804 . . . . . . . . . . . . . . . . 17 (𝑀 = π‘₯ β†’ (logβ€˜π‘€) = (logβ€˜π‘₯))
184183negeqd 11265 . . . . . . . . . . . . . . . 16 (𝑀 = π‘₯ β†’ -(logβ€˜π‘€) = -(logβ€˜π‘₯))
185 negex 11269 . . . . . . . . . . . . . . . 16 -(logβ€˜π‘₯) ∈ V
186184, 179, 185fvmpt 6907 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ℝ+ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯) = -(logβ€˜π‘₯))
187159, 186syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯) = -(logβ€˜π‘₯))
188187oveq2d 7323 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (𝑑 Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯)) = (𝑑 Β· -(logβ€˜π‘₯)))
189158recnd 11053 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ 𝑑 ∈ β„‚)
190160recnd 11053 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (logβ€˜π‘₯) ∈ β„‚)
191189, 190mulneg2d 11479 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (𝑑 Β· -(logβ€˜π‘₯)) = -(𝑑 Β· (logβ€˜π‘₯)))
192188, 191eqtrd 2776 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (𝑑 Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯)) = -(𝑑 Β· (logβ€˜π‘₯)))
193 fveq2 6804 . . . . . . . . . . . . . . . . 17 (𝑀 = 𝑦 β†’ (logβ€˜π‘€) = (logβ€˜π‘¦))
194193negeqd 11265 . . . . . . . . . . . . . . . 16 (𝑀 = 𝑦 β†’ -(logβ€˜π‘€) = -(logβ€˜π‘¦))
195 negex 11269 . . . . . . . . . . . . . . . 16 -(logβ€˜π‘¦) ∈ V
196194, 179, 195fvmpt 6907 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ+ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦) = -(logβ€˜π‘¦))
197165, 196syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦) = -(logβ€˜π‘¦))
198197oveq2d 7323 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((1 βˆ’ 𝑑) Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦)) = ((1 βˆ’ 𝑑) Β· -(logβ€˜π‘¦)))
199164recnd 11053 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (1 βˆ’ 𝑑) ∈ β„‚)
200166recnd 11053 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (logβ€˜π‘¦) ∈ β„‚)
201199, 200mulneg2d 11479 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((1 βˆ’ 𝑑) Β· -(logβ€˜π‘¦)) = -((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦)))
202198, 201eqtrd 2776 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((1 βˆ’ 𝑑) Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦)) = -((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦)))
203192, 202oveq12d 7325 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑑 Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦))) = (-(𝑑 Β· (logβ€˜π‘₯)) + -((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))))
204161recnd 11053 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ (𝑑 Β· (logβ€˜π‘₯)) ∈ β„‚)
205167recnd 11053 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦)) ∈ β„‚)
206204, 205negdid 11395 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ -((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))) = (-(𝑑 Β· (logβ€˜π‘₯)) + -((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))))
207203, 206eqtr4d 2779 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑑 Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦))) = -((𝑑 Β· (logβ€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· (logβ€˜π‘¦))))
208176, 182, 2073brtr4d 5113 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ∧ π‘₯ < 𝑦) ∧ 𝑑 ∈ (0(,)1)) β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((𝑑 Β· π‘₯) + ((1 βˆ’ 𝑑) Β· 𝑦))) < ((𝑑 Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘₯)) + ((1 βˆ’ 𝑑) Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘¦))))
209119, 123, 130, 208scvxcvx 26184 . . . . . . . 8 ((πœ‘ ∧ (𝑒 ∈ ℝ+ ∧ 𝑣 ∈ ℝ+ ∧ 𝑠 ∈ (0[,]1))) β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((𝑠 Β· 𝑒) + ((1 βˆ’ 𝑠) Β· 𝑣))) ≀ ((𝑠 Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘’)) + ((1 βˆ’ 𝑠) Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜π‘£))))
210119, 123, 130, 5, 137, 10, 153, 209jensen 26187 . . . . . . 7 (πœ‘ β†’ (((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))}))) ∈ ℝ+ ∧ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})))) ≀ ((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹))) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})))))
211210simprd 497 . . . . . 6 (πœ‘ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})))) ≀ ((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹))) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))}))))
212131adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (1 / (β™―β€˜π΄)) ∈ ℝ)
213139a1i 11 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝐴 Γ— {(1 / (β™―β€˜π΄))}) = (π‘˜ ∈ 𝐴 ↦ (1 / (β™―β€˜π΄))))
2145, 212, 11, 213, 36offval2 7585 . . . . . . . . . . . 12 (πœ‘ β†’ ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹) = (π‘˜ ∈ 𝐴 ↦ ((1 / (β™―β€˜π΄)) Β· (πΉβ€˜π‘˜))))
215214oveq2d 7323 . . . . . . . . . . 11 (πœ‘ β†’ (β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) = (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ ((1 / (β™―β€˜π΄)) Β· (πΉβ€˜π‘˜)))))
216 cnfldadd 20651 . . . . . . . . . . . 12 + = (+gβ€˜β„‚fld)
217 cnfldmul 20652 . . . . . . . . . . . 12 Β· = (.rβ€˜β„‚fld)
2182a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ β„‚fld ∈ Ring)
219109fmpttd 7021 . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜)):π΄βŸΆβ„‚)
220219, 5, 16fdmfifsupp 9186 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜)) finSupp 0)
22153, 1, 216, 217, 218, 5, 143, 109, 220gsummulc2 19895 . . . . . . . . . . 11 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ ((1 / (β™―β€˜π΄)) Β· (πΉβ€˜π‘˜)))) = ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜)))))
222 fss 6647 . . . . . . . . . . . . . . . 16 ((𝐹:π΄βŸΆβ„+ ∧ ℝ+ βŠ† ℝ) β†’ 𝐹:π΄βŸΆβ„)
22310, 118, 222sylancl 587 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
22410, 5, 16fdmfifsupp 9186 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐹 finSupp 0)
2251, 4, 5, 9, 223, 224gsumsubgcl 19570 . . . . . . . . . . . . . 14 (πœ‘ β†’ (β„‚fld Ξ£g 𝐹) ∈ ℝ)
226225recnd 11053 . . . . . . . . . . . . 13 (πœ‘ β†’ (β„‚fld Ξ£g 𝐹) ∈ β„‚)
227226, 24, 25divrec2d 11805 . . . . . . . . . . . 12 (πœ‘ β†’ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) = ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g 𝐹)))
228108oveq2d 7323 . . . . . . . . . . . 12 (πœ‘ β†’ ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g 𝐹)) = ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜)))))
229227, 228eqtr2d 2777 . . . . . . . . . . 11 (πœ‘ β†’ ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ (πΉβ€˜π‘˜)))) = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))
230215, 221, 2293eqtrd 2780 . . . . . . . . . 10 (πœ‘ β†’ (β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))
231230, 152oveq12d 7325 . . . . . . . . 9 (πœ‘ β†’ ((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))}))) = (((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) / 1))
232225, 23nndivred 12077 . . . . . . . . . . 11 (πœ‘ β†’ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) ∈ ℝ)
233232recnd 11053 . . . . . . . . . 10 (πœ‘ β†’ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) ∈ β„‚)
234233div1d 11793 . . . . . . . . 9 (πœ‘ β†’ (((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) / 1) = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))
235231, 234eqtrd 2776 . . . . . . . 8 (πœ‘ β†’ ((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))}))) = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))
236235fveq2d 6808 . . . . . . 7 (πœ‘ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})))) = ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
237 fveq2 6804 . . . . . . . . . 10 (𝑀 = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) β†’ (logβ€˜π‘€) = (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
238237negeqd 11265 . . . . . . . . 9 (𝑀 = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) β†’ -(logβ€˜π‘€) = -(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
239 negex 11269 . . . . . . . . 9 -(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) ∈ V
240238, 179, 239fvmpt 6907 . . . . . . . 8 (((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) ∈ ℝ+ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) = -(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
241115, 240syl 17 . . . . . . 7 (πœ‘ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) = -(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
242236, 241eqtrd 2776 . . . . . 6 (πœ‘ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€))β€˜((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· 𝐹)) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))})))) = -(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
24353, 1, 216, 217, 218, 5, 143, 31, 17gsummulc2 19895 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ ((1 / (β™―β€˜π΄)) Β· -(logβ€˜(πΉβ€˜π‘˜))))) = ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜))))))
244 negex 11269 . . . . . . . . . . . 12 -(logβ€˜(πΉβ€˜π‘˜)) ∈ V
245244a1i 11 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ -(logβ€˜(πΉβ€˜π‘˜)) ∈ V)
246 eqidd 2737 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) = (𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)))
247 fveq2 6804 . . . . . . . . . . . . 13 (𝑀 = (πΉβ€˜π‘˜) β†’ (logβ€˜π‘€) = (logβ€˜(πΉβ€˜π‘˜)))
248247negeqd 11265 . . . . . . . . . . . 12 (𝑀 = (πΉβ€˜π‘˜) β†’ -(logβ€˜π‘€) = -(logβ€˜(πΉβ€˜π‘˜)))
24911, 36, 246, 248fmptco 7033 . . . . . . . . . . 11 (πœ‘ β†’ ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹) = (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜))))
2505, 212, 245, 213, 249offval2 7585 . . . . . . . . . 10 (πœ‘ β†’ ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹)) = (π‘˜ ∈ 𝐴 ↦ ((1 / (β™―β€˜π΄)) Β· -(logβ€˜(πΉβ€˜π‘˜)))))
251250oveq2d 7323 . . . . . . . . 9 (πœ‘ β†’ (β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹))) = (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ ((1 / (β™―β€˜π΄)) Β· -(logβ€˜(πΉβ€˜π‘˜))))))
25219, 24, 25divrec2d 11805 . . . . . . . . 9 (πœ‘ β†’ ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) = ((1 / (β™―β€˜π΄)) Β· (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜))))))
253243, 251, 2523eqtr4d 2786 . . . . . . . 8 (πœ‘ β†’ (β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹))) = ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)))
254253, 152oveq12d 7325 . . . . . . 7 (πœ‘ β†’ ((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹))) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))}))) = (((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) / 1))
255117recnd 11053 . . . . . . . 8 (πœ‘ β†’ ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) ∈ β„‚)
256255div1d 11793 . . . . . . 7 (πœ‘ β†’ (((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) / 1) = ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)))
257254, 256eqtrd 2776 . . . . . 6 (πœ‘ β†’ ((β„‚fld Ξ£g ((𝐴 Γ— {(1 / (β™―β€˜π΄))}) ∘f Β· ((𝑀 ∈ ℝ+ ↦ -(logβ€˜π‘€)) ∘ 𝐹))) / (β„‚fld Ξ£g (𝐴 Γ— {(1 / (β™―β€˜π΄))}))) = ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)))
258211, 242, 2573brtr3d 5112 . . . . 5 (πœ‘ β†’ -(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) ≀ ((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)))
259116, 117, 258lenegcon1d 11607 . . . 4 (πœ‘ β†’ -((β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ -(logβ€˜(πΉβ€˜π‘˜)))) / (β™―β€˜π΄)) ≀ (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
260107, 259eqbrtrrd 5105 . . 3 (πœ‘ β†’ ((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))) ≀ (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))
261131, 104remulcld 11055 . . . 4 (πœ‘ β†’ ((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))) ∈ ℝ)
262 efle 15876 . . . 4 ((((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))) ∈ ℝ ∧ (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) ∈ ℝ) β†’ (((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))) ≀ (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) ↔ (expβ€˜((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹)))) ≀ (expβ€˜(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))))
263261, 116, 262syl2anc 585 . . 3 (πœ‘ β†’ (((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹))) ≀ (logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))) ↔ (expβ€˜((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹)))) ≀ (expβ€˜(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄))))))
264260, 263mpbid 231 . 2 (πœ‘ β†’ (expβ€˜((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹)))) ≀ (expβ€˜(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))))
265100rpcnd 12824 . . 3 (πœ‘ β†’ (𝑀 Ξ£g 𝐹) ∈ β„‚)
266100rpne0d 12827 . . 3 (πœ‘ β†’ (𝑀 Ξ£g 𝐹) β‰  0)
267265, 266, 143cxpefd 25916 . 2 (πœ‘ β†’ ((𝑀 Ξ£g 𝐹)↑𝑐(1 / (β™―β€˜π΄))) = (expβ€˜((1 / (β™―β€˜π΄)) Β· (logβ€˜(𝑀 Ξ£g 𝐹)))))
268115reeflogd 25828 . . 3 (πœ‘ β†’ (expβ€˜(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))) = ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))
269268eqcomd 2742 . 2 (πœ‘ β†’ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)) = (expβ€˜(logβ€˜((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))))
270264, 267, 2693brtr4d 5113 1 (πœ‘ β†’ ((𝑀 Ξ£g 𝐹)↑𝑐(1 / (β™―β€˜π΄))) ≀ ((β„‚fld Ξ£g 𝐹) / (β™―β€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104   β‰  wne 2941  Vcvv 3437   βˆ– cdif 3889   βŠ† wss 3892  βˆ…c0 4262  {csn 4565   class class class wbr 5081   ↦ cmpt 5164   Γ— cxp 5598   β†Ύ cres 5602   ∘ ccom 5604  βŸΆwf 6454  β€“1-1-ontoβ†’wf1o 6457  β€˜cfv 6458  (class class class)co 7307   ∘f cof 7563  Fincfn 8764  β„‚cc 10919  β„cr 10920  0cc0 10921  1c1 10922   + caddc 10924   Β· cmul 10926  +∞cpnf 11056   < clt 11059   ≀ cle 11060   βˆ’ cmin 11255  -cneg 11256   / cdiv 11682  β„•cn 12023  β„€cz 12369  β„+crp 12780  (,)cioo 13129  [,)cico 13131  [,]cicc 13132  β™―chash 14094  Ξ£csu 15446  expce 15820  Basecbs 16961   β†Ύs cress 16990  0gc0g 17199   Ξ£g cgsu 17200  Mndcmnd 18434   MndHom cmhm 18477  SubMndcsubmnd 18478  .gcmg 18749  SubGrpcsubg 18798   GrpHom cghm 18880   GrpIso cgim 18922  CMndccmn 19435  Abelcabl 19436  mulGrpcmgp 19769  Ringcrg 19832  CRingccrg 19833  DivRingcdr 20040  SubRingcsubrg 20069  β„‚fldccnfld 20646  β„fldcrefld 20858  logclog 25759  β†‘𝑐ccxp 25760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9447  ax-cnex 10977  ax-resscn 10978  ax-1cn 10979  ax-icn 10980  ax-addcl 10981  ax-addrcl 10982  ax-mulcl 10983  ax-mulrcl 10984  ax-mulcom 10985  ax-addass 10986  ax-mulass 10987  ax-distr 10988  ax-i2m1 10989  ax-1ne0 10990  ax-1rid 10991  ax-rnegex 10992  ax-rrecex 10993  ax-cnre 10994  ax-pre-lttri 10995  ax-pre-lttrn 10996  ax-pre-ltadd 10997  ax-pre-mulgt0 10998  ax-pre-sup 10999  ax-addf 11000  ax-mulf 11001
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3304  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-tp 4570  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-iin 4934  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-om 7745  df-1st 7863  df-2nd 7864  df-supp 8009  df-tpos 8073  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-2o 8329  df-er 8529  df-map 8648  df-pm 8649  df-ixp 8717  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-fsupp 9177  df-fi 9218  df-sup 9249  df-inf 9250  df-oi 9317  df-card 9745  df-pnf 11061  df-mnf 11062  df-xr 11063  df-ltxr 11064  df-le 11065  df-sub 11257  df-neg 11258  df-div 11683  df-nn 12024  df-2 12086  df-3 12087  df-4 12088  df-5 12089  df-6 12090  df-7 12091  df-8 12092  df-9 12093  df-n0 12284  df-z 12370  df-dec 12488  df-uz 12633  df-q 12739  df-rp 12781  df-xneg 12898  df-xadd 12899  df-xmul 12900  df-ioo 13133  df-ioc 13134  df-ico 13135  df-icc 13136  df-fz 13290  df-fzo 13433  df-fl 13562  df-mod 13640  df-seq 13772  df-exp 13833  df-fac 14038  df-bc 14067  df-hash 14095  df-shft 14827  df-cj 14859  df-re 14860  df-im 14861  df-sqrt 14995  df-abs 14996  df-limsup 15229  df-clim 15246  df-rlim 15247  df-sum 15447  df-ef 15826  df-sin 15828  df-cos 15829  df-pi 15831  df-struct 16897  df-sets 16914  df-slot 16932  df-ndx 16944  df-base 16962  df-ress 16991  df-plusg 17024  df-mulr 17025  df-starv 17026  df-sca 17027  df-vsca 17028  df-ip 17029  df-tset 17030  df-ple 17031  df-ds 17033  df-unif 17034  df-hom 17035  df-cco 17036  df-rest 17182  df-topn 17183  df-0g 17201  df-gsum 17202  df-topgen 17203  df-pt 17204  df-prds 17207  df-xrs 17262  df-qtop 17267  df-imas 17268  df-xps 17270  df-mre 17344  df-mrc 17345  df-acs 17347  df-mgm 18375  df-sgrp 18424  df-mnd 18435  df-mhm 18479  df-submnd 18480  df-grp 18629  df-minusg 18630  df-mulg 18750  df-subg 18801  df-ghm 18881  df-gim 18924  df-cntz 18972  df-cmn 19437  df-abl 19438  df-mgp 19770  df-ur 19787  df-ring 19834  df-cring 19835  df-oppr 19911  df-dvdsr 19932  df-unit 19933  df-invr 19963  df-dvr 19974  df-drng 20042  df-subrg 20071  df-psmet 20638  df-xmet 20639  df-met 20640  df-bl 20641  df-mopn 20642  df-fbas 20643  df-fg 20644  df-cnfld 20647  df-refld 20859  df-top 22092  df-topon 22109  df-topsp 22131  df-bases 22145  df-cld 22219  df-ntr 22220  df-cls 22221  df-nei 22298  df-lp 22336  df-perf 22337  df-cn 22427  df-cnp 22428  df-haus 22515  df-cmp 22587  df-tx 22762  df-hmeo 22955  df-fil 23046  df-fm 23138  df-flim 23139  df-flf 23140  df-xms 23522  df-ms 23523  df-tms 23524  df-cncf 24090  df-limc 25079  df-dv 25080  df-log 25761  df-cxp 25762
This theorem is referenced by:  amgm  26189  amgm2d  42022  amgm3d  42023  amgm4d  42024
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