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Theorem esumiun 30496
Description: Sum over a non necessarily disjoint indexed union. The inegality is strict in the case where the sets B(x) overlap. (Contributed by Thierry Arnoux, 21-Sep-2019.)
Hypotheses
Ref Expression
esumiun.0 (𝜑𝐴𝑉)
esumiun.1 ((𝜑𝑗𝐴) → 𝐵𝑊)
esumiun.2 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
Assertion
Ref Expression
esumiun (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘   𝐶,𝑗   𝑗,𝑊,𝑘   𝜑,𝑗,𝑘
Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)   𝑉(𝑗,𝑘)

Proof of Theorem esumiun
Dummy variables 𝑓 𝑙 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 esumiun.0 . . . 4 (𝜑𝐴𝑉)
2 esumiun.1 . . . 4 ((𝜑𝑗𝐴) → 𝐵𝑊)
31, 2aciunf1 29803 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6289 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 602 . . . . 5 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
6 f1f 6241 . . . . . . 7 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵 𝑗𝐴 ({𝑗} × 𝐵))
7 frn 6193 . . . . . . 7 (𝑓: 𝑗𝐴 𝐵 𝑗𝐴 ({𝑗} × 𝐵) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
86, 7syl 17 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
98adantr 466 . . . . 5 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
105, 9jca 501 . . . 4 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
1110eximi 1910 . . 3 (∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ∃𝑓((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
123, 11syl 17 . 2 (𝜑 → ∃𝑓((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
13 nfv 1995 . . . . . 6 𝑧(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
14 nfcv 2913 . . . . . 6 𝑧𝐶
15 nfcsb1v 3698 . . . . . 6 𝑘(2nd𝑧) / 𝑘𝐶
16 nfcv 2913 . . . . . 6 𝑧 𝑗𝐴 𝐵
17 nfcv 2913 . . . . . 6 𝑧ran 𝑓
18 nfcv 2913 . . . . . 6 𝑧𝑓
19 csbeq1a 3691 . . . . . 6 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
202ralrimiva 3115 . . . . . . . 8 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
21 iunexg 7290 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐵𝑊) → 𝑗𝐴 𝐵 ∈ V)
221, 20, 21syl2anc 573 . . . . . . 7 (𝜑 𝑗𝐴 𝐵 ∈ V)
2322adantr 466 . . . . . 6 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑗𝐴 𝐵 ∈ V)
24 simprl 754 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
25 f1ocnv 6290 . . . . . . . 8 (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑓:ran 𝑓1-1-onto 𝑗𝐴 𝐵)
2624, 25syl 17 . . . . . . 7 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓:ran 𝑓1-1-onto 𝑗𝐴 𝐵)
2726adantrlr 702 . . . . . 6 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓:ran 𝑓1-1-onto 𝑗𝐴 𝐵)
28 nfv 1995 . . . . . . . . 9 𝑗𝜑
29 nfcv 2913 . . . . . . . . . . . 12 𝑗𝑓
30 nfiu1 4684 . . . . . . . . . . . 12 𝑗 𝑗𝐴 𝐵
3129nfrn 5506 . . . . . . . . . . . 12 𝑗ran 𝑓
3229, 30, 31nff1o 6276 . . . . . . . . . . 11 𝑗 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓
33 nfv 1995 . . . . . . . . . . . 12 𝑗(2nd ‘(𝑓𝑙)) = 𝑙
3430, 33nfral 3094 . . . . . . . . . . 11 𝑗𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3532, 34nfan 1980 . . . . . . . . . 10 𝑗(𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
36 nfcv 2913 . . . . . . . . . . 11 𝑗ran 𝑓
37 nfiu1 4684 . . . . . . . . . . 11 𝑗 𝑗𝐴 ({𝑗} × 𝐵)
3836, 37nfss 3745 . . . . . . . . . 10 𝑗ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)
3935, 38nfan 1980 . . . . . . . . 9 𝑗((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
4028, 39nfan 1980 . . . . . . . 8 𝑗(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
41 nfv 1995 . . . . . . . 8 𝑗 𝑧 ∈ ran 𝑓
4240, 41nfan 1980 . . . . . . 7 𝑗((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
43 simpr 471 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓𝑘) = 𝑧)
4443fveq2d 6336 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (2nd ‘(𝑓𝑘)) = (2nd𝑧))
45 simplr 752 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → 𝑘 𝑗𝐴 𝐵)
46 simp-4r 770 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
4746simpld 482 . . . . . . . . . . . . 13 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4847simprd 483 . . . . . . . . . . . 12 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
4948ad2antrr 705 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
50 fveq2 6332 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (𝑓𝑙) = (𝑓𝑘))
5150fveq2d 6336 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
52 id 22 . . . . . . . . . . . . 13 (𝑙 = 𝑘𝑙 = 𝑘)
5351, 52eqeq12d 2786 . . . . . . . . . . . 12 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5453rspcva 3458 . . . . . . . . . . 11 ((𝑘 𝑗𝐴 𝐵 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5545, 49, 54syl2anc 573 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (2nd ‘(𝑓𝑘)) = 𝑘)
5644, 55eqtr3d 2807 . . . . . . . . 9 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (2nd𝑧) = 𝑘)
5747simpld 482 . . . . . . . . . . 11 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
5857ad2antrr 705 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
59 f1ocnvfv1 6675 . . . . . . . . . 10 ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑗𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6058, 45, 59syl2anc 573 . . . . . . . . 9 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓‘(𝑓𝑘)) = 𝑘)
6143fveq2d 6336 . . . . . . . . 9 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓‘(𝑓𝑘)) = (𝑓𝑧))
6256, 60, 613eqtr2rd 2812 . . . . . . . 8 (((((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 𝑗𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓𝑧) = (2nd𝑧))
63 f1ofn 6279 . . . . . . . . . 10 (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑓 Fn 𝑗𝐴 𝐵)
6457, 63syl 17 . . . . . . . . 9 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑓 Fn 𝑗𝐴 𝐵)
65 simpllr 760 . . . . . . . . 9 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑧 ∈ ran 𝑓)
66 fvelrnb 6385 . . . . . . . . . 10 (𝑓 Fn 𝑗𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧))
6766biimpa 462 . . . . . . . . 9 ((𝑓 Fn 𝑗𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧)
6864, 65, 67syl2anc 573 . . . . . . . 8 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧)
6962, 68r19.29a 3226 . . . . . . 7 (((((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (𝑓𝑧) = (2nd𝑧))
70 simprr 756 . . . . . . . . 9 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
7170sselda 3752 . . . . . . . 8 (((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 𝑗𝐴 ({𝑗} × 𝐵))
72 eliun 4658 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵))
7371, 72sylib 208 . . . . . . 7 (((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → ∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵))
7442, 69, 73r19.29af 3224 . . . . . 6 (((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (𝑓𝑧) = (2nd𝑧))
75 nfcv 2913 . . . . . . . . . 10 𝑗𝑘
7675, 30nfel 2926 . . . . . . . . 9 𝑗 𝑘 𝑗𝐴 𝐵
7728, 76nfan 1980 . . . . . . . 8 𝑗(𝜑𝑘 𝑗𝐴 𝐵)
78 esumiun.2 . . . . . . . . 9 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
7978adantllr 698 . . . . . . . 8 ((((𝜑𝑘 𝑗𝐴 𝐵) ∧ 𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
80 eliun 4658 . . . . . . . . . 10 (𝑘 𝑗𝐴 𝐵 ↔ ∃𝑗𝐴 𝑘𝐵)
8180biimpi 206 . . . . . . . . 9 (𝑘 𝑗𝐴 𝐵 → ∃𝑗𝐴 𝑘𝐵)
8281adantl 467 . . . . . . . 8 ((𝜑𝑘 𝑗𝐴 𝐵) → ∃𝑗𝐴 𝑘𝐵)
8377, 79, 82r19.29af 3224 . . . . . . 7 ((𝜑𝑘 𝑗𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
8483adantlr 694 . . . . . 6 (((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑘 𝑗𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
8513, 14, 15, 16, 17, 18, 19, 23, 27, 74, 84esumf1o 30452 . . . . 5 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 𝑗𝐴 𝐵𝐶 = Σ*𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶)
8685eqcomd 2777 . . . 4 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → Σ*𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶 = Σ*𝑘 𝑗𝐴 𝐵𝐶)
87 snex 5036 . . . . . . . . . 10 {𝑗} ∈ V
8887a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐴) → {𝑗} ∈ V)
89 xpexg 7107 . . . . . . . . 9 (({𝑗} ∈ V ∧ 𝐵𝑊) → ({𝑗} × 𝐵) ∈ V)
9088, 2, 89syl2anc 573 . . . . . . . 8 ((𝜑𝑗𝐴) → ({𝑗} × 𝐵) ∈ V)
9190ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
92 iunexg 7290 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
931, 91, 92syl2anc 573 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
9493adantr 466 . . . . 5 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
95 nfcv 2913 . . . . . . . . 9 𝑗𝑧
9695, 37nfel 2926 . . . . . . . 8 𝑗 𝑧 𝑗𝐴 ({𝑗} × 𝐵)
9728, 96nfan 1980 . . . . . . 7 𝑗(𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵))
98 nfcv 2913 . . . . . . . . 9 𝑗(2nd𝑧)
99 nfcv 2913 . . . . . . . . 9 𝑗𝐶
10098, 99nfcsb 3700 . . . . . . . 8 𝑗(2nd𝑧) / 𝑘𝐶
101 nfcv 2913 . . . . . . . 8 𝑗(0[,]+∞)
102100, 101nfel 2926 . . . . . . 7 𝑗(2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞)
103 simprr 756 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) ∈ 𝐵)
104 simplll 758 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
105 simplr 752 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑗𝐴)
10678ralrimiva 3115 . . . . . . . . 9 ((𝜑𝑗𝐴) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
107104, 105, 106syl2anc 573 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
108 rspcsbela 4150 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ (0[,]+∞)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
109103, 107, 108syl2anc 573 . . . . . . 7 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
110 xp1st 7347 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
111 elsni 4333 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
112110, 111syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
113 xp2nd 7348 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (2nd𝑧) ∈ 𝐵)
114112, 113jca 501 . . . . . . . . . 10 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
115114reximi 3159 . . . . . . . . 9 (∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11672, 115sylbi 207 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
117116adantl 467 . . . . . . 7 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11897, 102, 109, 117r19.29af2 3223 . . . . . 6 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
119118adantlr 694 . . . . 5 (((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
120 simprr 756 . . . . . 6 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
121120adantrlr 702 . . . . 5 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
12213, 94, 119, 121esummono 30456 . . . 4 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → Σ*𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶 ≤ Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
12386, 122eqbrtrrd 4810 . . 3 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
124 vex 3354 . . . . . . . . 9 𝑗 ∈ V
125 vex 3354 . . . . . . . . 9 𝑘 ∈ V
126124, 125op2ndd 7326 . . . . . . . 8 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = 𝑘)
127126eqcomd 2777 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
128127, 19syl 17 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = (2nd𝑧) / 𝑘𝐶)
129128eqcomd 2777 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) / 𝑘𝐶 = 𝐶)
13078anasss 457 . . . . 5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))
13115, 129, 1, 2, 130esum2d 30495 . . . 4 (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
132131adantr 466 . . 3 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
133123, 132breqtrrd 4814 . 2 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
13412, 133exlimddv 2015 1 (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  csb 3682  wss 3723  {csn 4316  cop 4322   ciun 4654   class class class wbr 4786   × cxp 5247  ccnv 5248  ran crn 5250   Fn wfn 6026  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  +∞cpnf 10273  cle 10277  [,]cicc 12383  Σ*cesum 30429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-reg 8653  ax-inf2 8702  ax-ac2 9487  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-r1 8791  df-rank 8792  df-card 8965  df-ac 9139  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ioc 12385  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14015  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-limsup 14410  df-clim 14427  df-rlim 14428  df-sum 14625  df-ef 15004  df-sin 15006  df-cos 15007  df-pi 15009  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-pt 16313  df-prds 16316  df-ordt 16369  df-xrs 16370  df-qtop 16375  df-imas 16376  df-xps 16378  df-mre 16454  df-mrc 16455  df-acs 16457  df-ps 17408  df-tsr 17409  df-plusf 17449  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-mulg 17749  df-subg 17799  df-cntz 17957  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-ring 18757  df-cring 18758  df-subrg 18988  df-abv 19027  df-lmod 19075  df-scaf 19076  df-sra 19387  df-rgmod 19388  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-fbas 19958  df-fg 19959  df-cnfld 19962  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-lp 21161  df-perf 21162  df-cn 21252  df-cnp 21253  df-haus 21340  df-tx 21586  df-hmeo 21779  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964  df-tmd 22096  df-tgp 22097  df-tsms 22150  df-trg 22183  df-xms 22345  df-ms 22346  df-tms 22347  df-nm 22607  df-ngp 22608  df-nrg 22610  df-nlm 22611  df-ii 22900  df-cncf 22901  df-limc 23850  df-dv 23851  df-log 24524  df-esum 30430
This theorem is referenced by:  omssubadd  30702
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