Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumiun Structured version   Visualization version   GIF version

Theorem esumiun 31410
Description: Sum over a nonnecessarily disjoint indexed union. The inequality 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 30419 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6617 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 617 . . . . 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 6565 . . . . . . 7 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵 𝑗𝐴 ({𝑗} × 𝐵))
76frnd 6510 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
87adantr 484 . . . . 5 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
95, 8jca 515 . . . 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)))
109eximi 1836 . . 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)))
113, 10syl 17 . 2 (𝜑 → ∃𝑓((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
12 nfv 1916 . . . . . 6 𝑧(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
13 nfcv 2982 . . . . . 6 𝑧𝐶
14 nfcsb1v 3890 . . . . . 6 𝑘(2nd𝑧) / 𝑘𝐶
15 nfcv 2982 . . . . . 6 𝑧 𝑗𝐴 𝐵
16 nfcv 2982 . . . . . 6 𝑧ran 𝑓
17 nfcv 2982 . . . . . 6 𝑧𝑓
18 csbeq1a 3880 . . . . . 6 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
192ralrimiva 3177 . . . . . . . 8 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
20 iunexg 7659 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐵𝑊) → 𝑗𝐴 𝐵 ∈ V)
211, 19, 20syl2anc 587 . . . . . . 7 (𝜑 𝑗𝐴 𝐵 ∈ V)
2221adantr 484 . . . . . 6 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑗𝐴 𝐵 ∈ V)
23 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
24 f1ocnv 6618 . . . . . . . 8 (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑓:ran 𝑓1-1-onto 𝑗𝐴 𝐵)
2523, 24syl 17 . . . . . . 7 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓:ran 𝑓1-1-onto 𝑗𝐴 𝐵)
2625adantrlr 722 . . . . . 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)
27 nfv 1916 . . . . . . . . 9 𝑗𝜑
28 nfcv 2982 . . . . . . . . . . . 12 𝑗𝑓
29 nfiu1 4939 . . . . . . . . . . . 12 𝑗 𝑗𝐴 𝐵
3028nfrn 5811 . . . . . . . . . . . 12 𝑗ran 𝑓
3128, 29, 30nff1o 6604 . . . . . . . . . . 11 𝑗 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓
32 nfv 1916 . . . . . . . . . . . 12 𝑗(2nd ‘(𝑓𝑙)) = 𝑙
3329, 32nfralw 3219 . . . . . . . . . . 11 𝑗𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3431, 33nfan 1901 . . . . . . . . . 10 𝑗(𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
35 nfcv 2982 . . . . . . . . . . 11 𝑗ran 𝑓
36 nfiu1 4939 . . . . . . . . . . 11 𝑗 𝑗𝐴 ({𝑗} × 𝐵)
3735, 36nfss 3945 . . . . . . . . . 10 𝑗ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)
3834, 37nfan 1901 . . . . . . . . 9 𝑗((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
3927, 38nfan 1901 . . . . . . . 8 𝑗(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
40 nfv 1916 . . . . . . . 8 𝑗 𝑧 ∈ ran 𝑓
4139, 40nfan 1901 . . . . . . 7 𝑗((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
42 simpr 488 . . . . . . . . . . 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)
4342fveq2d 6665 . . . . . . . . . 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))
44 simplr 768 . . . . . . . . . . 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)
45 simp-4r 783 . . . . . . . . . . . . . 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)))
4645simpld 498 . . . . . . . . . . . . 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))
4746simprd 499 . . . . . . . . . . . 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)
4847ad2antrr 725 . . . . . . . . . . 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)
49 2fveq3 6666 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
50 id 22 . . . . . . . . . . . . 13 (𝑙 = 𝑘𝑙 = 𝑘)
5149, 50eqeq12d 2840 . . . . . . . . . . . 12 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5251rspcva 3607 . . . . . . . . . . 11 ((𝑘 𝑗𝐴 𝐵 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5344, 48, 52syl2anc 587 . . . . . . . . . 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)
5443, 53eqtr3d 2861 . . . . . . . . 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)
5546simpld 498 . . . . . . . . . . 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)
5655ad2antrr 725 . . . . . . . . . 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)
57 f1ocnvfv1 7025 . . . . . . . . . 10 ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑗𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
5856, 44, 57syl2anc 587 . . . . . . . . 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)
5942fveq2d 6665 . . . . . . . . 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))
6054, 58, 593eqtr2rd 2866 . . . . . . . 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))
61 f1ofn 6607 . . . . . . . . . 10 (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑓 Fn 𝑗𝐴 𝐵)
6255, 61syl 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)
63 simpllr 775 . . . . . . . . 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)
64 fvelrnb 6717 . . . . . . . . . 10 (𝑓 Fn 𝑗𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧))
6564biimpa 480 . . . . . . . . 9 ((𝑓 Fn 𝑗𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧)
6662, 63, 65syl2anc 587 . . . . . . . 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)
6760, 66r19.29a 3281 . . . . . . 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))
68 simprr 772 . . . . . . . . 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))
6968sselda 3953 . . . . . . . 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))
70 eliun 4909 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵))
7169, 70sylib 221 . . . . . . 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))
7241, 67, 71r19.29af 3322 . . . . . 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))
73 nfcv 2982 . . . . . . . . . 10 𝑗𝑘
7473, 29nfel 2996 . . . . . . . . 9 𝑗 𝑘 𝑗𝐴 𝐵
7527, 74nfan 1901 . . . . . . . 8 𝑗(𝜑𝑘 𝑗𝐴 𝐵)
76 esumiun.2 . . . . . . . . 9 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
7776adantllr 718 . . . . . . . 8 ((((𝜑𝑘 𝑗𝐴 𝐵) ∧ 𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
78 eliun 4909 . . . . . . . . . 10 (𝑘 𝑗𝐴 𝐵 ↔ ∃𝑗𝐴 𝑘𝐵)
7978biimpi 219 . . . . . . . . 9 (𝑘 𝑗𝐴 𝐵 → ∃𝑗𝐴 𝑘𝐵)
8079adantl 485 . . . . . . . 8 ((𝜑𝑘 𝑗𝐴 𝐵) → ∃𝑗𝐴 𝑘𝐵)
8175, 77, 80r19.29af 3322 . . . . . . 7 ((𝜑𝑘 𝑗𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
8281adantlr 714 . . . . . 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[,]+∞))
8312, 13, 14, 15, 16, 17, 18, 22, 26, 72, 82esumf1o 31366 . . . . 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)
8483eqcomd 2830 . . . 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)
85 snex 5319 . . . . . . . . . 10 {𝑗} ∈ V
8685a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐴) → {𝑗} ∈ V)
8786, 2xpexd 7468 . . . . . . . 8 ((𝜑𝑗𝐴) → ({𝑗} × 𝐵) ∈ V)
8887ralrimiva 3177 . . . . . . 7 (𝜑 → ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
89 iunexg 7659 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
901, 88, 89syl2anc 587 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
9190adantr 484 . . . . 5 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
92 nfcv 2982 . . . . . . . . 9 𝑗𝑧
9392, 36nfel 2996 . . . . . . . 8 𝑗 𝑧 𝑗𝐴 ({𝑗} × 𝐵)
9427, 93nfan 1901 . . . . . . 7 𝑗(𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵))
95 nfcv 2982 . . . . . . . . 9 𝑗(2nd𝑧)
96 nfcv 2982 . . . . . . . . 9 𝑗𝐶
9795, 96nfcsbw 3892 . . . . . . . 8 𝑗(2nd𝑧) / 𝑘𝐶
98 nfcv 2982 . . . . . . . 8 𝑗(0[,]+∞)
9997, 98nfel 2996 . . . . . . 7 𝑗(2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞)
100 simprr 772 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) ∈ 𝐵)
101 simplll 774 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
102 simplr 768 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑗𝐴)
10376ralrimiva 3177 . . . . . . . . 9 ((𝜑𝑗𝐴) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
104101, 102, 103syl2anc 587 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
105 rspcsbela 4370 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ (0[,]+∞)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
106100, 104, 105syl2anc 587 . . . . . . 7 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
107 xp1st 7716 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
108 elsni 4567 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
109107, 108syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
110 xp2nd 7717 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (2nd𝑧) ∈ 𝐵)
111109, 110jca 515 . . . . . . . . . 10 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
112111reximi 3237 . . . . . . . . 9 (∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11370, 112sylbi 220 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
114113adantl 485 . . . . . . 7 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11594, 99, 106, 114r19.29af2 3321 . . . . . 6 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
116115adantlr 714 . . . . 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[,]+∞))
117 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
118117adantrlr 722 . . . . 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))
11912, 91, 116, 118esummono 31370 . . . 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)
12084, 119eqbrtrrd 5076 . . 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)
121 vex 3483 . . . . . . . . 9 𝑗 ∈ V
122 vex 3483 . . . . . . . . 9 𝑘 ∈ V
123121, 122op2ndd 7695 . . . . . . . 8 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = 𝑘)
124123eqcomd 2830 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
125124, 18syl 17 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = (2nd𝑧) / 𝑘𝐶)
126125eqcomd 2830 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) / 𝑘𝐶 = 𝐶)
12776anasss 470 . . . . 5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))
12814, 126, 1, 2, 127esum2d 31409 . . . 4 (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
129128adantr 484 . . 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)
130120, 129breqtrrd 5080 . 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)
13111, 130exlimddv 1937 1 (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  wral 3133  wrex 3134  Vcvv 3480  csb 3866  wss 3919  {csn 4550  cop 4556   ciun 4905   class class class wbr 5052   × cxp 5540  ccnv 5541  ran crn 5543   Fn wfn 6338  1-1wf1 6340  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  0cc0 10535  +∞cpnf 10670  cle 10674  [,]cicc 12738  Σ*cesum 31343
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-reg 9053  ax-inf2 9101  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  ax-addf 10614  ax-mulf 10615
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  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-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-fi 8872  df-sup 8903  df-inf 8904  df-oi 8971  df-r1 9190  df-rank 9191  df-card 9365  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-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ioo 12739  df-ioc 12740  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-mod 13242  df-seq 13374  df-exp 13435  df-fac 13639  df-bc 13668  df-hash 13696  df-shft 14426  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-limsup 14828  df-clim 14845  df-rlim 14846  df-sum 15043  df-ef 15421  df-sin 15423  df-cos 15424  df-pi 15426  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-pt 16718  df-prds 16721  df-ordt 16774  df-xrs 16775  df-qtop 16780  df-imas 16781  df-xps 16783  df-mre 16857  df-mrc 16858  df-acs 16860  df-ps 17810  df-tsr 17811  df-plusf 17851  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-cring 19300  df-subrg 19533  df-abv 19588  df-lmod 19636  df-scaf 19637  df-sra 19944  df-rgmod 19945  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-fbas 20542  df-fg 20543  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-lp 21744  df-perf 21745  df-cn 21835  df-cnp 21836  df-haus 21923  df-tx 22170  df-hmeo 22363  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-tmd 22680  df-tgp 22681  df-tsms 22735  df-trg 22768  df-xms 22930  df-ms 22931  df-tms 22932  df-nm 23192  df-ngp 23193  df-nrg 23195  df-nlm 23196  df-ii 23485  df-cncf 23486  df-limc 24472  df-dv 24473  df-log 25151  df-esum 31344
This theorem is referenced by:  omssubadd  31615
  Copyright terms: Public domain W3C validator