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Theorem esumiun 34095
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 32673 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6859 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 615 . . . . 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 6804 . . . . . . 7 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵 𝑗𝐴 ({𝑗} × 𝐵))
76frnd 6744 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
87adantr 480 . . . . 5 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
95, 8jca 511 . . . 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 1835 . . 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 1914 . . . . . 6 𝑧(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
13 nfcv 2905 . . . . . 6 𝑧𝐶
14 nfcsb1v 3923 . . . . . 6 𝑘(2nd𝑧) / 𝑘𝐶
15 nfcv 2905 . . . . . 6 𝑧 𝑗𝐴 𝐵
16 nfcv 2905 . . . . . 6 𝑧ran 𝑓
17 nfcv 2905 . . . . . 6 𝑧𝑓
18 csbeq1a 3913 . . . . . 6 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
192ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
20 iunexg 7988 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐵𝑊) → 𝑗𝐴 𝐵 ∈ V)
211, 19, 20syl2anc 584 . . . . . . 7 (𝜑 𝑗𝐴 𝐵 ∈ V)
2221adantr 480 . . . . . 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 771 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
24 f1ocnv 6860 . . . . . . . 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 723 . . . . . 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 1914 . . . . . . . . 9 𝑗𝜑
28 nfcv 2905 . . . . . . . . . . . 12 𝑗𝑓
29 nfiu1 5027 . . . . . . . . . . . 12 𝑗 𝑗𝐴 𝐵
3028nfrn 5963 . . . . . . . . . . . 12 𝑗ran 𝑓
3128, 29, 30nff1o 6846 . . . . . . . . . . 11 𝑗 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓
32 nfv 1914 . . . . . . . . . . . 12 𝑗(2nd ‘(𝑓𝑙)) = 𝑙
3329, 32nfralw 3311 . . . . . . . . . . 11 𝑗𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3431, 33nfan 1899 . . . . . . . . . 10 𝑗(𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
35 nfcv 2905 . . . . . . . . . . 11 𝑗ran 𝑓
36 nfiu1 5027 . . . . . . . . . . 11 𝑗 𝑗𝐴 ({𝑗} × 𝐵)
3735, 36nfss 3976 . . . . . . . . . 10 𝑗ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)
3834, 37nfan 1899 . . . . . . . . 9 𝑗((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
3927, 38nfan 1899 . . . . . . . 8 𝑗(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
40 nfv 1914 . . . . . . . 8 𝑗 𝑧 ∈ ran 𝑓
4139, 40nfan 1899 . . . . . . 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 484 . . . . . . . . . . 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 6910 . . . . . . . . . 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 769 . . . . . . . . . . 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 784 . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . 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 495 . . . . . . . . . . . 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 726 . . . . . . . . . . 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 6911 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
50 id 22 . . . . . . . . . . . . 13 (𝑙 = 𝑘𝑙 = 𝑘)
5149, 50eqeq12d 2753 . . . . . . . . . . . 12 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5251rspcva 3620 . . . . . . . . . . 11 ((𝑘 𝑗𝐴 𝐵 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5344, 48, 52syl2anc 584 . . . . . . . . . 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 2779 . . . . . . . . 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 494 . . . . . . . . . . 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 726 . . . . . . . . . 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 7296 . . . . . . . . . 10 ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑗𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
5856, 44, 57syl2anc 584 . . . . . . . . 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 6910 . . . . . . . . 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 2784 . . . . . . . 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 6849 . . . . . . . . . 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 776 . . . . . . . . 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 6969 . . . . . . . . . 10 (𝑓 Fn 𝑗𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧))
6564biimpa 476 . . . . . . . . 9 ((𝑓 Fn 𝑗𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧)
6662, 63, 65syl2anc 584 . . . . . . . 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 3162 . . . . . . 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 773 . . . . . . . . 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 3983 . . . . . . . 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 4995 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵))
7169, 70sylib 218 . . . . . . 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 3268 . . . . . 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 2905 . . . . . . . . . 10 𝑗𝑘
7473, 29nfel 2920 . . . . . . . . 9 𝑗 𝑘 𝑗𝐴 𝐵
7527, 74nfan 1899 . . . . . . . 8 𝑗(𝜑𝑘 𝑗𝐴 𝐵)
76 esumiun.2 . . . . . . . . 9 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
7776adantllr 719 . . . . . . . 8 ((((𝜑𝑘 𝑗𝐴 𝐵) ∧ 𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
78 eliun 4995 . . . . . . . . . 10 (𝑘 𝑗𝐴 𝐵 ↔ ∃𝑗𝐴 𝑘𝐵)
7978biimpi 216 . . . . . . . . 9 (𝑘 𝑗𝐴 𝐵 → ∃𝑗𝐴 𝑘𝐵)
8079adantl 481 . . . . . . . 8 ((𝜑𝑘 𝑗𝐴 𝐵) → ∃𝑗𝐴 𝑘𝐵)
8175, 77, 80r19.29af 3268 . . . . . . 7 ((𝜑𝑘 𝑗𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
8281adantlr 715 . . . . . 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 34051 . . . . 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 2743 . . . 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 vsnex 5434 . . . . . . . . . 10 {𝑗} ∈ V
8685a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐴) → {𝑗} ∈ V)
8786, 2xpexd 7771 . . . . . . . 8 ((𝜑𝑗𝐴) → ({𝑗} × 𝐵) ∈ V)
8887ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
89 iunexg 7988 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
901, 88, 89syl2anc 584 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
9190adantr 480 . . . . 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 2905 . . . . . . . . 9 𝑗𝑧
9392, 36nfel 2920 . . . . . . . 8 𝑗 𝑧 𝑗𝐴 ({𝑗} × 𝐵)
9427, 93nfan 1899 . . . . . . 7 𝑗(𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵))
95 nfcv 2905 . . . . . . . . 9 𝑗(2nd𝑧)
96 nfcv 2905 . . . . . . . . 9 𝑗𝐶
9795, 96nfcsbw 3925 . . . . . . . 8 𝑗(2nd𝑧) / 𝑘𝐶
98 nfcv 2905 . . . . . . . 8 𝑗(0[,]+∞)
9997, 98nfel 2920 . . . . . . 7 𝑗(2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞)
100 simprr 773 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) ∈ 𝐵)
101 simplll 775 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
102 simplr 769 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑗𝐴)
10376ralrimiva 3146 . . . . . . . . 9 ((𝜑𝑗𝐴) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
104101, 102, 103syl2anc 584 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
105 rspcsbela 4438 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ (0[,]+∞)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
106100, 104, 105syl2anc 584 . . . . . . 7 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
107 xp1st 8046 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
108 elsni 4643 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
109107, 108syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
110 xp2nd 8047 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (2nd𝑧) ∈ 𝐵)
111109, 110jca 511 . . . . . . . . . 10 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
112111reximi 3084 . . . . . . . . 9 (∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11370, 112sylbi 217 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
114113adantl 481 . . . . . . 7 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11594, 99, 106, 114r19.29af2 3267 . . . . . 6 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
116115adantlr 715 . . . . 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 773 . . . . . 6 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
118117adantrlr 723 . . . . 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 34055 . . . 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 5167 . . 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 3484 . . . . . . . . 9 𝑗 ∈ V
122 vex 3484 . . . . . . . . 9 𝑘 ∈ V
123121, 122op2ndd 8025 . . . . . . . 8 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = 𝑘)
124123eqcomd 2743 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
125124, 18syl 17 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = (2nd𝑧) / 𝑘𝐶)
126125eqcomd 2743 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) / 𝑘𝐶 = 𝐶)
12776anasss 466 . . . . 5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))
12814, 126, 1, 2, 127esum2d 34094 . . . 4 (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
129128adantr 480 . . 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 5171 . 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 1935 1 (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  csb 3899  wss 3951  {csn 4626  cop 4632   ciun 4991   class class class wbr 5143   × cxp 5683  ccnv 5684  ran crn 5686   Fn wfn 6556  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  0cc0 11155  +∞cpnf 11292  cle 11296  [,]cicc 13390  Σ*cesum 34028
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105  df-cos 16106  df-pi 16108  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-ordt 17546  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-ps 18611  df-tsr 18612  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-abv 20810  df-lmod 20860  df-scaf 20861  df-sra 21172  df-rgmod 21173  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-tmd 24080  df-tgp 24081  df-tsms 24135  df-trg 24168  df-xms 24330  df-ms 24331  df-tms 24332  df-nm 24595  df-ngp 24596  df-nrg 24598  df-nlm 24599  df-ii 24903  df-cncf 24904  df-limc 25901  df-dv 25902  df-log 26598  df-esum 34029
This theorem is referenced by:  omssubadd  34302
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