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Theorem esumiun 30606
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 29916 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6333 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 608 . . . . 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 6285 . . . . . . 7 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵 𝑗𝐴 ({𝑗} × 𝐵))
76frnd 6232 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
87adantr 472 . . . . 5 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
95, 8jca 507 . . . 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 1929 . . 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 2009 . . . . . 6 𝑧(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
13 nfcv 2907 . . . . . 6 𝑧𝐶
14 nfcsb1v 3709 . . . . . 6 𝑘(2nd𝑧) / 𝑘𝐶
15 nfcv 2907 . . . . . 6 𝑧 𝑗𝐴 𝐵
16 nfcv 2907 . . . . . 6 𝑧ran 𝑓
17 nfcv 2907 . . . . . 6 𝑧𝑓
18 csbeq1a 3702 . . . . . 6 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
192ralrimiva 3113 . . . . . . . 8 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
20 iunexg 7343 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐵𝑊) → 𝑗𝐴 𝐵 ∈ V)
211, 19, 20syl2anc 579 . . . . . . 7 (𝜑 𝑗𝐴 𝐵 ∈ V)
2221adantr 472 . . . . . 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 787 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
24 f1ocnv 6334 . . . . . . . 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 714 . . . . . 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 2009 . . . . . . . . 9 𝑗𝜑
28 nfcv 2907 . . . . . . . . . . . 12 𝑗𝑓
29 nfiu1 4708 . . . . . . . . . . . 12 𝑗 𝑗𝐴 𝐵
3028nfrn 5539 . . . . . . . . . . . 12 𝑗ran 𝑓
3128, 29, 30nff1o 6320 . . . . . . . . . . 11 𝑗 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓
32 nfv 2009 . . . . . . . . . . . 12 𝑗(2nd ‘(𝑓𝑙)) = 𝑙
3329, 32nfral 3092 . . . . . . . . . . 11 𝑗𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3431, 33nfan 1998 . . . . . . . . . 10 𝑗(𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
35 nfcv 2907 . . . . . . . . . . 11 𝑗ran 𝑓
36 nfiu1 4708 . . . . . . . . . . 11 𝑗 𝑗𝐴 ({𝑗} × 𝐵)
3735, 36nfss 3756 . . . . . . . . . 10 𝑗ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)
3834, 37nfan 1998 . . . . . . . . 9 𝑗((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
3927, 38nfan 1998 . . . . . . . 8 𝑗(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
40 nfv 2009 . . . . . . . 8 𝑗 𝑧 ∈ ran 𝑓
4139, 40nfan 1998 . . . . . . 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 477 . . . . . . . . . . 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 6381 . . . . . . . . . 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 785 . . . . . . . . . . 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 803 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 489 . . . . . . . . . . . 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 717 . . . . . . . . . . 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 6382 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
50 id 22 . . . . . . . . . . . . 13 (𝑙 = 𝑘𝑙 = 𝑘)
5149, 50eqeq12d 2780 . . . . . . . . . . . 12 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5251rspcva 3460 . . . . . . . . . . 11 ((𝑘 𝑗𝐴 𝐵 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5344, 48, 52syl2anc 579 . . . . . . . . . 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 2801 . . . . . . . . 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 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)) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
5655ad2antrr 717 . . . . . . . . . 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 6726 . . . . . . . . . 10 ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑗𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
5856, 44, 57syl2anc 579 . . . . . . . . 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 6381 . . . . . . . . 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 2806 . . . . . . . 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 6323 . . . . . . . . . 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 793 . . . . . . . . 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 6434 . . . . . . . . . 10 (𝑓 Fn 𝑗𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧))
6564biimpa 468 . . . . . . . . 9 ((𝑓 Fn 𝑗𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧)
6662, 63, 65syl2anc 579 . . . . . . . 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 3225 . . . . . . 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 789 . . . . . . . . 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 3763 . . . . . . . 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 4682 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵))
7169, 70sylib 209 . . . . . . 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 3223 . . . . . 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 2907 . . . . . . . . . 10 𝑗𝑘
7473, 29nfel 2920 . . . . . . . . 9 𝑗 𝑘 𝑗𝐴 𝐵
7527, 74nfan 1998 . . . . . . . 8 𝑗(𝜑𝑘 𝑗𝐴 𝐵)
76 esumiun.2 . . . . . . . . 9 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
7776adantllr 710 . . . . . . . 8 ((((𝜑𝑘 𝑗𝐴 𝐵) ∧ 𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
78 eliun 4682 . . . . . . . . . 10 (𝑘 𝑗𝐴 𝐵 ↔ ∃𝑗𝐴 𝑘𝐵)
7978biimpi 207 . . . . . . . . 9 (𝑘 𝑗𝐴 𝐵 → ∃𝑗𝐴 𝑘𝐵)
8079adantl 473 . . . . . . . 8 ((𝜑𝑘 𝑗𝐴 𝐵) → ∃𝑗𝐴 𝑘𝐵)
8175, 77, 80r19.29af 3223 . . . . . . 7 ((𝜑𝑘 𝑗𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
8281adantlr 706 . . . . . 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 30562 . . . . 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 2771 . . . 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 5066 . . . . . . . . . 10 {𝑗} ∈ V
8685a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐴) → {𝑗} ∈ V)
87 xpexg 7160 . . . . . . . . 9 (({𝑗} ∈ V ∧ 𝐵𝑊) → ({𝑗} × 𝐵) ∈ V)
8886, 2, 87syl2anc 579 . . . . . . . 8 ((𝜑𝑗𝐴) → ({𝑗} × 𝐵) ∈ V)
8988ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
90 iunexg 7343 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
911, 89, 90syl2anc 579 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
9291adantr 472 . . . . 5 ((𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
93 nfcv 2907 . . . . . . . . 9 𝑗𝑧
9493, 36nfel 2920 . . . . . . . 8 𝑗 𝑧 𝑗𝐴 ({𝑗} × 𝐵)
9527, 94nfan 1998 . . . . . . 7 𝑗(𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵))
96 nfcv 2907 . . . . . . . . 9 𝑗(2nd𝑧)
97 nfcv 2907 . . . . . . . . 9 𝑗𝐶
9896, 97nfcsb 3711 . . . . . . . 8 𝑗(2nd𝑧) / 𝑘𝐶
99 nfcv 2907 . . . . . . . 8 𝑗(0[,]+∞)
10098, 99nfel 2920 . . . . . . 7 𝑗(2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞)
101 simprr 789 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) ∈ 𝐵)
102 simplll 791 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
103 simplr 785 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑗𝐴)
10476ralrimiva 3113 . . . . . . . . 9 ((𝜑𝑗𝐴) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
105102, 103, 104syl2anc 579 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
106 rspcsbela 4170 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ (0[,]+∞)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
107101, 105, 106syl2anc 579 . . . . . . 7 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
108 xp1st 7400 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
109 elsni 4353 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
110108, 109syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
111 xp2nd 7401 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (2nd𝑧) ∈ 𝐵)
112110, 111jca 507 . . . . . . . . . 10 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
113112reximi 3157 . . . . . . . . 9 (∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11470, 113sylbi 208 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
115114adantl 473 . . . . . . 7 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11695, 100, 107, 115r19.29af2 3222 . . . . . 6 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
117116adantlr 706 . . . . 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[,]+∞))
118 simprr 789 . . . . . 6 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
119118adantrlr 714 . . . . 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))
12012, 92, 117, 119esummono 30566 . . . 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)
12184, 120eqbrtrrd 4835 . . 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)
122 vex 3353 . . . . . . . . 9 𝑗 ∈ V
123 vex 3353 . . . . . . . . 9 𝑘 ∈ V
124122, 123op2ndd 7379 . . . . . . . 8 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = 𝑘)
125124eqcomd 2771 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
126125, 18syl 17 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = (2nd𝑧) / 𝑘𝐶)
127126eqcomd 2771 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) / 𝑘𝐶 = 𝐶)
12876anasss 458 . . . . 5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))
12914, 127, 1, 2, 128esum2d 30605 . . . 4 (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
130129adantr 472 . . 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)
131121, 130breqtrrd 4839 . 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)
13211, 131exlimddv 2030 1 (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  csb 3693  wss 3734  {csn 4336  cop 4342   ciun 4678   class class class wbr 4811   × cxp 5277  ccnv 5278  ran crn 5280   Fn wfn 6065  1-1wf1 6067  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6844  1st c1st 7366  2nd c2nd 7367  0cc0 10191  +∞cpnf 10327  cle 10331  [,]cicc 12383  Σ*cesum 30539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-reg 8706  ax-inf2 8755  ax-ac2 9540  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269  ax-addf 10270  ax-mulf 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-om 7266  df-1st 7368  df-2nd 7369  df-supp 7500  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-ixp 8116  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fsupp 8485  df-fi 8526  df-sup 8557  df-inf 8558  df-oi 8624  df-r1 8844  df-rank 8845  df-card 9018  df-ac 9192  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-4 11339  df-5 11340  df-6 11341  df-7 11342  df-8 11343  df-9 11344  df-n0 11541  df-z 11627  df-dec 11744  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-ioo 12384  df-ioc 12385  df-ico 12386  df-icc 12387  df-fz 12537  df-fzo 12677  df-fl 12804  df-mod 12880  df-seq 13012  df-exp 13071  df-fac 13268  df-bc 13297  df-hash 13325  df-shft 14095  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-limsup 14490  df-clim 14507  df-rlim 14508  df-sum 14705  df-ef 15083  df-sin 15085  df-cos 15086  df-pi 15088  df-struct 16135  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-mulr 16231  df-starv 16232  df-sca 16233  df-vsca 16234  df-ip 16235  df-tset 16236  df-ple 16237  df-ds 16239  df-unif 16240  df-hom 16241  df-cco 16242  df-rest 16352  df-topn 16353  df-0g 16371  df-gsum 16372  df-topgen 16373  df-pt 16374  df-prds 16377  df-ordt 16430  df-xrs 16431  df-qtop 16436  df-imas 16437  df-xps 16439  df-mre 16515  df-mrc 16516  df-acs 16518  df-ps 17469  df-tsr 17470  df-plusf 17510  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-mhm 17604  df-submnd 17605  df-grp 17695  df-minusg 17696  df-sbg 17697  df-mulg 17811  df-subg 17858  df-cntz 18016  df-cmn 18464  df-abl 18465  df-mgp 18760  df-ur 18772  df-ring 18819  df-cring 18820  df-subrg 19050  df-abv 19089  df-lmod 19137  df-scaf 19138  df-sra 19449  df-rgmod 19450  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-fbas 20019  df-fg 20020  df-cnfld 20023  df-top 20981  df-topon 20998  df-topsp 21020  df-bases 21033  df-cld 21106  df-ntr 21107  df-cls 21108  df-nei 21185  df-lp 21223  df-perf 21224  df-cn 21314  df-cnp 21315  df-haus 21402  df-tx 21648  df-hmeo 21841  df-fil 21932  df-fm 22024  df-flim 22025  df-flf 22026  df-tmd 22158  df-tgp 22159  df-tsms 22212  df-trg 22245  df-xms 22407  df-ms 22408  df-tms 22409  df-nm 22669  df-ngp 22670  df-nrg 22672  df-nlm 22673  df-ii 22962  df-cncf 22963  df-limc 23924  df-dv 23925  df-log 24597  df-esum 30540
This theorem is referenced by:  omssubadd  30812
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