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Theorem esumiun 30130
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 29436 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6135 . . . . . 6 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 591 . . . . 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 6088 . . . . . . 7 (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) → 𝑓: 𝑗𝐴 𝐵 𝑗𝐴 ({𝑗} × 𝐵))
7 frn 6040 . . . . . . 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 481 . . . . 5 ((𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
105, 9jca 554 . . . 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 1760 . . 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 1841 . . . . . 6 𝑧(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
14 nfcv 2762 . . . . . 6 𝑧𝐶
15 nfcsb1v 3542 . . . . . 6 𝑘(2nd𝑧) / 𝑘𝐶
16 nfcv 2762 . . . . . 6 𝑧 𝑗𝐴 𝐵
17 nfcv 2762 . . . . . 6 𝑧ran 𝑓
18 nfcv 2762 . . . . . 6 𝑧𝑓
19 csbeq1a 3535 . . . . . 6 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
202ralrimiva 2963 . . . . . . . 8 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
21 iunexg 7128 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐵𝑊) → 𝑗𝐴 𝐵 ∈ V)
221, 20, 21syl2anc 692 . . . . . . 7 (𝜑 𝑗𝐴 𝐵 ∈ V)
2322adantr 481 . . . . . 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 793 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓)
25 f1ocnv 6136 . . . . . . . 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 758 . . . . . 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 1841 . . . . . . . . 9 𝑗𝜑
29 nfcv 2762 . . . . . . . . . . . 12 𝑗𝑓
30 nfiu1 4541 . . . . . . . . . . . 12 𝑗 𝑗𝐴 𝐵
3129nfrn 5357 . . . . . . . . . . . 12 𝑗ran 𝑓
3229, 30, 31nff1o 6122 . . . . . . . . . . 11 𝑗 𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓
33 nfv 1841 . . . . . . . . . . . 12 𝑗(2nd ‘(𝑓𝑙)) = 𝑙
3430, 33nfral 2942 . . . . . . . . . . 11 𝑗𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3532, 34nfan 1826 . . . . . . . . . 10 𝑗(𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
36 nfcv 2762 . . . . . . . . . . 11 𝑗ran 𝑓
37 nfiu1 4541 . . . . . . . . . . 11 𝑗 𝑗𝐴 ({𝑗} × 𝐵)
3836, 37nfss 3588 . . . . . . . . . 10 𝑗ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)
3935, 38nfan 1826 . . . . . . . . 9 𝑗((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
4028, 39nfan 1826 . . . . . . . 8 𝑗(𝜑 ∧ ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵)))
41 nfv 1841 . . . . . . . 8 𝑗 𝑧 ∈ ran 𝑓
4240, 41nfan 1826 . . . . . . 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 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)
4443fveq2d 6182 . . . . . . . . . 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 791 . . . . . . . . . . 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 806 . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 761 . . . . . . . . . . 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 6178 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (𝑓𝑙) = (𝑓𝑘))
5150fveq2d 6182 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
52 id 22 . . . . . . . . . . . . 13 (𝑙 = 𝑘𝑙 = 𝑘)
5351, 52eqeq12d 2635 . . . . . . . . . . . 12 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5453rspcva 3302 . . . . . . . . . . 11 ((𝑘 𝑗𝐴 𝐵 ∧ ∀𝑙 𝑗𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5545, 49, 54syl2anc 692 . . . . . . . . . 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 2656 . . . . . . . . 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 475 . . . . . . . . . . 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 761 . . . . . . . . . 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 6517 . . . . . . . . . 10 ((𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑗𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6058, 45, 59syl2anc 692 . . . . . . . . 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 6182 . . . . . . . . 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 2661 . . . . . . . 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 6125 . . . . . . . . . 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 798 . . . . . . . . 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 6230 . . . . . . . . . 10 (𝑓 Fn 𝑗𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧))
6766biimpa 501 . . . . . . . . 9 ((𝑓 Fn 𝑗𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑗𝐴 𝐵(𝑓𝑘) = 𝑧)
6864, 65, 67syl2anc 692 . . . . . . . 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 3074 . . . . . . 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 795 . . . . . . . . 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 3595 . . . . . . . 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 4515 . . . . . . . 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 3072 . . . . . 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 2762 . . . . . . . . . 10 𝑗𝑘
7675, 30nfel 2774 . . . . . . . . 9 𝑗 𝑘 𝑗𝐴 𝐵
7728, 76nfan 1826 . . . . . . . 8 𝑗(𝜑𝑘 𝑗𝐴 𝐵)
78 esumiun.2 . . . . . . . . 9 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
7978adantllr 754 . . . . . . . 8 ((((𝜑𝑘 𝑗𝐴 𝐵) ∧ 𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))
80 eliun 4515 . . . . . . . . . 10 (𝑘 𝑗𝐴 𝐵 ↔ ∃𝑗𝐴 𝑘𝐵)
8180biimpi 206 . . . . . . . . 9 (𝑘 𝑗𝐴 𝐵 → ∃𝑗𝐴 𝑘𝐵)
8281adantl 482 . . . . . . . 8 ((𝜑𝑘 𝑗𝐴 𝐵) → ∃𝑗𝐴 𝑘𝐵)
8377, 79, 82r19.29af 3072 . . . . . . 7 ((𝜑𝑘 𝑗𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
8483adantlr 750 . . . . . 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 30086 . . . . 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 2626 . . . 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 4899 . . . . . . . . . 10 {𝑗} ∈ V
8887a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐴) → {𝑗} ∈ V)
89 xpexg 6945 . . . . . . . . 9 (({𝑗} ∈ V ∧ 𝐵𝑊) → ({𝑗} × 𝐵) ∈ V)
9088, 2, 89syl2anc 692 . . . . . . . 8 ((𝜑𝑗𝐴) → ({𝑗} × 𝐵) ∈ V)
9190ralrimiva 2963 . . . . . . 7 (𝜑 → ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
92 iunexg 7128 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑗𝐴 ({𝑗} × 𝐵) ∈ V) → 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
931, 91, 92syl2anc 692 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) ∈ V)
9493adantr 481 . . . . 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 2762 . . . . . . . . 9 𝑗𝑧
9695, 37nfel 2774 . . . . . . . 8 𝑗 𝑧 𝑗𝐴 ({𝑗} × 𝐵)
9728, 96nfan 1826 . . . . . . 7 𝑗(𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵))
98 nfcv 2762 . . . . . . . . 9 𝑗(2nd𝑧)
99 nfcv 2762 . . . . . . . . 9 𝑗𝐶
10098, 99nfcsb 3544 . . . . . . . 8 𝑗(2nd𝑧) / 𝑘𝐶
101 nfcv 2762 . . . . . . . 8 𝑗(0[,]+∞)
102100, 101nfel 2774 . . . . . . 7 𝑗(2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞)
103 simprr 795 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) ∈ 𝐵)
104 simplll 797 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
105 simplr 791 . . . . . . . . 9 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑗𝐴)
10678ralrimiva 2963 . . . . . . . . 9 ((𝜑𝑗𝐴) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
107104, 105, 106syl2anc 692 . . . . . . . 8 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 𝐶 ∈ (0[,]+∞))
108 rspcsbela 3997 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ (0[,]+∞)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
109103, 107, 108syl2anc 692 . . . . . . 7 ((((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) ∧ 𝑗𝐴) ∧ ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
110 xp1st 7183 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
111 elsni 4185 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
112110, 111syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
113 xp2nd 7184 . . . . . . . . . . 11 (𝑧 ∈ ({𝑗} × 𝐵) → (2nd𝑧) ∈ 𝐵)
114112, 113jca 554 . . . . . . . . . 10 (𝑧 ∈ ({𝑗} × 𝐵) → ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
115114reximi 3008 . . . . . . . . 9 (∃𝑗𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11672, 115sylbi 207 . . . . . . . 8 (𝑧 𝑗𝐴 ({𝑗} × 𝐵) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
117116adantl 482 . . . . . . 7 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → ∃𝑗𝐴 ((1st𝑧) = 𝑗 ∧ (2nd𝑧) ∈ 𝐵))
11897, 102, 109, 117r19.29af2 3071 . . . . . 6 ((𝜑𝑧 𝑗𝐴 ({𝑗} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ (0[,]+∞))
119118adantlr 750 . . . . 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 795 . . . . . 6 ((𝜑 ∧ (𝑓: 𝑗𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))) → ran 𝑓 𝑗𝐴 ({𝑗} × 𝐵))
121120adantrlr 758 . . . . 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 30090 . . . 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 4668 . . 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 3198 . . . . . . . . 9 𝑗 ∈ V
125 vex 3198 . . . . . . . . 9 𝑘 ∈ V
126124, 125op2ndd 7164 . . . . . . . 8 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = 𝑘)
127126eqcomd 2626 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
128127, 19syl 17 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = (2nd𝑧) / 𝑘𝐶)
129128eqcomd 2626 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) / 𝑘𝐶 = 𝐶)
13078anasss 678 . . . . 5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))
13115, 129, 1, 2, 130esum2d 30129 . . . 4 (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)(2nd𝑧) / 𝑘𝐶)
132131adantr 481 . . 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 4672 . 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 1861 1 (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  csb 3526  wss 3567  {csn 4168  cop 4174   ciun 4511   class class class wbr 4644   × cxp 5102  ccnv 5103  ran crn 5105   Fn wfn 5871  wf 5872  1-1wf1 5873  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  0cc0 9921  +∞cpnf 10056  cle 10060  [,]cicc 12163  Σ*cesum 30063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-inf2 8523  ax-ac2 9270  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000  ax-mulf 10001
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-fi 8302  df-sup 8333  df-inf 8334  df-oi 8400  df-r1 8612  df-rank 8613  df-card 8750  df-ac 8924  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ioo 12164  df-ioc 12165  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-fac 13044  df-bc 13073  df-hash 13101  df-shft 13788  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-limsup 14183  df-clim 14200  df-rlim 14201  df-sum 14398  df-ef 14779  df-sin 14781  df-cos 14782  df-pi 14784  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-starv 15937  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-unif 15946  df-hom 15947  df-cco 15948  df-rest 16064  df-topn 16065  df-0g 16083  df-gsum 16084  df-topgen 16085  df-pt 16086  df-prds 16089  df-ordt 16142  df-xrs 16143  df-qtop 16148  df-imas 16149  df-xps 16151  df-mre 16227  df-mrc 16228  df-acs 16230  df-ps 17181  df-tsr 17182  df-plusf 17222  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-mhm 17316  df-submnd 17317  df-grp 17406  df-minusg 17407  df-sbg 17408  df-mulg 17522  df-subg 17572  df-cntz 17731  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-cring 18531  df-subrg 18759  df-abv 18798  df-lmod 18846  df-scaf 18847  df-sra 19153  df-rgmod 19154  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-fbas 19724  df-fg 19725  df-cnfld 19728  df-top 20680  df-topon 20697  df-topsp 20718  df-bases 20731  df-cld 20804  df-ntr 20805  df-cls 20806  df-nei 20883  df-lp 20921  df-perf 20922  df-cn 21012  df-cnp 21013  df-haus 21100  df-tx 21346  df-hmeo 21539  df-fil 21631  df-fm 21723  df-flim 21724  df-flf 21725  df-tmd 21857  df-tgp 21858  df-tsms 21911  df-trg 21944  df-xms 22106  df-ms 22107  df-tms 22108  df-nm 22368  df-ngp 22369  df-nrg 22371  df-nlm 22372  df-ii 22661  df-cncf 22662  df-limc 23611  df-dv 23612  df-log 24284  df-esum 30064
This theorem is referenced by:  omssubadd  30336
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