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Theorem fsumiunle 30571
Description: Upper bound for a sum of nonnegative terms over an indexed union. The inequality may be strict if the indexed union is non-disjoint, since in the right hand side, a summand may be counted several times. (Contributed by Thierry Arnoux, 1-Jan-2021.)
Hypotheses
Ref Expression
fsumiunle.1 (𝜑𝐴 ∈ Fin)
fsumiunle.2 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
fsumiunle.3 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
fsumiunle.4 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)
Assertion
Ref Expression
fsumiunle (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘,𝑥   𝐵,𝑘   𝑥,𝐶   𝜑,𝑘,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑘)

Proof of Theorem fsumiunle
Dummy variables 𝑓 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsumiunle.1 . . . 4 (𝜑𝐴 ∈ Fin)
2 fsumiunle.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)
31, 2aciunf1 30426 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6601 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 617 . . . . 5 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
6 f1f 6549 . . . . . . 7 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵 𝑥𝐴 ({𝑥} × 𝐵))
76frnd 6494 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
87adantr 484 . . . . 5 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
95, 8jca 515 . . . 4 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
109eximi 1836 . . 3 (∃𝑓(𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ∃𝑓((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
113, 10syl 17 . 2 (𝜑 → ∃𝑓((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
12 csbeq1a 3842 . . . . . . 7 (𝑘 = 𝑦𝐶 = 𝑦 / 𝑘𝐶)
13 nfcv 2955 . . . . . . 7 𝑦 𝑥𝐴 𝐵
14 nfcv 2955 . . . . . . 7 𝑘 𝑥𝐴 𝐵
15 nfcv 2955 . . . . . . 7 𝑦𝐶
16 nfcsb1v 3852 . . . . . . 7 𝑘𝑦 / 𝑘𝐶
1712, 13, 14, 15, 16cbvsum 15044 . . . . . 6 Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑦 𝑥𝐴 𝐵𝑦 / 𝑘𝐶
18 csbeq1 3831 . . . . . . 7 (𝑦 = (2nd𝑧) → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
19 snfi 8577 . . . . . . . . . . . 12 {𝑥} ∈ Fin
20 xpfi 8773 . . . . . . . . . . . 12 (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
2119, 2, 20sylancr 590 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ({𝑥} × 𝐵) ∈ Fin)
2221ralrimiva 3149 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
23 iunfi 8796 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
241, 22, 23syl2anc 587 . . . . . . . . 9 (𝜑 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
2524adantr 484 . . . . . . . 8 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
26 simprr 772 . . . . . . . 8 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
2725, 26ssfid 8725 . . . . . . 7 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ∈ Fin)
28 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
29 f1ocnv 6602 . . . . . . . . 9 (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑓:ran 𝑓1-1-onto 𝑥𝐴 𝐵)
3028, 29syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓:ran 𝑓1-1-onto 𝑥𝐴 𝐵)
3130adantrlr 722 . . . . . . 7 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓:ran 𝑓1-1-onto 𝑥𝐴 𝐵)
32 nfv 1915 . . . . . . . . . 10 𝑥𝜑
33 nfcv 2955 . . . . . . . . . . . . 13 𝑥𝑓
34 nfiu1 4915 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
3533nfrn 5788 . . . . . . . . . . . . 13 𝑥ran 𝑓
3633, 34, 35nff1o 6588 . . . . . . . . . . . 12 𝑥 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓
37 nfv 1915 . . . . . . . . . . . . 13 𝑥(2nd ‘(𝑓𝑙)) = 𝑙
3834, 37nfralw 3189 . . . . . . . . . . . 12 𝑥𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3936, 38nfan 1900 . . . . . . . . . . 11 𝑥(𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
40 nfcv 2955 . . . . . . . . . . . 12 𝑥ran 𝑓
41 nfiu1 4915 . . . . . . . . . . . 12 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
4240, 41nfss 3907 . . . . . . . . . . 11 𝑥ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)
4339, 42nfan 1900 . . . . . . . . . 10 𝑥((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
4432, 43nfan 1900 . . . . . . . . 9 𝑥(𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
45 nfv 1915 . . . . . . . . 9 𝑥 𝑧 ∈ ran 𝑓
4644, 45nfan 1900 . . . . . . . 8 𝑥((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
47 simpr 488 . . . . . . . . . . . 12 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓𝑘) = 𝑧)
4847fveq2d 6649 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (2nd ‘(𝑓𝑘)) = (2nd𝑧))
49 simplr 768 . . . . . . . . . . . 12 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → 𝑘 𝑥𝐴 𝐵)
50 simp-4r 783 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
5150simpld 498 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
5251simprd 499 . . . . . . . . . . . . 13 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
5352ad2antrr 725 . . . . . . . . . . . 12 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
54 2fveq3 6650 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
55 id 22 . . . . . . . . . . . . . 14 (𝑙 = 𝑘𝑙 = 𝑘)
5654, 55eqeq12d 2814 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5756rspcva 3569 . . . . . . . . . . . 12 ((𝑘 𝑥𝐴 𝐵 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5849, 53, 57syl2anc 587 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (2nd ‘(𝑓𝑘)) = 𝑘)
5948, 58eqtr3d 2835 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (2nd𝑧) = 𝑘)
6051simpld 498 . . . . . . . . . . . 12 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
6160ad2antrr 725 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
62 f1ocnvfv1 7011 . . . . . . . . . . 11 ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑥𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6361, 49, 62syl2anc 587 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓‘(𝑓𝑘)) = 𝑘)
6447fveq2d 6649 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓‘(𝑓𝑘)) = (𝑓𝑧))
6559, 63, 643eqtr2rd 2840 . . . . . . . . 9 (((((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 𝑥𝐴 𝐵) ∧ (𝑓𝑘) = 𝑧) → (𝑓𝑧) = (2nd𝑧))
66 f1ofn 6591 . . . . . . . . . . 11 (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑓 Fn 𝑥𝐴 𝐵)
6760, 66syl 17 . . . . . . . . . 10 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑓 Fn 𝑥𝐴 𝐵)
68 simpllr 775 . . . . . . . . . 10 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑧 ∈ ran 𝑓)
69 fvelrnb 6701 . . . . . . . . . . 11 (𝑓 Fn 𝑥𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧))
7069biimpa 480 . . . . . . . . . 10 ((𝑓 Fn 𝑥𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧)
7167, 68, 70syl2anc 587 . . . . . . . . 9 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧)
7265, 71r19.29a 3248 . . . . . . . 8 (((((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (𝑓𝑧) = (2nd𝑧))
7326sselda 3915 . . . . . . . . 9 (((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 𝑥𝐴 ({𝑥} × 𝐵))
74 eliun 4885 . . . . . . . . 9 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
7573, 74sylib 221 . . . . . . . 8 (((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
7646, 72, 75r19.29af 3289 . . . . . . 7 (((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (𝑓𝑧) = (2nd𝑧))
77 nfv 1915 . . . . . . . . . 10 𝑘(𝜑𝑦 𝑥𝐴 𝐵)
78 nfcv 2955 . . . . . . . . . . 11 𝑘
7916, 78nfel 2969 . . . . . . . . . 10 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
8077, 79nfim 1897 . . . . . . . . 9 𝑘((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
81 eleq1w 2872 . . . . . . . . . . 11 (𝑘 = 𝑦 → (𝑘 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
8281anbi2d 631 . . . . . . . . . 10 (𝑘 = 𝑦 → ((𝜑𝑘 𝑥𝐴 𝐵) ↔ (𝜑𝑦 𝑥𝐴 𝐵)))
8312eleq1d 2874 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑘𝐶 ∈ ℂ))
8482, 83imbi12d 348 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
85 nfcv 2955 . . . . . . . . . . . 12 𝑥𝑘
8685, 34nfel 2969 . . . . . . . . . . 11 𝑥 𝑘 𝑥𝐴 𝐵
8732, 86nfan 1900 . . . . . . . . . 10 𝑥(𝜑𝑘 𝑥𝐴 𝐵)
88 fsumiunle.3 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
8988adantllr 718 . . . . . . . . . . 11 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
9089recnd 10658 . . . . . . . . . 10 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
91 eliun 4885 . . . . . . . . . . . 12 (𝑘 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑘𝐵)
9291biimpi 219 . . . . . . . . . . 11 (𝑘 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑘𝐵)
9392adantl 485 . . . . . . . . . 10 ((𝜑𝑘 𝑥𝐴 𝐵) → ∃𝑥𝐴 𝑘𝐵)
9487, 90, 93r19.29af 3289 . . . . . . . . 9 ((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ)
9580, 84, 94chvarfv 2240 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
9695adantlr 714 . . . . . . 7 (((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
9718, 27, 31, 76, 96fsumf1o 15072 . . . . . 6 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑦 𝑥𝐴 𝐵𝑦 / 𝑘𝐶 = Σ𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶)
9817, 97syl5eq 2845 . . . . 5 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶)
9998eqcomd 2804 . . . 4 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶 = Σ𝑘 𝑥𝐴 𝐵𝐶)
100 nfcv 2955 . . . . . . . . 9 𝑥𝑧
101100, 41nfel 2969 . . . . . . . 8 𝑥 𝑧 𝑥𝐴 ({𝑥} × 𝐵)
10232, 101nfan 1900 . . . . . . 7 𝑥(𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵))
103 xp2nd 7704 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
104103adantl 485 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) ∈ 𝐵)
10588ralrimiva 3149 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
106105adantlr 714 . . . . . . . . 9 (((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
107106adantr 484 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℝ)
108 nfcsb1v 3852 . . . . . . . . . . 11 𝑘(2nd𝑧) / 𝑘𝐶
109108nfel1 2971 . . . . . . . . . 10 𝑘(2nd𝑧) / 𝑘𝐶 ∈ ℝ
110 csbeq1a 3842 . . . . . . . . . . 11 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
111110eleq1d 2874 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (𝐶 ∈ ℝ ↔ (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
112109, 111rspc 3559 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 𝐶 ∈ ℝ → (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
113112imp 410 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ ℝ) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
114104, 107, 113syl2anc 587 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
11574biimpi 219 . . . . . . . 8 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
116115adantl 485 . . . . . . 7 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
117102, 114, 116r19.29af 3289 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
118117adantlr 714 . . . . 5 (((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
119 xp1st 7703 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) ∈ {𝑥})
120 elsni 4542 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑥} → (1st𝑧) = 𝑥)
121119, 120syl 17 . . . . . . . . . 10 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) = 𝑥)
122121, 103jca 515 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵))
123 simplll 774 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
124 simplr 768 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑥𝐴)
125 fsumiunle.4 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)
126125ralrimiva 3149 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 0 ≤ 𝐶)
127123, 124, 126syl2anc 587 . . . . . . . . 9 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
128122, 127sylan2 595 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
129 nfcv 2955 . . . . . . . . . . 11 𝑘0
130 nfcv 2955 . . . . . . . . . . 11 𝑘
131129, 130, 108nfbr 5077 . . . . . . . . . 10 𝑘0 ≤ (2nd𝑧) / 𝑘𝐶
132110breq2d 5042 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ (2nd𝑧) / 𝑘𝐶))
133131, 132rspc 3559 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 0 ≤ 𝐶 → 0 ≤ (2nd𝑧) / 𝑘𝐶))
134133imp 410 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 0 ≤ 𝐶) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
135104, 128, 134syl2anc 587 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
136102, 135, 116r19.29af 3289 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
137136adantlr 714 . . . . 5 (((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
13825, 118, 137, 26fsumless 15143 . . . 4 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑧 ∈ ran 𝑓(2nd𝑧) / 𝑘𝐶 ≤ Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
13999, 138eqbrtrrd 5054 . . 3 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
140 nfcv 2955 . . . . . . . 8 𝑦𝐵
141 nfcv 2955 . . . . . . . 8 𝑘𝐵
14212, 140, 141, 15, 16cbvsum 15044 . . . . . . 7 Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶
143142a1i 11 . . . . . 6 (𝜑 → Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶)
144143sumeq2sdv 15053 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶)
145 vex 3444 . . . . . . . . . 10 𝑥 ∈ V
146 vex 3444 . . . . . . . . . 10 𝑦 ∈ V
147145, 146op2ndd 7682 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
148147eqcomd 2804 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝑧))
149148csbeq1d 3832 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
150149eqcomd 2804 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘𝐶 = 𝑦 / 𝑘𝐶)
151 nfv 1915 . . . . . . . . 9 𝑘((𝜑𝑥𝐴) ∧ 𝑦𝐵)
15216nfel1 2971 . . . . . . . . 9 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
153151, 152nfim 1897 . . . . . . . 8 𝑘(((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
154 eleq1w 2872 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝑘𝐵𝑦𝐵))
155154anbi2d 631 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑥𝐴) ∧ 𝑘𝐵) ↔ ((𝜑𝑥𝐴) ∧ 𝑦𝐵)))
156155, 83imbi12d 348 . . . . . . . 8 (𝑘 = 𝑦 → ((((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
15788recnd 10658 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
158153, 156, 157chvarfv 2240 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
159158anasss 470 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 / 𝑘𝐶 ∈ ℂ)
160150, 1, 2, 159fsum2d 15118 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
161144, 160eqtrd 2833 . . . 4 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
162161adantr 484 . . 3 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
163139, 162breqtrrd 5058 . 2 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
16411, 163exlimddv 1936 1 (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2111  wral 3106  wrex 3107  csb 3828  wss 3881  {csn 4525  cop 4531   ciun 4881   class class class wbr 5030   × cxp 5517  ccnv 5518  ran crn 5520   Fn wfn 6319  1-1wf1 6321  1-1-ontowf1o 6323  cfv 6324  1st c1st 7669  2nd c2nd 7670  Fincfn 8492  cc 10524  cr 10525  0cc0 10526  cle 10665  Σcsu 15034
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088  ax-ac2 9874  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-r1 9177  df-rank 9178  df-card 9352  df-ac 9527  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035
This theorem is referenced by:  hgt750lema  32038
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