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

Theorem fsumiunle 30298
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 30170 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6455 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 605 . . . . 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 6404 . . . . . . 7 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵 𝑥𝐴 ({𝑥} × 𝐵))
76frnd 6351 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
87adantr 473 . . . . 5 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
95, 8jca 504 . . . 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 1797 . . 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 3795 . . . . . . 7 (𝑘 = 𝑦𝐶 = 𝑦 / 𝑘𝐶)
13 nfcv 2932 . . . . . . 7 𝑦 𝑥𝐴 𝐵
14 nfcv 2932 . . . . . . 7 𝑘 𝑥𝐴 𝐵
15 nfcv 2932 . . . . . . 7 𝑦𝐶
16 nfcsb1v 3804 . . . . . . 7 𝑘𝑦 / 𝑘𝐶
1712, 13, 14, 15, 16cbvsum 14912 . . . . . 6 Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑦 𝑥𝐴 𝐵𝑦 / 𝑘𝐶
18 csbeq1 3789 . . . . . . 7 (𝑦 = (2nd𝑧) → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
19 snfi 8391 . . . . . . . . . . . 12 {𝑥} ∈ Fin
20 xpfi 8584 . . . . . . . . . . . 12 (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
2119, 2, 20sylancr 578 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ({𝑥} × 𝐵) ∈ Fin)
2221ralrimiva 3132 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
23 iunfi 8607 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
241, 22, 23syl2anc 576 . . . . . . . . 9 (𝜑 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
2524adantr 473 . . . . . . . 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 760 . . . . . . . 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 8536 . . . . . . 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 758 . . . . . . . . 9 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
29 f1ocnv 6456 . . . . . . . . 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 710 . . . . . . 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 1873 . . . . . . . . . 10 𝑥𝜑
33 nfcv 2932 . . . . . . . . . . . . 13 𝑥𝑓
34 nfiu1 4823 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
3533nfrn 5667 . . . . . . . . . . . . 13 𝑥ran 𝑓
3633, 34, 35nff1o 6442 . . . . . . . . . . . 12 𝑥 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓
37 nfv 1873 . . . . . . . . . . . . 13 𝑥(2nd ‘(𝑓𝑙)) = 𝑙
3834, 37nfral 3174 . . . . . . . . . . . 12 𝑥𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3936, 38nfan 1862 . . . . . . . . . . 11 𝑥(𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
40 nfcv 2932 . . . . . . . . . . . 12 𝑥ran 𝑓
41 nfiu1 4823 . . . . . . . . . . . 12 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
4240, 41nfss 3851 . . . . . . . . . . 11 𝑥ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)
4339, 42nfan 1862 . . . . . . . . . 10 𝑥((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
4432, 43nfan 1862 . . . . . . . . 9 𝑥(𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
45 nfv 1873 . . . . . . . . 9 𝑥 𝑧 ∈ ran 𝑓
4644, 45nfan 1862 . . . . . . . 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 477 . . . . . . . . . . . 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 6503 . . . . . . . . . . 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 756 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 713 . . . . . . . . . . . 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 6504 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
55 id 22 . . . . . . . . . . . . . 14 (𝑙 = 𝑘𝑙 = 𝑘)
5654, 55eqeq12d 2793 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5756rspcva 3533 . . . . . . . . . . . 12 ((𝑘 𝑥𝐴 𝐵 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5849, 53, 57syl2anc 576 . . . . . . . . . . 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 2816 . . . . . . . . . 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 487 . . . . . . . . . . . 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 713 . . . . . . . . . . 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 6858 . . . . . . . . . . 11 ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑥𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6361, 49, 62syl2anc 576 . . . . . . . . . 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 6503 . . . . . . . . . 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 2821 . . . . . . . . 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 6445 . . . . . . . . . . 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 763 . . . . . . . . . 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 6556 . . . . . . . . . . 11 (𝑓 Fn 𝑥𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧))
7069biimpa 469 . . . . . . . . . 10 ((𝑓 Fn 𝑥𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧)
7167, 68, 70syl2anc 576 . . . . . . . . 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 3234 . . . . . . . 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 3858 . . . . . . . . 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 4796 . . . . . . . . 9 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
7573, 74sylib 210 . . . . . . . 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 3272 . . . . . . 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 1873 . . . . . . . . . 10 𝑘(𝜑𝑦 𝑥𝐴 𝐵)
78 nfcv 2932 . . . . . . . . . . 11 𝑘
7916, 78nfel 2944 . . . . . . . . . 10 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
8077, 79nfim 1859 . . . . . . . . 9 𝑘((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
81 eleq1w 2848 . . . . . . . . . . 11 (𝑘 = 𝑦 → (𝑘 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
8281anbi2d 619 . . . . . . . . . 10 (𝑘 = 𝑦 → ((𝜑𝑘 𝑥𝐴 𝐵) ↔ (𝜑𝑦 𝑥𝐴 𝐵)))
8312eleq1d 2850 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑘𝐶 ∈ ℂ))
8482, 83imbi12d 337 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
85 nfcv 2932 . . . . . . . . . . . 12 𝑥𝑘
8685, 34nfel 2944 . . . . . . . . . . 11 𝑥 𝑘 𝑥𝐴 𝐵
8732, 86nfan 1862 . . . . . . . . . 10 𝑥(𝜑𝑘 𝑥𝐴 𝐵)
88 fsumiunle.3 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
8988adantllr 706 . . . . . . . . . . 11 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
9089recnd 10468 . . . . . . . . . 10 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
91 eliun 4796 . . . . . . . . . . . 12 (𝑘 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑘𝐵)
9291biimpi 208 . . . . . . . . . . 11 (𝑘 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑘𝐵)
9392adantl 474 . . . . . . . . . 10 ((𝜑𝑘 𝑥𝐴 𝐵) → ∃𝑥𝐴 𝑘𝐵)
9487, 90, 93r19.29af 3272 . . . . . . . . 9 ((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ)
9580, 84, 94chvar 2326 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
9695adantlr 702 . . . . . . 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 14940 . . . . . 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 2826 . . . . 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 2784 . . . 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 2932 . . . . . . . . 9 𝑥𝑧
101100, 41nfel 2944 . . . . . . . 8 𝑥 𝑧 𝑥𝐴 ({𝑥} × 𝐵)
10232, 101nfan 1862 . . . . . . 7 𝑥(𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵))
103 xp2nd 7534 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
104103adantl 474 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) ∈ 𝐵)
10588ralrimiva 3132 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
106105adantlr 702 . . . . . . . . 9 (((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
107106adantr 473 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℝ)
108 nfcsb1v 3804 . . . . . . . . . . 11 𝑘(2nd𝑧) / 𝑘𝐶
109108nfel1 2946 . . . . . . . . . 10 𝑘(2nd𝑧) / 𝑘𝐶 ∈ ℝ
110 csbeq1a 3795 . . . . . . . . . . 11 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
111110eleq1d 2850 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (𝐶 ∈ ℝ ↔ (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
112109, 111rspc 3529 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 𝐶 ∈ ℝ → (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
113112imp 398 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ ℝ) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
114104, 107, 113syl2anc 576 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
11574biimpi 208 . . . . . . . 8 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
116115adantl 474 . . . . . . 7 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
117102, 114, 116r19.29af 3272 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
118117adantlr 702 . . . . 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 7533 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) ∈ {𝑥})
120 elsni 4458 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑥} → (1st𝑧) = 𝑥)
121119, 120syl 17 . . . . . . . . . 10 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) = 𝑥)
122121, 103jca 504 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵))
123 simplll 762 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
124 simplr 756 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑥𝐴)
125 fsumiunle.4 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)
126125ralrimiva 3132 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 0 ≤ 𝐶)
127123, 124, 126syl2anc 576 . . . . . . . . 9 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
128122, 127sylan2 583 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
129 nfcv 2932 . . . . . . . . . . 11 𝑘0
130 nfcv 2932 . . . . . . . . . . 11 𝑘
131129, 130, 108nfbr 4976 . . . . . . . . . 10 𝑘0 ≤ (2nd𝑧) / 𝑘𝐶
132110breq2d 4941 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ (2nd𝑧) / 𝑘𝐶))
133131, 132rspc 3529 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 0 ≤ 𝐶 → 0 ≤ (2nd𝑧) / 𝑘𝐶))
134133imp 398 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 0 ≤ 𝐶) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
135104, 128, 134syl2anc 576 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
136102, 135, 116r19.29af 3272 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
137136adantlr 702 . . . . 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 15011 . . . 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 4953 . . 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 2932 . . . . . . . 8 𝑦𝐵
141 nfcv 2932 . . . . . . . 8 𝑘𝐵
14212, 140, 141, 15, 16cbvsum 14912 . . . . . . 7 Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶
143142a1i 11 . . . . . 6 (𝜑 → Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶)
144143sumeq2sdv 14921 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶)
145 vex 3418 . . . . . . . . . 10 𝑥 ∈ V
146 vex 3418 . . . . . . . . . 10 𝑦 ∈ V
147145, 146op2ndd 7512 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
148147eqcomd 2784 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝑧))
149148csbeq1d 3793 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
150149eqcomd 2784 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘𝐶 = 𝑦 / 𝑘𝐶)
151 nfv 1873 . . . . . . . . 9 𝑘((𝜑𝑥𝐴) ∧ 𝑦𝐵)
15216nfel1 2946 . . . . . . . . 9 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
153151, 152nfim 1859 . . . . . . . 8 𝑘(((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
154 eleq1w 2848 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝑘𝐵𝑦𝐵))
155154anbi2d 619 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑥𝐴) ∧ 𝑘𝐵) ↔ ((𝜑𝑥𝐴) ∧ 𝑦𝐵)))
156155, 83imbi12d 337 . . . . . . . 8 (𝑘 = 𝑦 → ((((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
15788recnd 10468 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
158153, 156, 157chvar 2326 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
159158anasss 459 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 / 𝑘𝐶 ∈ ℂ)
160150, 1, 2, 159fsum2d 14986 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
161144, 160eqtrd 2814 . . . 4 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
162161adantr 473 . . 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 4957 . 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 1894 1 (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wex 1742  wcel 2050  wral 3088  wrex 3089  csb 3786  wss 3829  {csn 4441  cop 4447   ciun 4792   class class class wbr 4929   × cxp 5405  ccnv 5406  ran crn 5408   Fn wfn 6183  1-1wf1 6185  1-1-ontowf1o 6187  cfv 6188  1st c1st 7499  2nd c2nd 7500  Fincfn 8306  cc 10333  cr 10334  0cc0 10335  cle 10475  Σcsu 14903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-reg 8851  ax-inf2 8898  ax-ac2 9683  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-sup 8701  df-oi 8769  df-r1 8987  df-rank 8988  df-card 9162  df-ac 9336  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-3 11504  df-n0 11708  df-z 11794  df-uz 12059  df-rp 12205  df-ico 12560  df-fz 12709  df-fzo 12850  df-seq 13185  df-exp 13245  df-hash 13506  df-cj 14319  df-re 14320  df-im 14321  df-sqrt 14455  df-abs 14456  df-clim 14706  df-sum 14904
This theorem is referenced by:  hgt750lema  31582
  Copyright terms: Public domain W3C validator