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Theorem fsumiunle 30472
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 30336 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6619 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 614 . . . . 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 6568 . . . . . . 7 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵 𝑥𝐴 ({𝑥} × 𝐵))
76frnd 6514 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
87adantr 481 . . . . 5 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
95, 8jca 512 . . . 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 1826 . . 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 3894 . . . . . . 7 (𝑘 = 𝑦𝐶 = 𝑦 / 𝑘𝐶)
13 nfcv 2974 . . . . . . 7 𝑦 𝑥𝐴 𝐵
14 nfcv 2974 . . . . . . 7 𝑘 𝑥𝐴 𝐵
15 nfcv 2974 . . . . . . 7 𝑦𝐶
16 nfcsb1v 3904 . . . . . . 7 𝑘𝑦 / 𝑘𝐶
1712, 13, 14, 15, 16cbvsum 15040 . . . . . 6 Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑦 𝑥𝐴 𝐵𝑦 / 𝑘𝐶
18 csbeq1 3883 . . . . . . 7 (𝑦 = (2nd𝑧) → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
19 snfi 8582 . . . . . . . . . . . 12 {𝑥} ∈ Fin
20 xpfi 8777 . . . . . . . . . . . 12 (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
2119, 2, 20sylancr 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ({𝑥} × 𝐵) ∈ Fin)
2221ralrimiva 3179 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
23 iunfi 8800 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
241, 22, 23syl2anc 584 . . . . . . . . 9 (𝜑 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
2524adantr 481 . . . . . . . 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 769 . . . . . . . 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 8729 . . . . . . 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 767 . . . . . . . . 9 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
29 f1ocnv 6620 . . . . . . . . 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 719 . . . . . . 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 1906 . . . . . . . . . 10 𝑥𝜑
33 nfcv 2974 . . . . . . . . . . . . 13 𝑥𝑓
34 nfiu1 4944 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
3533nfrn 5817 . . . . . . . . . . . . 13 𝑥ran 𝑓
3633, 34, 35nff1o 6606 . . . . . . . . . . . 12 𝑥 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓
37 nfv 1906 . . . . . . . . . . . . 13 𝑥(2nd ‘(𝑓𝑙)) = 𝑙
3834, 37nfralw 3222 . . . . . . . . . . . 12 𝑥𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3936, 38nfan 1891 . . . . . . . . . . 11 𝑥(𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
40 nfcv 2974 . . . . . . . . . . . 12 𝑥ran 𝑓
41 nfiu1 4944 . . . . . . . . . . . 12 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
4240, 41nfss 3957 . . . . . . . . . . 11 𝑥ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)
4339, 42nfan 1891 . . . . . . . . . 10 𝑥((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
4432, 43nfan 1891 . . . . . . . . 9 𝑥(𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
45 nfv 1906 . . . . . . . . 9 𝑥 𝑧 ∈ ran 𝑓
4644, 45nfan 1891 . . . . . . . 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 485 . . . . . . . . . . . 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 6667 . . . . . . . . . . 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 765 . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 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 722 . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
55 id 22 . . . . . . . . . . . . . 14 (𝑙 = 𝑘𝑙 = 𝑘)
5654, 55eqeq12d 2834 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5756rspcva 3618 . . . . . . . . . . . 12 ((𝑘 𝑥𝐴 𝐵 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5849, 53, 57syl2anc 584 . . . . . . . . . . 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 2855 . . . . . . . . . 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 495 . . . . . . . . . . . 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 722 . . . . . . . . . . 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 7024 . . . . . . . . . . 11 ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑥𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6361, 49, 62syl2anc 584 . . . . . . . . . 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 6667 . . . . . . . . . 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 2860 . . . . . . . . 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 6609 . . . . . . . . . . 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 772 . . . . . . . . . 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 6719 . . . . . . . . . . 11 (𝑓 Fn 𝑥𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧))
7069biimpa 477 . . . . . . . . . 10 ((𝑓 Fn 𝑥𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧)
7167, 68, 70syl2anc 584 . . . . . . . . 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 3286 . . . . . . . 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 3964 . . . . . . . . 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 4914 . . . . . . . . 9 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
7573, 74sylib 219 . . . . . . . 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 3328 . . . . . . 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 1906 . . . . . . . . . 10 𝑘(𝜑𝑦 𝑥𝐴 𝐵)
78 nfcv 2974 . . . . . . . . . . 11 𝑘
7916, 78nfel 2989 . . . . . . . . . 10 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
8077, 79nfim 1888 . . . . . . . . 9 𝑘((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
81 eleq1w 2892 . . . . . . . . . . 11 (𝑘 = 𝑦 → (𝑘 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
8281anbi2d 628 . . . . . . . . . 10 (𝑘 = 𝑦 → ((𝜑𝑘 𝑥𝐴 𝐵) ↔ (𝜑𝑦 𝑥𝐴 𝐵)))
8312eleq1d 2894 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑘𝐶 ∈ ℂ))
8482, 83imbi12d 346 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
85 nfcv 2974 . . . . . . . . . . . 12 𝑥𝑘
8685, 34nfel 2989 . . . . . . . . . . 11 𝑥 𝑘 𝑥𝐴 𝐵
8732, 86nfan 1891 . . . . . . . . . 10 𝑥(𝜑𝑘 𝑥𝐴 𝐵)
88 fsumiunle.3 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
8988adantllr 715 . . . . . . . . . . 11 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
9089recnd 10657 . . . . . . . . . 10 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
91 eliun 4914 . . . . . . . . . . . 12 (𝑘 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑘𝐵)
9291biimpi 217 . . . . . . . . . . 11 (𝑘 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑘𝐵)
9392adantl 482 . . . . . . . . . 10 ((𝜑𝑘 𝑥𝐴 𝐵) → ∃𝑥𝐴 𝑘𝐵)
9487, 90, 93r19.29af 3328 . . . . . . . . 9 ((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ)
9580, 84, 94chvarfv 2232 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
9695adantlr 711 . . . . . . 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 15068 . . . . . 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 2865 . . . . 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 2824 . . . 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 2974 . . . . . . . . 9 𝑥𝑧
101100, 41nfel 2989 . . . . . . . 8 𝑥 𝑧 𝑥𝐴 ({𝑥} × 𝐵)
10232, 101nfan 1891 . . . . . . 7 𝑥(𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵))
103 xp2nd 7711 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
104103adantl 482 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) ∈ 𝐵)
10588ralrimiva 3179 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
106105adantlr 711 . . . . . . . . 9 (((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
107106adantr 481 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℝ)
108 nfcsb1v 3904 . . . . . . . . . . 11 𝑘(2nd𝑧) / 𝑘𝐶
109108nfel1 2991 . . . . . . . . . 10 𝑘(2nd𝑧) / 𝑘𝐶 ∈ ℝ
110 csbeq1a 3894 . . . . . . . . . . 11 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
111110eleq1d 2894 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (𝐶 ∈ ℝ ↔ (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
112109, 111rspc 3608 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 𝐶 ∈ ℝ → (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
113112imp 407 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ ℝ) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
114104, 107, 113syl2anc 584 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
11574biimpi 217 . . . . . . . 8 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
116115adantl 482 . . . . . . 7 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
117102, 114, 116r19.29af 3328 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
118117adantlr 711 . . . . 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 7710 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) ∈ {𝑥})
120 elsni 4574 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑥} → (1st𝑧) = 𝑥)
121119, 120syl 17 . . . . . . . . . 10 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) = 𝑥)
122121, 103jca 512 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵))
123 simplll 771 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
124 simplr 765 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑥𝐴)
125 fsumiunle.4 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)
126125ralrimiva 3179 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 0 ≤ 𝐶)
127123, 124, 126syl2anc 584 . . . . . . . . 9 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
128122, 127sylan2 592 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
129 nfcv 2974 . . . . . . . . . . 11 𝑘0
130 nfcv 2974 . . . . . . . . . . 11 𝑘
131129, 130, 108nfbr 5104 . . . . . . . . . 10 𝑘0 ≤ (2nd𝑧) / 𝑘𝐶
132110breq2d 5069 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ (2nd𝑧) / 𝑘𝐶))
133131, 132rspc 3608 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 0 ≤ 𝐶 → 0 ≤ (2nd𝑧) / 𝑘𝐶))
134133imp 407 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 0 ≤ 𝐶) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
135104, 128, 134syl2anc 584 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
136102, 135, 116r19.29af 3328 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
137136adantlr 711 . . . . 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 15139 . . . 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 5081 . . 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 2974 . . . . . . . 8 𝑦𝐵
141 nfcv 2974 . . . . . . . 8 𝑘𝐵
14212, 140, 141, 15, 16cbvsum 15040 . . . . . . 7 Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶
143142a1i 11 . . . . . 6 (𝜑 → Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶)
144143sumeq2sdv 15049 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶)
145 vex 3495 . . . . . . . . . 10 𝑥 ∈ V
146 vex 3495 . . . . . . . . . 10 𝑦 ∈ V
147145, 146op2ndd 7689 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
148147eqcomd 2824 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝑧))
149148csbeq1d 3884 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
150149eqcomd 2824 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘𝐶 = 𝑦 / 𝑘𝐶)
151 nfv 1906 . . . . . . . . 9 𝑘((𝜑𝑥𝐴) ∧ 𝑦𝐵)
15216nfel1 2991 . . . . . . . . 9 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
153151, 152nfim 1888 . . . . . . . 8 𝑘(((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
154 eleq1w 2892 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝑘𝐵𝑦𝐵))
155154anbi2d 628 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑥𝐴) ∧ 𝑘𝐵) ↔ ((𝜑𝑥𝐴) ∧ 𝑦𝐵)))
156155, 83imbi12d 346 . . . . . . . 8 (𝑘 = 𝑦 → ((((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
15788recnd 10657 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
158153, 156, 157chvarfv 2232 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
159158anasss 467 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 / 𝑘𝐶 ∈ ℂ)
160150, 1, 2, 159fsum2d 15114 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
161144, 160eqtrd 2853 . . . 4 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
162161adantr 481 . . 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 5085 . 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 1927 1 (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  csb 3880  wss 3933  {csn 4557  cop 4563   ciun 4910   class class class wbr 5057   × cxp 5546  ccnv 5547  ran crn 5549   Fn wfn 6343  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  1st c1st 7676  2nd c2nd 7677  Fincfn 8497  cc 10523  cr 10524  0cc0 10525  cle 10664  Σcsu 15030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092  ax-ac2 9873  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-r1 9181  df-rank 9182  df-card 9356  df-ac 9530  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031
This theorem is referenced by:  hgt750lema  31827
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