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Theorem fsumiunle 32831
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 32673 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6859 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 615 . . . . 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 6804 . . . . . . 7 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵 𝑥𝐴 ({𝑥} × 𝐵))
76frnd 6744 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
87adantr 480 . . . . 5 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
95, 8jca 511 . . . 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 1835 . . 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 3913 . . . . . . 7 (𝑘 = 𝑦𝐶 = 𝑦 / 𝑘𝐶)
13 nfcv 2905 . . . . . . 7 𝑦𝐶
14 nfcsb1v 3923 . . . . . . 7 𝑘𝑦 / 𝑘𝐶
1512, 13, 14cbvsum 15731 . . . . . 6 Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑦 𝑥𝐴 𝐵𝑦 / 𝑘𝐶
16 csbeq1 3902 . . . . . . 7 (𝑦 = (2nd𝑧) → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
17 snfi 9083 . . . . . . . . . . . 12 {𝑥} ∈ Fin
18 xpfi 9358 . . . . . . . . . . . 12 (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
1917, 2, 18sylancr 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ({𝑥} × 𝐵) ∈ Fin)
2019ralrimiva 3146 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
21 iunfi 9383 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
221, 20, 21syl2anc 584 . . . . . . . . 9 (𝜑 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
2322adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
24 simprr 773 . . . . . . . 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))
2523, 24ssfid 9301 . . . . . . 7 ((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ∈ Fin)
26 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
27 f1ocnv 6860 . . . . . . . . 9 (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑓:ran 𝑓1-1-onto 𝑥𝐴 𝐵)
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓:ran 𝑓1-1-onto 𝑥𝐴 𝐵)
2928adantrlr 723 . . . . . . 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)
30 nfv 1914 . . . . . . . . . 10 𝑥𝜑
31 nfcv 2905 . . . . . . . . . . . . 13 𝑥𝑓
32 nfiu1 5027 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
3331nfrn 5963 . . . . . . . . . . . . 13 𝑥ran 𝑓
3431, 32, 33nff1o 6846 . . . . . . . . . . . 12 𝑥 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓
35 nfv 1914 . . . . . . . . . . . . 13 𝑥(2nd ‘(𝑓𝑙)) = 𝑙
3632, 35nfralw 3311 . . . . . . . . . . . 12 𝑥𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3734, 36nfan 1899 . . . . . . . . . . 11 𝑥(𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
38 nfcv 2905 . . . . . . . . . . . 12 𝑥ran 𝑓
39 nfiu1 5027 . . . . . . . . . . . 12 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
4038, 39nfss 3976 . . . . . . . . . . 11 𝑥ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)
4137, 40nfan 1899 . . . . . . . . . 10 𝑥((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
4230, 41nfan 1899 . . . . . . . . 9 𝑥(𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
43 nfv 1914 . . . . . . . . 9 𝑥 𝑧 ∈ ran 𝑓
4442, 43nfan 1899 . . . . . . . 8 𝑥((𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
45 simpr 484 . . . . . . . . . . . 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)
4645fveq2d 6910 . . . . . . . . . . 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))
47 simplr 769 . . . . . . . . . . . 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)
48 simp-4r 784 . . . . . . . . . . . . . . 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)))
4948simpld 494 . . . . . . . . . . . . . 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))
5049simprd 495 . . . . . . . . . . . . 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)
5150ad2antrr 726 . . . . . . . . . . . 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)
52 2fveq3 6911 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
53 id 22 . . . . . . . . . . . . . 14 (𝑙 = 𝑘𝑙 = 𝑘)
5452, 53eqeq12d 2753 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5554rspcva 3620 . . . . . . . . . . . 12 ((𝑘 𝑥𝐴 𝐵 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5647, 51, 55syl2anc 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)
5746, 56eqtr3d 2779 . . . . . . . . . 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)
5849simpld 494 . . . . . . . . . . . 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)
5958ad2antrr 726 . . . . . . . . . . 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)
60 f1ocnvfv1 7296 . . . . . . . . . . 11 ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑥𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6159, 47, 60syl2anc 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)
6245fveq2d 6910 . . . . . . . . . 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))
6357, 61, 623eqtr2rd 2784 . . . . . . . . 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))
64 f1ofn 6849 . . . . . . . . . . 11 (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑓 Fn 𝑥𝐴 𝐵)
6558, 64syl 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)
66 simpllr 776 . . . . . . . . . 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)
67 fvelrnb 6969 . . . . . . . . . . 11 (𝑓 Fn 𝑥𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧))
6867biimpa 476 . . . . . . . . . 10 ((𝑓 Fn 𝑥𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧)
6965, 66, 68syl2anc 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)
7063, 69r19.29a 3162 . . . . . . . 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))
7124sselda 3983 . . . . . . . . 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))
72 eliun 4995 . . . . . . . . 9 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
7371, 72sylib 218 . . . . . . . 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))
7444, 70, 73r19.29af 3268 . . . . . . 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))
75 nfv 1914 . . . . . . . . . 10 𝑘(𝜑𝑦 𝑥𝐴 𝐵)
76 nfcv 2905 . . . . . . . . . . 11 𝑘
7714, 76nfel 2920 . . . . . . . . . 10 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
7875, 77nfim 1896 . . . . . . . . 9 𝑘((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
79 eleq1w 2824 . . . . . . . . . . 11 (𝑘 = 𝑦 → (𝑘 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
8079anbi2d 630 . . . . . . . . . 10 (𝑘 = 𝑦 → ((𝜑𝑘 𝑥𝐴 𝐵) ↔ (𝜑𝑦 𝑥𝐴 𝐵)))
8112eleq1d 2826 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑘𝐶 ∈ ℂ))
8280, 81imbi12d 344 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
83 nfcv 2905 . . . . . . . . . . . 12 𝑥𝑘
8483, 32nfel 2920 . . . . . . . . . . 11 𝑥 𝑘 𝑥𝐴 𝐵
8530, 84nfan 1899 . . . . . . . . . 10 𝑥(𝜑𝑘 𝑥𝐴 𝐵)
86 fsumiunle.3 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
8786adantllr 719 . . . . . . . . . . 11 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
8887recnd 11289 . . . . . . . . . 10 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
89 eliun 4995 . . . . . . . . . . . 12 (𝑘 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑘𝐵)
9089biimpi 216 . . . . . . . . . . 11 (𝑘 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑘𝐵)
9190adantl 481 . . . . . . . . . 10 ((𝜑𝑘 𝑥𝐴 𝐵) → ∃𝑥𝐴 𝑘𝐵)
9285, 88, 91r19.29af 3268 . . . . . . . . 9 ((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ)
9378, 82, 92chvarfv 2240 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
9493adantlr 715 . . . . . . 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)
9516, 25, 29, 74, 94fsumf1o 15759 . . . . . 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)
9615, 95eqtrid 2789 . . . . 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)
9796eqcomd 2743 . . . 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)
98 nfcv 2905 . . . . . . . . 9 𝑥𝑧
9998, 39nfel 2920 . . . . . . . 8 𝑥 𝑧 𝑥𝐴 ({𝑥} × 𝐵)
10030, 99nfan 1899 . . . . . . 7 𝑥(𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵))
101 xp2nd 8047 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
102101adantl 481 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) ∈ 𝐵)
10386ralrimiva 3146 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
104103adantlr 715 . . . . . . . . 9 (((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
105104adantr 480 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℝ)
106 nfcsb1v 3923 . . . . . . . . . . 11 𝑘(2nd𝑧) / 𝑘𝐶
107106nfel1 2922 . . . . . . . . . 10 𝑘(2nd𝑧) / 𝑘𝐶 ∈ ℝ
108 csbeq1a 3913 . . . . . . . . . . 11 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
109108eleq1d 2826 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (𝐶 ∈ ℝ ↔ (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
110107, 109rspc 3610 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 𝐶 ∈ ℝ → (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
111110imp 406 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ ℝ) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
112102, 105, 111syl2anc 584 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
11372biimpi 216 . . . . . . . 8 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
114113adantl 481 . . . . . . 7 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
115100, 112, 114r19.29af 3268 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
116115adantlr 715 . . . . 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)
117 xp1st 8046 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) ∈ {𝑥})
118 elsni 4643 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑥} → (1st𝑧) = 𝑥)
119117, 118syl 17 . . . . . . . . . 10 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) = 𝑥)
120119, 101jca 511 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵))
121 simplll 775 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
122 simplr 769 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑥𝐴)
123 fsumiunle.4 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)
124123ralrimiva 3146 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 0 ≤ 𝐶)
125121, 122, 124syl2anc 584 . . . . . . . . 9 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
126120, 125sylan2 593 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
127 nfcv 2905 . . . . . . . . . . 11 𝑘0
128 nfcv 2905 . . . . . . . . . . 11 𝑘
129127, 128, 106nfbr 5190 . . . . . . . . . 10 𝑘0 ≤ (2nd𝑧) / 𝑘𝐶
130108breq2d 5155 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ (2nd𝑧) / 𝑘𝐶))
131129, 130rspc 3610 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 0 ≤ 𝐶 → 0 ≤ (2nd𝑧) / 𝑘𝐶))
132131imp 406 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 0 ≤ 𝐶) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
133102, 126, 132syl2anc 584 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
134100, 133, 114r19.29af 3268 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
135134adantlr 715 . . . . 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)
13623, 116, 135, 24fsumless 15832 . . . 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)
13797, 136eqbrtrrd 5167 . . 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)
13812, 13, 14cbvsum 15731 . . . . . . 7 Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶
139138a1i 11 . . . . . 6 (𝜑 → Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶)
140139sumeq2sdv 15739 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶)
141 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
142 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
143141, 142op2ndd 8025 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
144143eqcomd 2743 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝑧))
145144csbeq1d 3903 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
146145eqcomd 2743 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘𝐶 = 𝑦 / 𝑘𝐶)
147 nfv 1914 . . . . . . . . 9 𝑘((𝜑𝑥𝐴) ∧ 𝑦𝐵)
14814nfel1 2922 . . . . . . . . 9 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
149147, 148nfim 1896 . . . . . . . 8 𝑘(((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
150 eleq1w 2824 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝑘𝐵𝑦𝐵))
151150anbi2d 630 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑥𝐴) ∧ 𝑘𝐵) ↔ ((𝜑𝑥𝐴) ∧ 𝑦𝐵)))
152151, 81imbi12d 344 . . . . . . . 8 (𝑘 = 𝑦 → ((((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
15386recnd 11289 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
154149, 152, 153chvarfv 2240 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
155154anasss 466 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 / 𝑘𝐶 ∈ ℂ)
156146, 1, 2, 155fsum2d 15807 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
157140, 156eqtrd 2777 . . . 4 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
158157adantr 480 . . 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)
159137, 158breqtrrd 5171 . 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)
16011, 159exlimddv 1935 1 (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  csb 3899  wss 3951  {csn 4626  cop 4632   ciun 4991   class class class wbr 5143   × cxp 5683  ccnv 5684  ran crn 5686   Fn wfn 6556  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  1st c1st 8012  2nd c2nd 8013  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  cle 11296  Σcsu 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  hgt750lema  34672
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