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Theorem fsumiunle 32682
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 32535 . . 3 (𝜑 → ∃𝑓(𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙))
4 f1f1orn 6849 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
54anim1i 613 . . . . 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 6793 . . . . . . 7 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → 𝑓: 𝑥𝐴 𝐵 𝑥𝐴 ({𝑥} × 𝐵))
76frnd 6731 . . . . . 6 (𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
87adantr 479 . . . . 5 ((𝑓: 𝑥𝐴 𝐵1-1 𝑥𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
95, 8jca 510 . . . 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 1829 . . 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 3903 . . . . . . 7 (𝑘 = 𝑦𝐶 = 𝑦 / 𝑘𝐶)
13 nfcv 2891 . . . . . . 7 𝑦 𝑥𝐴 𝐵
14 nfcv 2891 . . . . . . 7 𝑘 𝑥𝐴 𝐵
15 nfcv 2891 . . . . . . 7 𝑦𝐶
16 nfcsb1v 3914 . . . . . . 7 𝑘𝑦 / 𝑘𝐶
1712, 13, 14, 15, 16cbvsum 15682 . . . . . 6 Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑦 𝑥𝐴 𝐵𝑦 / 𝑘𝐶
18 csbeq1 3892 . . . . . . 7 (𝑦 = (2nd𝑧) → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
19 snfi 9071 . . . . . . . . . . . 12 {𝑥} ∈ Fin
20 xpfi 9346 . . . . . . . . . . . 12 (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
2119, 2, 20sylancr 585 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ({𝑥} × 𝐵) ∈ Fin)
2221ralrimiva 3135 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
23 iunfi 9371 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin) → 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
241, 22, 23syl2anc 582 . . . . . . . . 9 (𝜑 𝑥𝐴 ({𝑥} × 𝐵) ∈ Fin)
2524adantr 479 . . . . . . . 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 771 . . . . . . . 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 9295 . . . . . . 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 769 . . . . . . . . 9 ((𝜑 ∧ (𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))) → 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓)
29 f1ocnv 6850 . . . . . . . . 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 721 . . . . . . 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 1909 . . . . . . . . . 10 𝑥𝜑
33 nfcv 2891 . . . . . . . . . . . . 13 𝑥𝑓
34 nfiu1 5031 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
3533nfrn 5954 . . . . . . . . . . . . 13 𝑥ran 𝑓
3633, 34, 35nff1o 6836 . . . . . . . . . . . 12 𝑥 𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓
37 nfv 1909 . . . . . . . . . . . . 13 𝑥(2nd ‘(𝑓𝑙)) = 𝑙
3834, 37nfralw 3298 . . . . . . . . . . . 12 𝑥𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙
3936, 38nfan 1894 . . . . . . . . . . 11 𝑥(𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙)
40 nfcv 2891 . . . . . . . . . . . 12 𝑥ran 𝑓
41 nfiu1 5031 . . . . . . . . . . . 12 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
4240, 41nfss 3969 . . . . . . . . . . 11 𝑥ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)
4339, 42nfan 1894 . . . . . . . . . 10 𝑥((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵))
4432, 43nfan 1894 . . . . . . . . 9 𝑥(𝜑 ∧ ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) ∧ ran 𝑓 𝑥𝐴 ({𝑥} × 𝐵)))
45 nfv 1909 . . . . . . . . 9 𝑥 𝑧 ∈ ran 𝑓
4644, 45nfan 1894 . . . . . . . 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 483 . . . . . . . . . . . 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 6900 . . . . . . . . . . 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 767 . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → (2nd ‘(𝑓𝑙)) = (2nd ‘(𝑓𝑘)))
55 id 22 . . . . . . . . . . . . . 14 (𝑙 = 𝑘𝑙 = 𝑘)
5654, 55eqeq12d 2741 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ((2nd ‘(𝑓𝑙)) = 𝑙 ↔ (2nd ‘(𝑓𝑘)) = 𝑘))
5756rspcva 3604 . . . . . . . . . . . 12 ((𝑘 𝑥𝐴 𝐵 ∧ ∀𝑙 𝑥𝐴 𝐵(2nd ‘(𝑓𝑙)) = 𝑙) → (2nd ‘(𝑓𝑘)) = 𝑘)
5849, 53, 57syl2anc 582 . . . . . . . . . . 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 2767 . . . . . . . . . 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 493 . . . . . . . . . . . 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 724 . . . . . . . . . . 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 7285 . . . . . . . . . . 11 ((𝑓: 𝑥𝐴 𝐵1-1-onto→ran 𝑓𝑘 𝑥𝐴 𝐵) → (𝑓‘(𝑓𝑘)) = 𝑘)
6361, 49, 62syl2anc 582 . . . . . . . . . 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 6900 . . . . . . . . . 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 2772 . . . . . . . . 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 6839 . . . . . . . . . . 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 774 . . . . . . . . . 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 6958 . . . . . . . . . . 11 (𝑓 Fn 𝑥𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧))
7069biimpa 475 . . . . . . . . . 10 ((𝑓 Fn 𝑥𝐴 𝐵𝑧 ∈ ran 𝑓) → ∃𝑘 𝑥𝐴 𝐵(𝑓𝑘) = 𝑧)
7167, 68, 70syl2anc 582 . . . . . . . . 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 3151 . . . . . . . 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 3976 . . . . . . . . 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 5001 . . . . . . . . 9 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
7573, 74sylib 217 . . . . . . . 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 3255 . . . . . . 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 1909 . . . . . . . . . 10 𝑘(𝜑𝑦 𝑥𝐴 𝐵)
78 nfcv 2891 . . . . . . . . . . 11 𝑘
7916, 78nfel 2906 . . . . . . . . . 10 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
8077, 79nfim 1891 . . . . . . . . 9 𝑘((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
81 eleq1w 2808 . . . . . . . . . . 11 (𝑘 = 𝑦 → (𝑘 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
8281anbi2d 628 . . . . . . . . . 10 (𝑘 = 𝑦 → ((𝜑𝑘 𝑥𝐴 𝐵) ↔ (𝜑𝑦 𝑥𝐴 𝐵)))
8312eleq1d 2810 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑘𝐶 ∈ ℂ))
8482, 83imbi12d 343 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
85 nfcv 2891 . . . . . . . . . . . 12 𝑥𝑘
8685, 34nfel 2906 . . . . . . . . . . 11 𝑥 𝑘 𝑥𝐴 𝐵
8732, 86nfan 1894 . . . . . . . . . 10 𝑥(𝜑𝑘 𝑥𝐴 𝐵)
88 fsumiunle.3 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
8988adantllr 717 . . . . . . . . . . 11 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)
9089recnd 11279 . . . . . . . . . 10 ((((𝜑𝑘 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
91 eliun 5001 . . . . . . . . . . . 12 (𝑘 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑘𝐵)
9291biimpi 215 . . . . . . . . . . 11 (𝑘 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑘𝐵)
9392adantl 480 . . . . . . . . . 10 ((𝜑𝑘 𝑥𝐴 𝐵) → ∃𝑥𝐴 𝑘𝐵)
9487, 90, 93r19.29af 3255 . . . . . . . . 9 ((𝜑𝑘 𝑥𝐴 𝐵) → 𝐶 ∈ ℂ)
9580, 84, 94chvarfv 2228 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
9695adantlr 713 . . . . . . 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 15710 . . . . . 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, 97eqtrid 2777 . . . . 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 2731 . . . 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 2891 . . . . . . . . 9 𝑥𝑧
101100, 41nfel 2906 . . . . . . . 8 𝑥 𝑧 𝑥𝐴 ({𝑥} × 𝐵)
10232, 101nfan 1894 . . . . . . 7 𝑥(𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵))
103 xp2nd 8027 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
104103adantl 480 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) ∈ 𝐵)
10588ralrimiva 3135 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
106105adantlr 713 . . . . . . . . 9 (((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℝ)
107106adantr 479 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℝ)
108 nfcsb1v 3914 . . . . . . . . . . 11 𝑘(2nd𝑧) / 𝑘𝐶
109108nfel1 2908 . . . . . . . . . 10 𝑘(2nd𝑧) / 𝑘𝐶 ∈ ℝ
110 csbeq1a 3903 . . . . . . . . . . 11 (𝑘 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑘𝐶)
111110eleq1d 2810 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (𝐶 ∈ ℝ ↔ (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
112109, 111rspc 3594 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 𝐶 ∈ ℝ → (2nd𝑧) / 𝑘𝐶 ∈ ℝ))
113112imp 405 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 𝐶 ∈ ℝ) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
114104, 107, 113syl2anc 582 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
11574biimpi 215 . . . . . . . 8 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
116115adantl 480 . . . . . . 7 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
117102, 114, 116r19.29af 3255 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → (2nd𝑧) / 𝑘𝐶 ∈ ℝ)
118117adantlr 713 . . . . 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 8026 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) ∈ {𝑥})
120 elsni 4647 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑥} → (1st𝑧) = 𝑥)
121119, 120syl 17 . . . . . . . . . 10 (𝑧 ∈ ({𝑥} × 𝐵) → (1st𝑧) = 𝑥)
122121, 103jca 510 . . . . . . . . 9 (𝑧 ∈ ({𝑥} × 𝐵) → ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵))
123 simplll 773 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝜑)
124 simplr 767 . . . . . . . . . 10 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → 𝑥𝐴)
125 fsumiunle.4 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)
126125ralrimiva 3135 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑘𝐵 0 ≤ 𝐶)
127123, 124, 126syl2anc 582 . . . . . . . . 9 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ ((1st𝑧) = 𝑥 ∧ (2nd𝑧) ∈ 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
128122, 127sylan2 591 . . . . . . . 8 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘𝐵 0 ≤ 𝐶)
129 nfcv 2891 . . . . . . . . . . 11 𝑘0
130 nfcv 2891 . . . . . . . . . . 11 𝑘
131129, 130, 108nfbr 5196 . . . . . . . . . 10 𝑘0 ≤ (2nd𝑧) / 𝑘𝐶
132110breq2d 5161 . . . . . . . . . 10 (𝑘 = (2nd𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ (2nd𝑧) / 𝑘𝐶))
133131, 132rspc 3594 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐵 → (∀𝑘𝐵 0 ≤ 𝐶 → 0 ≤ (2nd𝑧) / 𝑘𝐶))
134133imp 405 . . . . . . . 8 (((2nd𝑧) ∈ 𝐵 ∧ ∀𝑘𝐵 0 ≤ 𝐶) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
135104, 128, 134syl2anc 582 . . . . . . 7 ((((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) ∧ 𝑥𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
136102, 135, 116r19.29af 3255 . . . . . 6 ((𝜑𝑧 𝑥𝐴 ({𝑥} × 𝐵)) → 0 ≤ (2nd𝑧) / 𝑘𝐶)
137136adantlr 713 . . . . 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 15783 . . . 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 5173 . . 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 2891 . . . . . . . 8 𝑦𝐵
141 nfcv 2891 . . . . . . . 8 𝑘𝐵
14212, 140, 141, 15, 16cbvsum 15682 . . . . . . 7 Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶
143142a1i 11 . . . . . 6 (𝜑 → Σ𝑘𝐵 𝐶 = Σ𝑦𝐵 𝑦 / 𝑘𝐶)
144143sumeq2sdv 15691 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶)
145 vex 3465 . . . . . . . . . 10 𝑥 ∈ V
146 vex 3465 . . . . . . . . . 10 𝑦 ∈ V
147145, 146op2ndd 8005 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
148147eqcomd 2731 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝑧))
149148csbeq1d 3893 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 / 𝑘𝐶 = (2nd𝑧) / 𝑘𝐶)
150149eqcomd 2731 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) / 𝑘𝐶 = 𝑦 / 𝑘𝐶)
151 nfv 1909 . . . . . . . . 9 𝑘((𝜑𝑥𝐴) ∧ 𝑦𝐵)
15216nfel1 2908 . . . . . . . . 9 𝑘𝑦 / 𝑘𝐶 ∈ ℂ
153151, 152nfim 1891 . . . . . . . 8 𝑘(((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
154 eleq1w 2808 . . . . . . . . . 10 (𝑘 = 𝑦 → (𝑘𝐵𝑦𝐵))
155154anbi2d 628 . . . . . . . . 9 (𝑘 = 𝑦 → (((𝜑𝑥𝐴) ∧ 𝑘𝐵) ↔ ((𝜑𝑥𝐴) ∧ 𝑦𝐵)))
156155, 83imbi12d 343 . . . . . . . 8 (𝑘 = 𝑦 → ((((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)))
15788recnd 11279 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
158153, 156, 157chvarfv 2228 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → 𝑦 / 𝑘𝐶 ∈ ℂ)
159158anasss 465 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 / 𝑘𝐶 ∈ ℂ)
160150, 1, 2, 159fsum2d 15758 . . . . 5 (𝜑 → Σ𝑥𝐴 Σ𝑦𝐵 𝑦 / 𝑘𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
161144, 160eqtrd 2765 . . . 4 (𝜑 → Σ𝑥𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑥𝐴 ({𝑥} × 𝐵)(2nd𝑧) / 𝑘𝐶)
162161adantr 479 . . 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 5177 . 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 1930 1 (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  wral 3050  wrex 3059  csb 3889  wss 3944  {csn 4630  cop 4636   ciun 4997   class class class wbr 5149   × cxp 5676  ccnv 5677  ran crn 5679   Fn wfn 6544  1-1wf1 6546  1-1-ontowf1o 6548  cfv 6549  1st c1st 7992  2nd c2nd 7993  Fincfn 8964  cc 11143  cr 11144  0cc0 11145  cle 11286  Σcsu 15673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9622  ax-inf2 9671  ax-ac2 10493  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-oi 9540  df-r1 9794  df-rank 9795  df-card 9969  df-ac 10146  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-ico 13370  df-fz 13525  df-fzo 13668  df-seq 14008  df-exp 14068  df-hash 14331  df-cj 15087  df-re 15088  df-im 15089  df-sqrt 15223  df-abs 15224  df-clim 15473  df-sum 15674
This theorem is referenced by:  hgt750lema  34422
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