Theorem esumiun 34484

Description: Sum over a nonnecessarily disjoint indexed union. The inequality is strict in the case where the sets B(x) overlap. (Contributed by Thierry Arnoux, 21-Sep-2019.)

Hypotheses

Ref	Expression
esumiun.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumiun.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
esumiun.2	⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))

Assertion

Ref	Expression
esumiun	⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)

Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘   𝐶,𝑗   𝑗,𝑊,𝑘   𝜑,𝑗,𝑘

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)   𝑉(𝑗,𝑘)

Proof of Theorem esumiun

Dummy variables 𝑓 𝑙 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		esumiun.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		esumiun.1	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
3	1, 2	aciunf1 33008	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6832	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 626	. . . . 5 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
6		f1f 6774	. . . . . . 7 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
7	6	frnd 6714	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
8	7	adantr 485	. . . . 5 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
9	5, 8	jca 520	. . . 4 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
10	9	eximi 1865	. . 3 ⊢ (∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ∃𝑓((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
11	3, 10	syl 18	. 2 ⊢ (𝜑 → ∃𝑓((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
12		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
13		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑧𝐶
14		nfcsb1v 3877	. . . . . 6 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
15		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 𝐵
16		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑧ran 𝑓
17		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑧◡𝑓
18		csbeq1a 3867	. . . . . 6 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
19	2	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
20		iunexg 7956	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
21	1, 19, 20	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
22	21	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
23		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
24		f1ocnv 6833	. . . . . . . 8 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
25	23, 24	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
26	25	adantrlr 735	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
27		nfv 1944	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
28		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑓
29		nfiu1 4992	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
30	28	nfrn 5942	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗ran 𝑓
31	28, 29, 30	nff1o 6818	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
32		nfv 1944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(2nd ‘(𝑓‘𝑙)) = 𝑙
33	29, 32	nfralw 3312	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
34	31, 33	nfan 1929	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
35		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ran 𝑓
36		nfiu1 4992	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
37	35, 36	nfss 3930	. . . . . . . . . 10 ⊢ Ⅎ𝑗ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
38	34, 37	nfan 1929	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
39	27, 38	nfan 1929	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
40		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ran 𝑓
41	39, 40	nfan 1929	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
42		simpr 489	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (𝑓‘𝑘) = 𝑧)
43	42	fveq2d 6885	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = (2nd ‘𝑧))
44		simplr 780	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
45		simp-4r 795	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
46	45	simpld 499	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
47	46	simprd 500	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
48	47	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
49		2fveq3 6886	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
50		id 23	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
51	49, 50	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
52	51	rspcva 3579	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
53	44, 48, 52	syl2anc 595	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
54	43, 53	eqtr3d 2800	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘𝑧) = 𝑘)
55	46	simpld 499	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
56	55	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
57		f1ocnvfv1 7274	. . . . . . . . . 10 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
58	56, 44, 57	syl2anc 595	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
59	42	fveq2d 6885	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = (◡𝑓‘𝑧))
60	54, 58, 59	3eqtr2rd 2805	. . . . . . . 8 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
61		f1ofn 6821	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → 𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵)
62	55, 61	syl 18	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵)
63		simpllr 787	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑧 ∈ ran 𝑓)
64		fvelrnb 6941	. . . . . . . . . 10 ⊢ (𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧))
65	64	biimpa 481	. . . . . . . . 9 ⊢ ((𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
66	62, 63, 65	syl2anc 595	. . . . . . . 8 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
67	60, 66	r19.29a 3173	. . . . . . 7 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
68		simprr 784	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
69	68	sselda 3937	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
70		eliun 4960	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
71	69, 70	sylib 221	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
72	41, 67, 71	r19.29af 3274	. . . . . 6 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
73		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑘
74	73, 29	nfel 2939	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
75	27, 74	nfan 1929	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
76		esumiun.2	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
77	76	adantllr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
78		eliun 4960	. . . . . . . . 9 ⊢ (𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
79	78	bilani 509	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
80	75, 77, 79	r19.29af 3274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
81	80	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
82	12, 13, 14, 15, 16, 17, 18, 22, 26, 72, 81	esumf1o 34440	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 = Σ*𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
83	82	eqcomd 2769	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶)
84		vsnex 5406	. . . . . . . . . 10 ⊢ {𝑗} ∈ V
85	84	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ∈ V)
86	85, 2	xpexd 7746	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
87	86	ralrimiva 3157	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
88		iunexg 7956	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
89	1, 87, 88	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
90	89	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
91		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑧
92	91, 36	nfel 2939	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
93	27, 92	nfan 1929	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
94		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑗(2nd ‘𝑧)
95		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐶
96	94, 95	nfcsbw 3879	. . . . . . . 8 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶
97		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑗(0[,]+∞)
98	96, 97	nfel 2939	. . . . . . 7 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞)
99		simprr 784	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
100		simplll 786	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
101		simplr 780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑗 ∈ 𝐴)
102	76	ralrimiva 3157	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
103	100, 101, 102	syl2anc 595	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
104		rspcsbela 4403	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
105	99, 103, 104	syl2anc 595	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
106		xp1st 8014	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
107		elsni 4606	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
108	106, 107	syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
109		xp2nd 8015	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
110	108, 109	jca 520	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
111	110	reximi 3103	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
112	70, 111	sylbi 220	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
113	112	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
114	93, 98, 105, 113	r19.29af2 3273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
115	114	adantlr 727	. . . . 5 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
116		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
117	116	adantrlr 735	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
118	12, 90, 115, 117	esummono 34444	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ≤ Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
119	83, 118	eqbrtrrd 5135	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
120		vex 3459	. . . . . . . . 9 ⊢ 𝑗 ∈ V
121		vex 3459	. . . . . . . . 9 ⊢ 𝑘 ∈ V
122	120, 121	op2ndd 7993	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = 𝑘)
123	122	eqcomd 2769	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
124	123, 18	syl 18	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
125	124	eqcomd 2769	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = 𝐶)
126	76	anasss 471	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
127	14, 125, 1, 2, 126	esum2d 34483	. . . 4 ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
128	127	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
129	119, 128	breqtrrd 5139	. 2 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)
130	11, 129	exlimddv 1965	1 ⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ⦋csb 3853   ⊆ wss 3905  {csn 4589  ⟨cop 4595  ∪ ciun 4956   class class class wbr 5109   × cxp 5659  ^◡ccnv 5660  ran crn 5662   Fn wfn 6531  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  0cc0 11095  +∞cpnf 11235   ≤ cle 11239  [,]cicc 13370  Σ^*cesum 34417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606  ax-ac2 10442  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174  ax-mulf 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-fi 9367  df-sup 9398  df-inf 9399  df-oi 9468  df-r1 9732  df-rank 9733  df-card 9921  df-ac 10096  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-q 12968  df-rp 13012  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13371  df-ioc 13372  df-ico 13373  df-icc 13374  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-fac 14306  df-bc 14335  df-hash 14363  df-shft 15100  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-limsup 15518  df-clim 15535  df-rlim 15536  df-sum 15734  df-ef 16116  df-sin 16118  df-cos 16119  df-pi 16121  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-rest 17470  df-topn 17471  df-0g 17489  df-gsum 17490  df-topgen 17491  df-pt 17492  df-prds 17495  df-ordt 17550  df-xrs 17551  df-qtop 17556  df-imas 17557  df-xps 17559  df-mre 17633  df-mrc 17634  df-acs 17636  df-ps 18617  df-tsr 18618  df-plusf 18692  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-cntz 19382  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-subrng 20645  df-subrg 20669  df-abv 20912  df-lmod 20983  df-scaf 20984  df-sra 21294  df-rgmod 21295  df-psmet 21514  df-xmet 21515  df-met 21516  df-bl 21517  df-mopn 21518  df-fbas 21519  df-fg 21520  df-cnfld 21523  df-top 23051  df-topon 23068  df-topsp 23090  df-bases 23103  df-cld 23176  df-ntr 23177  df-cls 23178  df-nei 23255  df-lp 23293  df-perf 23294  df-cn 23384  df-cnp 23385  df-haus 23472  df-tx 23719  df-hmeo 23912  df-fil 24003  df-fm 24095  df-flim 24096  df-flf 24097  df-tmd 24229  df-tgp 24230  df-tsms 24284  df-trg 24317  df-xms 24477  df-ms 24478  df-tms 24479  df-nm 24739  df-ngp 24740  df-nrg 24742  df-nlm 24743  df-ii 25036  df-cncf 25037  df-limc 26025  df-dv 26026  df-log 26721  df-esum 34418

This theorem is referenced by:  omssubadd  34690