Theorem esumiun 34391

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 32865	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6818	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 624	. . . . 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 6760	. . . . . . 7 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
7	6	frnd 6700	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
8	7	adantr 484	. . . . 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 519	. . . 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 1855	. . 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 17	. 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 1934	. . . . . 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 2924	. . . . . 6 ⊢ Ⅎ𝑧𝐶
14		nfcsb1v 3876	. . . . . 6 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
15		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 𝐵
16		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑧ran 𝑓
17		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑧◡𝑓
18		csbeq1a 3866	. . . . . 6 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
19	2	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
20		iunexg 7944	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
21	1, 19, 20	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
22	21	adantr 484	. . . . . 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 780	. . . . . . . 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 6819	. . . . . . . 8 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
25	23, 24	syl 17	. . . . . . 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 733	. . . . . 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 1934	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
28		nfcv 2924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑓
29		nfiu1 4985	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
30	28	nfrn 5928	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗ran 𝑓
31	28, 29, 30	nff1o 6804	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
32		nfv 1934	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(2nd ‘(𝑓‘𝑙)) = 𝑙
33	29, 32	nfralw 3309	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
34	31, 33	nfan 1919	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
35		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ran 𝑓
36		nfiu1 4985	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
37	35, 36	nfss 3929	. . . . . . . . . 10 ⊢ Ⅎ𝑗ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
38	34, 37	nfan 1919	. . . . . . . . 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 1919	. . . . . . . 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 1934	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ran 𝑓
41	39, 40	nfan 1919	. . . . . . 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 488	. . . . . . . . . . 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 6871	. . . . . . . . . 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 778	. . . . . . . . . . 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 793	. . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 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 499	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 6872	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
50		id 22	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
51	49, 50	eqeq12d 2778	. . . . . . . . . . . 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 593	. . . . . . . . . 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 2799	. . . . . . . . 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 498	. . . . . . . . . . 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 736	. . . . . . . . . 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 7260	. . . . . . . . . 10 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
58	56, 44, 57	syl2anc 593	. . . . . . . . 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 6871	. . . . . . . . 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 2804	. . . . . . . 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 6807	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → 𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵)
62	55, 61	syl 17	. . . . . . . . 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 785	. . . . . . . . 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 6927	. . . . . . . . . 10 ⊢ (𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧))
65	64	biimpa 480	. . . . . . . . 9 ⊢ ((𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
66	62, 63, 65	syl2anc 593	. . . . . . . 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 3170	. . . . . . 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 782	. . . . . . . . 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 3936	. . . . . . . 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 4953	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
71	69, 70	sylib 220	. . . . . . 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 3271	. . . . . 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 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑘
74	73, 29	nfel 2938	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
75	27, 74	nfan 1919	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
76		esumiun.2	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
77	76	adantllr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
78		eliun 4953	. . . . . . . . 9 ⊢ (𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
79	78	bilani 508	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
80	75, 77, 79	r19.29af 3271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
81	80	adantlr 725	. . . . . 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 34347	. . . . 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 2768	. . . 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 5392	. . . . . . . . . 10 ⊢ {𝑗} ∈ V
85	84	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ∈ V)
86	85, 2	xpexd 7734	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
87	86	ralrimiva 3154	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
88		iunexg 7944	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
89	1, 87, 88	syl2anc 593	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
90	89	adantr 484	. . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑧
92	91, 36	nfel 2938	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
93	27, 92	nfan 1919	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
94		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑗(2nd ‘𝑧)
95		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐶
96	94, 95	nfcsbw 3878	. . . . . . . 8 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶
97		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑗(0[,]+∞)
98	96, 97	nfel 2938	. . . . . . 7 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞)
99		simprr 782	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
100		simplll 784	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
101		simplr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑗 ∈ 𝐴)
102	76	ralrimiva 3154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
103	100, 101, 102	syl2anc 593	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
104		rspcsbela 4392	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
105	99, 103, 104	syl2anc 593	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
106		xp1st 8002	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
107		elsni 4599	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
108	106, 107	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
109		xp2nd 8003	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
110	108, 109	jca 519	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
111	110	reximi 3100	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
112	70, 111	sylbi 219	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
113	112	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
114	93, 98, 105, 113	r19.29af2 3270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
115	114	adantlr 725	. . . . 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 782	. . . . . 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 733	. . . . 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 34351	. . . 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 5124	. . 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 3458	. . . . . . . . 9 ⊢ 𝑗 ∈ V
121		vex 3458	. . . . . . . . 9 ⊢ 𝑘 ∈ V
122	120, 121	op2ndd 7981	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = 𝑘)
123	122	eqcomd 2768	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
124	123, 18	syl 17	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
125	124	eqcomd 2768	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = 𝐶)
126	76	anasss 470	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
127	14, 125, 1, 2, 126	esum2d 34390	. . . 4 ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
128	127	adantr 484	. . 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 5128	. 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 1955	1 ⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454  ⦋csb 3852   ⊆ wss 3904  {csn 4582  ⟨cop 4588  ∪ ciun 4949   class class class wbr 5100   × cxp 5645  ^◡ccnv 5646  ran crn 5648   Fn wfn 6516  –1-1→wf1 6518  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969  0cc0 11073  +∞cpnf 11213   ≤ cle 11217  [,]cicc 13352  Σ^*cesum 34324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-reg 9540  ax-inf2 9596  ax-ac2 10420  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151  ax-addf 11152  ax-mulf 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-fi 9357  df-sup 9388  df-inf 9389  df-oi 9458  df-r1 9722  df-rank 9723  df-card 9897  df-ac 10072  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-q 12950  df-rp 12994  df-xneg 13114  df-xadd 13115  df-xmul 13116  df-ioo 13353  df-ioc 13354  df-ico 13355  df-icc 13356  df-fz 13513  df-fzo 13660  df-fl 13802  df-mod 13880  df-seq 14015  df-exp 14075  df-fac 14287  df-bc 14316  df-hash 14344  df-shft 15080  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-limsup 15498  df-clim 15515  df-rlim 15516  df-sum 15714  df-ef 16097  df-sin 16099  df-cos 16100  df-pi 16102  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-starv 17301  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-unif 17309  df-hom 17310  df-cco 17311  df-rest 17451  df-topn 17452  df-0g 17470  df-gsum 17471  df-topgen 17472  df-pt 17473  df-prds 17476  df-ordt 17531  df-xrs 17532  df-qtop 17537  df-imas 17538  df-xps 17540  df-mre 17614  df-mrc 17615  df-acs 17617  df-ps 18598  df-tsr 18599  df-plusf 18673  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-submnd 18818  df-grp 18978  df-minusg 18979  df-sbg 18980  df-mulg 19110  df-subg 19165  df-cntz 19357  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20232  df-ring 20285  df-cring 20286  df-subrng 20596  df-subrg 20620  df-abv 20858  df-lmod 20929  df-scaf 20930  df-sra 21240  df-rgmod 21241  df-psmet 21416  df-xmet 21417  df-met 21418  df-bl 21419  df-mopn 21420  df-fbas 21421  df-fg 21422  df-cnfld 21425  df-top 22954  df-topon 22971  df-topsp 22993  df-bases 23006  df-cld 23079  df-ntr 23080  df-cls 23081  df-nei 23158  df-lp 23196  df-perf 23197  df-cn 23287  df-cnp 23288  df-haus 23375  df-tx 23622  df-hmeo 23815  df-fil 23906  df-fm 23998  df-flim 23999  df-flf 24000  df-tmd 24132  df-tgp 24133  df-tsms 24187  df-trg 24220  df-xms 24380  df-ms 24381  df-tms 24382  df-nm 24642  df-ngp 24643  df-nrg 24645  df-nlm 24646  df-ii 24939  df-cncf 24940  df-limc 25928  df-dv 25929  df-log 26621  df-esum 34325

This theorem is referenced by:  omssubadd  34597