Theorem esumiun 34238

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 32736	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6791	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 616	. . . . 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 6736	. . . . . . 7 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
7	6	frnd 6676	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
8	7	adantr 480	. . . . 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 511	. . . 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 1837	. . 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 1916	. . . . . 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 2898	. . . . . 6 ⊢ Ⅎ𝑧𝐶
14		nfcsb1v 3861	. . . . . 6 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
15		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 𝐵
16		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑧ran 𝑓
17		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑧◡𝑓
18		csbeq1a 3851	. . . . . 6 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
19	2	ralrimiva 3129	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
20		iunexg 7916	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
21	1, 19, 20	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
22	21	adantr 480	. . . . . 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 771	. . . . . . . 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 6792	. . . . . . . 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 724	. . . . . 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 1916	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
28		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑓
29		nfiu1 4969	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
30	28	nfrn 5907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗ran 𝑓
31	28, 29, 30	nff1o 6778	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
32		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(2nd ‘(𝑓‘𝑙)) = 𝑙
33	29, 32	nfralw 3284	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
34	31, 33	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
35		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ran 𝑓
36		nfiu1 4969	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
37	35, 36	nfss 3914	. . . . . . . . . 10 ⊢ Ⅎ𝑗ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
38	34, 37	nfan 1901	. . . . . . . . 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 1901	. . . . . . . 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 1916	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ran 𝑓
41	39, 40	nfan 1901	. . . . . . 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 484	. . . . . . . . . . 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 6844	. . . . . . . . . 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 769	. . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 6845	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
50		id 22	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
51	49, 50	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
52	51	rspcva 3562	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
53	44, 48, 52	syl2anc 585	. . . . . . . . . 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 2773	. . . . . . . . 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 494	. . . . . . . . . . 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 727	. . . . . . . . . 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 7231	. . . . . . . . . 10 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
58	56, 44, 57	syl2anc 585	. . . . . . . . 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 6844	. . . . . . . . 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 2778	. . . . . . . 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 6781	. . . . . . . . . 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 776	. . . . . . . . 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 6900	. . . . . . . . . 10 ⊢ (𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧))
65	64	biimpa 476	. . . . . . . . 9 ⊢ ((𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
66	62, 63, 65	syl2anc 585	. . . . . . . 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 3145	. . . . . . 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 773	. . . . . . . . 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 3921	. . . . . . . 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 4937	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
71	69, 70	sylib 218	. . . . . . 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 3246	. . . . . 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 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑘
74	73, 29	nfel 2913	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
75	27, 74	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
76		esumiun.2	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
77	76	adantllr 720	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
78		eliun 4937	. . . . . . . . . 10 ⊢ (𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
79	78	biimpi 216	. . . . . . . . 9 ⊢ (𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
80	79	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
81	75, 77, 80	r19.29af 3246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
82	81	adantlr 716	. . . . . 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[,]+∞))
83	12, 13, 14, 15, 16, 17, 18, 22, 26, 72, 82	esumf1o 34194	. . . . 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)
84	83	eqcomd 2742	. . . 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)
85		vsnex 5377	. . . . . . . . . 10 ⊢ {𝑗} ∈ V
86	85	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ∈ V)
87	86, 2	xpexd 7705	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
88	87	ralrimiva 3129	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
89		iunexg 7916	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
90	1, 88, 89	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
91	90	adantr 480	. . . . 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)
92		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑧
93	92, 36	nfel 2913	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
94	27, 93	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
95		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗(2nd ‘𝑧)
96		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐶
97	95, 96	nfcsbw 3863	. . . . . . . 8 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶
98		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑗(0[,]+∞)
99	97, 98	nfel 2913	. . . . . . 7 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞)
100		simprr 773	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
101		simplll 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
102		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑗 ∈ 𝐴)
103	76	ralrimiva 3129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
104	101, 102, 103	syl2anc 585	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
105		rspcsbela 4378	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
106	100, 104, 105	syl2anc 585	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
107		xp1st 7974	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
108		elsni 4584	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
109	107, 108	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
110		xp2nd 7975	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
111	109, 110	jca 511	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
112	111	reximi 3075	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
113	70, 112	sylbi 217	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
114	113	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
115	94, 99, 106, 114	r19.29af2 3245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
116	115	adantlr 716	. . . . 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[,]+∞))
117		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
118	117	adantrlr 724	. . . . 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))
119	12, 91, 116, 118	esummono 34198	. . . 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)
120	84, 119	eqbrtrrd 5109	. . 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)
121		vex 3433	. . . . . . . . 9 ⊢ 𝑗 ∈ V
122		vex 3433	. . . . . . . . 9 ⊢ 𝑘 ∈ V
123	121, 122	op2ndd 7953	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = 𝑘)
124	123	eqcomd 2742	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
125	124, 18	syl 17	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
126	125	eqcomd 2742	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = 𝐶)
127	76	anasss 466	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
128	14, 126, 1, 2, 127	esum2d 34237	. . . 4 ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
129	128	adantr 480	. . 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)
130	120, 129	breqtrrd 5113	. 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)
131	11, 130	exlimddv 1937	1 ⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429  ⦋csb 3837   ⊆ wss 3889  {csn 4567  ⟨cop 4573  ∪ ciun 4933   class class class wbr 5085   × cxp 5629  ^◡ccnv 5630  ran crn 5632   Fn wfn 6493  –1-1→wf1 6495  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  0cc0 11038  +∞cpnf 11176   ≤ cle 11180  [,]cicc 13301  Σ^*cesum 34171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-r1 9688  df-rank 9689  df-card 9863  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ioo 13302  df-ioc 13303  df-ico 13304  df-icc 13305  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-seq 13964  df-exp 14024  df-fac 14236  df-bc 14265  df-hash 14293  df-shft 15029  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-limsup 15433  df-clim 15450  df-rlim 15451  df-sum 15649  df-ef 16032  df-sin 16034  df-cos 16035  df-pi 16037  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-ordt 17465  df-xrs 17466  df-qtop 17471  df-imas 17472  df-xps 17474  df-mre 17548  df-mrc 17549  df-acs 17551  df-ps 18532  df-tsr 18533  df-plusf 18607  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-subrng 20523  df-subrg 20547  df-abv 20786  df-lmod 20857  df-scaf 20858  df-sra 21168  df-rgmod 21169  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-fbas 21349  df-fg 21350  df-cnfld 21353  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-lp 23101  df-perf 23102  df-cn 23192  df-cnp 23193  df-haus 23280  df-tx 23527  df-hmeo 23720  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905  df-tmd 24037  df-tgp 24038  df-tsms 24092  df-trg 24125  df-xms 24285  df-ms 24286  df-tms 24287  df-nm 24547  df-ngp 24548  df-nrg 24550  df-nlm 24551  df-ii 24844  df-cncf 24845  df-limc 25833  df-dv 25834  df-log 26520  df-esum 34172

This theorem is referenced by:  omssubadd  34444