Theorem fsumiunle 32889

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	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumiunle.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
fsumiunle.3	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
fsumiunle.4	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶)

Assertion

Ref	Expression
fsumiunle	⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		fsumiunle.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		fsumiunle.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
3	1, 2	aciunf1 32721	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6784	. . . . . 6 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 616	. . . . 5 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
6		f1f 6729	. . . . . . 7 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
7	6	frnd 6669	. . . . . 6 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
8	7	adantr 480	. . . . 5 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
9	5, 8	jca 511	. . . 4 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
10	9	eximi 1837	. . 3 ⊢ (∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ∃𝑓((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
11	3, 10	syl 17	. 2 ⊢ (𝜑 → ∃𝑓((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
12		csbeq1a 3862	. . . . . . 7 ⊢ (𝑘 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
13		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝐶
14		nfcsb1v 3872	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶
15	12, 13, 14	cbvsum 15620	. . . . . 6 ⊢ Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶
16		csbeq1 3851	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
17		snfi 8982	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ Fin
18		xpfi 9222	. . . . . . . . . . . 12 ⊢ (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
19	17, 2, 18	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × 𝐵) ∈ Fin)
20	19	ralrimiva 3127	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
21		iunfi 9245	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
22	1, 20, 21	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
24		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
25	23, 24	ssfid 9171	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ∈ Fin)
26		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
27		f1ocnv 6785	. . . . . . . . 9 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑥 ∈ 𝐴 𝐵)
28	26, 27	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑥 ∈ 𝐴 𝐵)
29	28	adantrlr 724	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑥 ∈ 𝐴 𝐵)
30		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
31		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑓
32		nfiu1 4981	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
33	31	nfrn 5900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran 𝑓
34	31, 32, 33	nff1o 6771	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
35		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(2nd ‘(𝑓‘𝑙)) = 𝑙
36	32, 35	nfralw 3282	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
37	34, 36	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
38		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥ran 𝑓
39		nfiu1 4981	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
40	38, 39	nfss 3925	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
41	37, 40	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
42	30, 41	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
43		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ ran 𝑓
44	42, 43	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
45		simpr 484	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (𝑓‘𝑘) = 𝑧)
46	45	fveq2d 6837	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = (2nd ‘𝑧))
47		simplr 769	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
48		simp-4r 784	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
49	48	simpld 494	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
50	49	simprd 495	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
51	50	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
52		2fveq3 6838	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
54	52, 53	eqeq12d 2751	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
55	54	rspcva 3573	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
56	47, 51, 55	syl2anc 585	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
57	46, 56	eqtr3d 2772	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘𝑧) = 𝑘)
58	49	simpld 494	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
59	58	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
60		f1ocnvfv1 7222	. . . . . . . . . . 11 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
61	59, 47, 60	syl2anc 585	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
62	45	fveq2d 6837	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = (◡𝑓‘𝑧))
63	57, 61, 62	3eqtr2rd 2777	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
64		f1ofn 6774	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → 𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵)
65	58, 64	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵)
66		simpllr 776	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑧 ∈ ran 𝑓)
67		fvelrnb 6893	. . . . . . . . . . 11 ⊢ (𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧))
68	67	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
69	65, 66, 68	syl2anc 585	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∃𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
70	63, 69	r19.29a 3143	. . . . . . . 8 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
71	24	sselda 3932	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
72		eliun 4949	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
73	71, 72	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
74	44, 70, 73	r19.29af 3244	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
75		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
76		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℂ
77	14, 76	nfel 2912	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
78	75, 77	nfim 1898	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
79		eleq1w 2818	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
80	79	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)))
81	12	eleq1d 2820	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ))
82	80, 81	imbi12d 344	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)))
83		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑘
84	83, 32	nfel 2912	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
85	30, 84	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
86		fsumiunle.3	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
87	86	adantllr 720	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
88	87	recnd 11162	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
89		eliun 4949	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵)
90	89	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵)
91	90	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵)
92	85, 88, 91	r19.29af 3244	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ)
93	78, 82, 92	chvarfv 2246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
94	93	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
95	16, 25, 29, 74, 94	fsumf1o 15648	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
96	15, 95	eqtrid 2782	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
97	96	eqcomd 2741	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶)
98		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
99	98, 39	nfel 2912	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
100	30, 99	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
101		xp2nd 7966	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
102	101	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
103	86	ralrimiva 3127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
104	103	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
105	104	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
106		nfcsb1v 3872	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
107	106	nfel1 2914	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ
108		csbeq1a 3862	. . . . . . . . . . 11 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
109	108	eleq1d 2820	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (𝐶 ∈ ℝ ↔ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ))
110	107, 109	rspc 3563	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ))
111	110	imp 406	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
112	102, 105, 111	syl2anc 585	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
113	72	biimpi 216	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
114	113	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
115	100, 112, 114	r19.29af 3244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
116	115	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
117		xp1st 7965	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) ∈ {𝑥})
118		elsni 4596	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ {𝑥} → (1st ‘𝑧) = 𝑥)
119	117, 118	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) = 𝑥)
120	119, 101	jca 511	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵))
121		simplll 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
122		simplr 769	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑥 ∈ 𝐴)
123		fsumiunle.4	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶)
124	123	ralrimiva 3127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
125	121, 122, 124	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
126	120, 125	sylan2 594	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
127		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
128		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ≤
129	127, 128, 106	nfbr 5144	. . . . . . . . . 10 ⊢ Ⅎ𝑘0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶
130	108	breq2d 5109	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶))
131	129, 130	rspc 3563	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) ∈ 𝐵 → (∀𝑘 ∈ 𝐵 0 ≤ 𝐶 → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶))
132	131	imp 406	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 0 ≤ 𝐶) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
133	102, 126, 132	syl2anc 585	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
134	100, 133, 114	r19.29af 3244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
135	134	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
136	23, 116, 135, 24	fsumless 15721	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ≤ Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
137	97, 136	eqbrtrrd 5121	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
138	12, 13, 14	cbvsum 15620	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶
139	138	a1i 11	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶)
140	139	sumeq2sdv 15628	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶)
141		vex 3443	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
142		vex 3443	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
143	141, 142	op2ndd 7944	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
144	143	eqcomd 2741	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝑧))
145	144	csbeq1d 3852	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
146	145	eqcomd 2741	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
147		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)
148	14	nfel1 2914	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
149	147, 148	nfim 1898	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
150		eleq1w 2818	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
151	150	anbi2d 631	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)))
152	151, 81	imbi12d 344	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)))
153	86	recnd 11162	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
154	149, 152, 153	chvarfv 2246	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
155	154	anasss 466	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
156	146, 1, 2, 155	fsum2d 15696	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
157	140, 156	eqtrd 2770	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
158	157	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
159	137, 158	breqtrrd 5125	. 2 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)
160	11, 159	exlimddv 1937	1 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  ⦋csb 3848   ⊆ wss 3900  {csn 4579  ⟨cop 4585  ∪ ciun 4945   class class class wbr 5097   × cxp 5621  ^◡ccnv 5622  ran crn 5624   Fn wfn 6486  –1-1→wf1 6488  –1-1-onto→wf1o 6490  ‘cfv 6491  1^st c1st 7931  2^nd c2nd 7932  Fincfn 8885  ℂcc 11026  ℝcr 11027  0cc0 11028   ≤ cle 11169  Σcsu 15611

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-reg 9499  ax-inf2 9552  ax-ac2 10375  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-sup 9347  df-oi 9417  df-r1 9678  df-rank 9679  df-card 9853  df-ac 10028  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-z 12491  df-uz 12754  df-rp 12908  df-ico 13269  df-fz 13426  df-fzo 13573  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612

This theorem is referenced by:  hgt750lema  34793