Theorem fsumiunle 33487

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 33326	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6855	. . . . . 6 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓)
5	4	anim1i 627	. . . . 5 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
6		f1f 6797	. . . . . . 7 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
7	6	frnd 6737	. . . . . 6 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
8	7	adantr 486	. . . . 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 521	. . . 4 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
10	9	eximi 1868	. . 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⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
11	3, 10	syl 18	. 2 ⊢ (𝜑 → ∃𝑓((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
12		csbeq1a 3865	. . . . . . 7 ⊢ (𝑘 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
13		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦𝐶
14		nfcsb1v 3875	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶
15	12, 13, 14	cbvsum 15888	. . . . . 6 ⊢ Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶
16		csbeq1 3854	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
17		snfi 9096	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ Fin
18		xpfi 9336	. . . . . . . . . . . 12 ⊢ (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
19	17, 2, 18	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × 𝐵) ∈ Fin)
20	19	ralrimiva 3158	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
21		iunfi 9357	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
22	1, 20, 21	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
24		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 9285	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ∈ Fin)
26		simprl 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓)
27		f1ocnv 6856	. . . . . . . . 9 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 → ◡𝑓:ran 𝑓⟶1-1onto∪ 𝑥 ∈ 𝐴 𝐵)
28	26, 27	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ◡𝑓:ran 𝑓⟶1-1onto∪ 𝑥 ∈ 𝐴 𝐵)
29	28	adantrlr 736	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ◡𝑓:ran 𝑓⟶1-1onto∪ 𝑥 ∈ 𝐴 𝐵)
30		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
31		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑓
32		nfiu1 4989	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
33	31	nfrn 5940	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran 𝑓
34	31, 32, 33	nff1o 6841	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓
35		nfv 1949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(2nd ‘(𝑓‘𝑙)) = 𝑙
36	32, 35	nfralw 3313	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
37	34, 36	nfan 1934	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
38		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥ran 𝑓
39		nfiu1 4989	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
40	38, 39	nfss 3906	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
41	37, 40	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
42	30, 41	nfan 1934	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
43		nfv 1949	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ ran 𝑓
44	42, 43	nfan 1934	. . . . . . . 8 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
45		simpr 490	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 6909	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 781	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 796	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
49	48	simpld 500	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
50	49	simprd 501	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 739	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 6910	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
53		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
54	52, 53	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
55	54	rspcva 3578	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
56	47, 51, 55	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 2800	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 500	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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⟶1-1ontoran 𝑓)
59	58	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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⟶1-1ontoran 𝑓)
60		f1ocnvfv1 7304	. . . . . . . . . . 11 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
61	59, 47, 60	syl2anc 596	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 6909	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 2805	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 6844	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 → 𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵)
65	58, 64	syl 18	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 788	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 6965	. . . . . . . . . . 11 ⊢ (𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧))
68	67	biimpa 482	. . . . . . . . . 10 ⊢ ((𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
69	65, 66, 68	syl2anc 596	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 3174	. . . . . . . 8 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 3913	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 4958	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
73	71, 72	sylib 221	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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 3275	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
75		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
76		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℂ
77	14, 76	nfel 2940	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
78	75, 77	nfim 1931	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
79		eleq1w 2847	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
80	79	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)))
81	12	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ))
82	80, 81	imbi12d 347	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)))
83		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑘
84	83, 32	nfel 2940	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
85	30, 84	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
86		fsumiunle.3	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
87	86	adantllr 732	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
88	87	recnd 11361	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
89		eliun 4958	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵)
90	89	bilani 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵)
91	85, 88, 90	r19.29af 3275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ)
92	78, 82, 91	chvarfv 2280	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
93	92	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
94	16, 25, 29, 74, 93	fsumf1o 15915	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
95	15, 94	eqtrid 2810	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
96	95	eqcomd 2769	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
97		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
98	97, 39	nfel 2940	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
99	30, 98	nfan 1934	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
100		xp2nd 8055	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
101	100	adantl 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
102	86	ralrimiva 3158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
103	102	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
104	103	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
105		nfcsb1v 3875	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
106	105	nfel1 2942	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ
107		csbeq1a 3865	. . . . . . . . . . 11 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
108	107	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (𝐶 ∈ ℝ ↔ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ))
109	106, 108	rspc 3568	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ))
110	109	imp 412	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
111	101, 104, 110	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
112	72	bilani 510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
113	99, 111, 112	r19.29af 3275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
114	113	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
115		xp1st 8054	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) ∈ {𝑥})
116		elsni 4601	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ {𝑥} → (1st ‘𝑧) = 𝑥)
117	115, 116	syl 18	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) = 𝑥)
118	117, 100	jca 521	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵))
119		simplll 787	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
120		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑥 ∈ 𝐴)
121		fsumiunle.4	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶)
122	121	ralrimiva 3158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
123	119, 120, 122	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
124	118, 123	sylan2 605	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
125		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
126		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ≤
127	125, 126, 105	nfbr 5155	. . . . . . . . . 10 ⊢ Ⅎ𝑘0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶
128	107	breq2d 5118	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶))
129	127, 128	rspc 3568	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) ∈ 𝐵 → (∀𝑘 ∈ 𝐵 0 ≤ 𝐶 → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶))
130	129	imp 412	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 0 ≤ 𝐶) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
131	101, 124, 130	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
132	99, 131, 112	r19.29af 3275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
133	132	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
134	23, 114, 133, 24	fsumless 15989	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
135	96, 134	eqbrtrrd 5132	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
136	12, 13, 14	cbvsum 15888	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶
137	136	a1i 11	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶)
138	137	sumeq2sdv 15896	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶)
139		vex 3458	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
140		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
141	139, 140	op2ndd 8033	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
142	141	eqcomd 2769	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝑧))
143	142	csbeq1d 3855	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
144	143	eqcomd 2769	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
145		nfv 1949	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)
146	14	nfel1 2942	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
147	145, 146	nfim 1931	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
148		eleq1w 2847	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
149	148	anbi2d 642	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)))
150	149, 81	imbi12d 347	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)))
151	86	recnd 11361	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
152	147, 150, 151	chvarfv 2280	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
153	152	anasss 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
154	144, 1, 2, 153	fsum2d 15963	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
155	138, 154	eqtrd 2798	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
156	155	adantr 486	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 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)
157	135, 156	breqtrrd 5136	. 2 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶1-1ontoran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)
158	11, 157	exlimddv 1970	1 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⦋csb 3851   ⊆ wss 3900  {csn 4588  ⟨cop 4838  ∪ ciun 4954   class class class wbr 5106   × cxp 5653  ^◡ccnv 5654  ran crn 5656   Fn wfn 6550  ⟶1-1wf1 6552  ⟶1-1ontowf1o 6554  ‘cfv 6555  1^st c1st 8020  2^nd c2nd 8021  Fincfn 8998  ℂcc 11222  ℝcr 11223  0cc0 11224   ≤ cle 11368  Σcsu 15879

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-ac2 10565  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-oi 9528  df-card 10044  df-ac 10219  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-z 12718  df-uz 12990  df-rp 13145  df-ico 13506  df-fz 13664  df-fzo 13813  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-sum 15880

This theorem is used by:  hgt750lema  35353