Theorem fsumiunle 32833

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 32681	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6873	. . . . . 6 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 614	. . . . 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 6817	. . . . . . 7 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
7	6	frnd 6755	. . . . . 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 1833	. . 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 3935	. . . . . . 7 ⊢ (𝑘 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
13		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑦𝐶
14		nfcsb1v 3946	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶
15	12, 13, 14	cbvsum 15743	. . . . . 6 ⊢ Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶
16		csbeq1 3924	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
17		snfi 9109	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ Fin
18		xpfi 9386	. . . . . . . . . . . 12 ⊢ (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
19	17, 2, 18	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × 𝐵) ∈ Fin)
20	19	ralrimiva 3152	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
21		iunfi 9411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
22	1, 20, 21	syl2anc 583	. . . . . . . . 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 772	. . . . . . . 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 9329	. . . . . . 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 770	. . . . . . . . 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 6874	. . . . . . . . 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 722	. . . . . . 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 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
31		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑓
32		nfiu1 5050	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
33	31	nfrn 5977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran 𝑓
34	31, 32, 33	nff1o 6860	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
35		nfv 1913	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(2nd ‘(𝑓‘𝑙)) = 𝑙
36	32, 35	nfralw 3317	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
37	34, 36	nfan 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
38		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥ran 𝑓
39		nfiu1 5050	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
40	38, 39	nfss 4001	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
41	37, 40	nfan 1898	. . . . . . . . . 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 1898	. . . . . . . . 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 1913	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ ran 𝑓
44	42, 43	nfan 1898	. . . . . . . 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 6924	. . . . . . . . . . 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 768	. . . . . . . . . . . 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 783	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 6925	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
54	52, 53	eqeq12d 2756	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
55	54	rspcva 3633	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
56	47, 51, 55	syl2anc 583	. . . . . . . . . . 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 2782	. . . . . . . . . 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 725	. . . . . . . . . . 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 7312	. . . . . . . . . . 11 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
61	59, 47, 60	syl2anc 583	. . . . . . . . . 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 6924	. . . . . . . . . 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 2787	. . . . . . . . 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 6863	. . . . . . . . . . 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 775	. . . . . . . . . 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 6982	. . . . . . . . . . 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 583	. . . . . . . . 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 3168	. . . . . . . 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 4008	. . . . . . . . 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 5019	. . . . . . . . 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 3274	. . . . . . 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 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
76		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℂ
77	14, 76	nfel 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
78	75, 77	nfim 1895	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
79		eleq1w 2827	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
80	79	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)))
81	12	eleq1d 2829	. . . . . . . . . 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 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑘
84	83, 32	nfel 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
85	30, 84	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
86		fsumiunle.3	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
87	86	adantllr 718	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
88	87	recnd 11318	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
89		eliun 5019	. . . . . . . . . . . 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 3274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ)
93	78, 82, 92	chvarfv 2241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
94	93	adantlr 714	. . . . . . 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 15771	. . . . . 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 2792	. . . . 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 2746	. . . 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 2908	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
99	98, 39	nfel 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
100	30, 99	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
101		xp2nd 8063	. . . . . . . . 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 3152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
104	103	adantlr 714	. . . . . . . . 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 3946	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
107	106	nfel1 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ
108		csbeq1a 3935	. . . . . . . . . . 11 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
109	108	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (𝐶 ∈ ℝ ↔ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ))
110	107, 109	rspc 3623	. . . . . . . . 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 583	. . . . . . 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 3274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
116	115	adantlr 714	. . . . 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 8062	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) ∈ {𝑥})
118		elsni 4665	. . . . . . . . . . 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 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
122		simplr 768	. . . . . . . . . 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 3152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
125	121, 122, 124	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
126	120, 125	sylan2 592	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
127		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
128		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ≤
129	127, 128, 106	nfbr 5213	. . . . . . . . . 10 ⊢ Ⅎ𝑘0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶
130	108	breq2d 5178	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶))
131	129, 130	rspc 3623	. . . . . . . . 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 583	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
134	100, 133, 114	r19.29af 3274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
135	134	adantlr 714	. . . . 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 15844	. . . 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 5190	. . 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 15743	. . . . . . 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 15751	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶)
141		vex 3492	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
142		vex 3492	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
143	141, 142	op2ndd 8041	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
144	143	eqcomd 2746	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝑧))
145	144	csbeq1d 3925	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
146	145	eqcomd 2746	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
147		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)
148	14	nfel1 2925	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
149	147, 148	nfim 1895	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
150		eleq1w 2827	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
151	150	anbi2d 629	. . . . . . . . 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 11318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
154	149, 152, 153	chvarfv 2241	. . . . . . 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 15819	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
157	140, 156	eqtrd 2780	. . . 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 5194	. 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 1934	1 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ⦋csb 3921   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ∪ ciun 5015   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699  ran crn 5701   Fn wfn 6568  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184   ≤ cle 11325  Σcsu 15734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-r1 9833  df-rank 9834  df-card 10008  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735

This theorem is referenced by:  hgt750lema  34634