Theorem fsumiunle 32831

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 32673	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6859	. . . . . 6 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 615	. . . . 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 6804	. . . . . . 7 ⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
7	6	frnd 6744	. . . . . 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 1835	. . 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 3913	. . . . . . 7 ⊢ (𝑘 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
13		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑦𝐶
14		nfcsb1v 3923	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶
15	12, 13, 14	cbvsum 15731	. . . . . 6 ⊢ Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶
16		csbeq1 3902	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
17		snfi 9083	. . . . . . . . . . . 12 ⊢ {𝑥} ∈ Fin
18		xpfi 9358	. . . . . . . . . . . 12 ⊢ (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin)
19	17, 2, 18	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × 𝐵) ∈ Fin)
20	19	ralrimiva 3146	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
21		iunfi 9383	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin)
22	1, 20, 21	syl2anc 584	. . . . . . . . 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 9301	. . . . . . 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 6860	. . . . . . . . 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 723	. . . . . . 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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
31		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑓
32		nfiu1 5027	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
33	31	nfrn 5963	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran 𝑓
34	31, 32, 33	nff1o 6846	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
35		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(2nd ‘(𝑓‘𝑙)) = 𝑙
36	32, 35	nfralw 3311	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
37	34, 36	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
38		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥ran 𝑓
39		nfiu1 5027	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
40	38, 39	nfss 3976	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
41	37, 40	nfan 1899	. . . . . . . . . 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 1899	. . . . . . . . 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 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ ran 𝑓
44	42, 43	nfan 1899	. . . . . . . 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 6910	. . . . . . . . . . 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 726	. . . . . . . . . . . 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 6911	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
54	52, 53	eqeq12d 2753	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
55	54	rspcva 3620	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
56	47, 51, 55	syl2anc 584	. . . . . . . . . . 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 2779	. . . . . . . . . 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 726	. . . . . . . . . . 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 7296	. . . . . . . . . . 11 ⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
61	59, 47, 60	syl2anc 584	. . . . . . . . . 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 6910	. . . . . . . . . 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 2784	. . . . . . . . 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 6849	. . . . . . . . . . 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 6969	. . . . . . . . . . 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 584	. . . . . . . . 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 3162	. . . . . . . 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 3983	. . . . . . . . 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 4995	. . . . . . . . 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 3268	. . . . . . 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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
76		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℂ
77	14, 76	nfel 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
78	75, 77	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
79		eleq1w 2824	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
80	79	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)))
81	12	eleq1d 2826	. . . . . . . . . 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 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑘
84	83, 32	nfel 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
85	30, 84	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
86		fsumiunle.3	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
87	86	adantllr 719	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)
88	87	recnd 11289	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
89		eliun 4995	. . . . . . . . . . . 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 3268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ)
93	78, 82, 92	chvarfv 2240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
94	93	adantlr 715	. . . . . . 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 15759	. . . . . 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 2789	. . . . 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 2743	. . . 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 2905	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
99	98, 39	nfel 2920	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
100	30, 99	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
101		xp2nd 8047	. . . . . . . . 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 3146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ)
104	103	adantlr 715	. . . . . . . . 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 3923	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
107	106	nfel1 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ
108		csbeq1a 3913	. . . . . . . . . . 11 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
109	108	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (𝐶 ∈ ℝ ↔ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ))
110	107, 109	rspc 3610	. . . . . . . . 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 584	. . . . . . 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 3268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)
116	115	adantlr 715	. . . . 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 8046	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) ∈ {𝑥})
118		elsni 4643	. . . . . . . . . . 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 3146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
125	121, 122, 124	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
126	120, 125	sylan2 593	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶)
127		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
128		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ≤
129	127, 128, 106	nfbr 5190	. . . . . . . . . 10 ⊢ Ⅎ𝑘0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶
130	108	breq2d 5155	. . . . . . . . . 10 ⊢ (𝑘 = (2nd ‘𝑧) → (0 ≤ 𝐶 ↔ 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶))
131	129, 130	rspc 3610	. . . . . . . . 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 584	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
134	100, 133, 114	r19.29af 3268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤ ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
135	134	adantlr 715	. . . . 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 15832	. . . 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 5167	. . 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 15731	. . . . . . 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 15739	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶)
141		vex 3484	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
142		vex 3484	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
143	141, 142	op2ndd 8025	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
144	143	eqcomd 2743	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝑧))
145	144	csbeq1d 3903	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
146	145	eqcomd 2743	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = ⦋𝑦 / 𝑘⦌𝐶)
147		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)
148	14	nfel1 2922	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ
149	147, 148	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)
150		eleq1w 2824	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
151	150	anbi2d 630	. . . . . . . . 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 11289	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
154	149, 152, 153	chvarfv 2240	. . . . . . 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 15807	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
157	140, 156	eqtrd 2777	. . . 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 5171	. 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 1935	1 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  ⦋csb 3899   ⊆ wss 3951  {csn 4626  ⟨cop 4632  ∪ ciun 4991   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684  ran crn 5686   Fn wfn 6556  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013  Fincfn 8985  ℂcc 11153  ℝcr 11154  0cc0 11155   ≤ cle 11296  Σcsu 15722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723

This theorem is referenced by:  hgt750lema  34672