Theorem sge0split 46825

Description: Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0split.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sge0split.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
sge0split.u	⊢ 𝑈 = (𝐴 ∪ 𝐵)
sge0split.in0	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
sge0split.f	⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))

Assertion

Ref	Expression
sge0split	⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))

Proof of Theorem sge0split

Dummy variables 𝑎 𝑏 𝑥 𝑧 𝑦 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0split.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝐴 ∈ 𝑉)
3		sge0split.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝐵 ∈ 𝑊)
5		sge0split.u	. . . 4 ⊢ 𝑈 = (𝐴 ∪ 𝐵)
6		sge0split.in0	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (𝐴 ∩ 𝐵) = ∅)
8		sge0split.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝐹:𝑈⟶(0[,]+∞))
10		simpr 484	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) ∈ ℝ)
11	2, 4, 5, 7, 9, 10	sge0resplit 46822	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
12		unexg 7686	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
13	1, 3, 12	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
14	5, 13	eqeltrid 2839	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ V)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝑈 ∈ V)
16	15, 9, 10	sge0ssre 46813	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ)
17	15, 9, 10	sge0ssre 46813	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ)
18		rexadd 13173	. . . . 5 ⊢ (((Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ ∧ (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
19	16, 17, 18	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
20	19	eqcomd 2741	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
21	11, 20	eqtrd 2770	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
22		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → 𝜑)
23		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → ¬ (Σ^‘𝐹) ∈ ℝ)
24	14, 8	sge0repnf 46802	. . . . . 6 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞))
25	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞))
26	23, 25	mtbid 324	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → ¬ ¬ (Σ^‘𝐹) = +∞)
27	26	notnotrd 133	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = +∞)
28	14, 8	sge0xrcl 46801	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
29	28	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ ℝ*)
30		ssun1 4109	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31	30, 5	sseqtrri 3966	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑈
32	31	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
33	8, 32	fssresd 6696	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
34	1, 33	sge0xrcl 46801	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ*)
35		iccssxr 13372	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
36		ssun2 4110	. . . . . . . . . . 11 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
37	36, 5	sseqtrri 3966	. . . . . . . . . 10 ⊢ 𝐵 ⊆ 𝑈
38	37	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
39	8, 38	fssresd 6696	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
40	3, 39	sge0cl 46797	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐵)) ∈ (0[,]+∞))
41	35, 40	sselid 3915	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ*)
42	34, 41	xaddcld 13242	. . . . 5 ⊢ (𝜑 → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ∈ ℝ*)
43	42	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ∈ ℝ*)
44		pnfxr 11188	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
45		eqid 2735	. . . . . . . . 9 ⊢ +∞ = +∞
46		xreqle 45738	. . . . . . . . 9 ⊢ ((+∞ ∈ ℝ* ∧ +∞ = +∞) → +∞ ≤ +∞)
47	44, 45, 46	mp2an 693	. . . . . . . 8 ⊢ +∞ ≤ +∞
48	47	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ ≤ +∞)
49	14	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝑈 ∈ V)
50	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝐹:𝑈⟶(0[,]+∞))
51		rnresss 5971	. . . . . . . . . . 11 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
52	51	sseli 3913	. . . . . . . . . 10 ⊢ (+∞ ∈ ran (𝐹 ↾ 𝐴) → +∞ ∈ ran 𝐹)
53	52	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ ∈ ran 𝐹)
54	49, 50, 53	sge0pnfval 46789	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) = +∞)
55		xrge0neqmnf 13394	. . . . . . . . . . . . . 14 ⊢ ((Σ^‘(𝐹 ↾ 𝐵)) ∈ (0[,]+∞) → (Σ^‘(𝐹 ↾ 𝐵)) ≠ -∞)
56	40, 55	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐵)) ≠ -∞)
57		xaddpnf2 13168	. . . . . . . . . . . . 13 ⊢ (((Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ* ∧ (Σ^‘(𝐹 ↾ 𝐵)) ≠ -∞) → (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
58	41, 56, 57	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
59	58	eqcomd 2741	. . . . . . . . . . 11 ⊢ (𝜑 → +∞ = (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
60	59	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ = (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
61	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝐴 ∈ 𝑉)
62	33	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
63		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ ∈ ran (𝐹 ↾ 𝐴))
64	61, 62, 63	sge0pnfval 46789	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘(𝐹 ↾ 𝐴)) = +∞)
65	64	oveq1d 7371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
66	60, 54, 65	3eqtr4d 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
67	66, 54	eqtr3d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
68	54, 67	breq12d 5087	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ((Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ↔ +∞ ≤ +∞))
69	48, 68	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
70	47	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → +∞ ≤ +∞)
71	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝑈 ∈ V)
72	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
73		rnresss 5971	. . . . . . . . . . . . 13 ⊢ ran (𝐹 ↾ 𝐵) ⊆ ran 𝐹
74	73	sseli 3913	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ran (𝐹 ↾ 𝐵) → +∞ ∈ ran 𝐹)
75	74	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → +∞ ∈ ran 𝐹)
76	71, 72, 75	sge0pnfval 46789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) = +∞)
77	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐵 ∈ 𝑊)
78	39	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
79		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → +∞ ∈ ran (𝐹 ↾ 𝐵))
80	77, 78, 79	sge0pnfval 46789	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) = +∞)
81	80	oveq2d 7372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞))
82	1, 33	sge0cl 46797	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐴)) ∈ (0[,]+∞))
83		xrge0neqmnf 13394	. . . . . . . . . . . . . 14 ⊢ ((Σ^‘(𝐹 ↾ 𝐴)) ∈ (0[,]+∞) → (Σ^‘(𝐹 ↾ 𝐴)) ≠ -∞)
84	82, 83	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐴)) ≠ -∞)
85		xaddpnf1 13167	. . . . . . . . . . . . 13 ⊢ (((Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ* ∧ (Σ^‘(𝐹 ↾ 𝐴)) ≠ -∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞) = +∞)
86	34, 84, 85	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞) = +∞)
87	86	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞) = +∞)
88	81, 87	eqtrd 2770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
89	76, 88	breq12d 5087	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ↔ +∞ ≤ +∞))
90	70, 89	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
91	90	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
92		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
93		vex 3431	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
94		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
95	94	elrnmpt 5902	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
96	93, 95	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
97	92, 96	sylib 218	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
98		simp3 1139	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
99		inss1 4167	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝑥 ∩ 𝐴)
100		inss2 4168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
101	99, 100	sstri 3926	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ 𝐴
102		inss2 4168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝑥 ∩ 𝐵)
103		inss2 4168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
104	102, 103	sstri 3926	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ 𝐵
105	101, 104	ssini 4170	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝐴 ∩ 𝐵)
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝐴 ∩ 𝐵))
107	106, 6	sseqtrd 3953	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ ∅)
108		ss0 4332	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ ∅ → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
110	109	ad3antrrr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
111		indi 4214	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵))
112	111	eqcomi 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)) = (𝑥 ∩ (𝐴 ∪ 𝐵))
113	112	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)) = (𝑥 ∩ (𝐴 ∪ 𝐵)))
114	5	eqcomi 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∪ 𝐵) = 𝑈
115	114	ineq2i 4148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = (𝑥 ∩ 𝑈)
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ (𝐴 ∪ 𝐵)) = (𝑥 ∩ 𝑈))
117		elinel1 4132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
118		elpwi 4538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ 𝒫 𝑈 → 𝑥 ⊆ 𝑈)
119	117, 118	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ⊆ 𝑈)
120		dfss2 3903	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ⊆ 𝑈 ↔ (𝑥 ∩ 𝑈) = 𝑥)
121	120	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → (𝑥 ∩ 𝑈) = 𝑥)
122	119, 121	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝑈) = 𝑥)
123	113, 116, 122	3eqtrrd 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
124	123	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
125		elinel2 4133	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
126	125	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
127		rge0ssre 13398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0[,)+∞) ⊆ ℝ
128	8	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
129		pm4.56 991	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((¬ +∞ ∈ ran (𝐹 ↾ 𝐴) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ↔ ¬ (+∞ ∈ ran (𝐹 ↾ 𝐴) ∨ +∞ ∈ ran (𝐹 ↾ 𝐵)))
130	129	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((¬ +∞ ∈ ran (𝐹 ↾ 𝐴) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ (+∞ ∈ ran (𝐹 ↾ 𝐴) ∨ +∞ ∈ ran (𝐹 ↾ 𝐵)))
131		elun 4085	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (+∞ ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) ↔ (+∞ ∈ ran (𝐹 ↾ 𝐴) ∨ +∞ ∈ ran (𝐹 ↾ 𝐵)))
132	130, 131	sylnibr 329	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ +∞ ∈ ran (𝐹 ↾ 𝐴) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)))
133	132	adantll 715	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)))
134		rnresun 45598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))
135	134	eqcomi 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) = ran (𝐹 ↾ (𝐴 ∪ 𝐵))
136	135	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) = ran (𝐹 ↾ (𝐴 ∪ 𝐵)))
137	114	reseq2i 5930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = (𝐹 ↾ 𝑈)
138	137	rneqi 5881	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran (𝐹 ↾ 𝑈)
139	138	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran (𝐹 ↾ 𝑈))
140		ffn 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:𝑈⟶(0[,]+∞) → 𝐹 Fn 𝑈)
141		fnresdm 6606	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 Fn 𝑈 → (𝐹 ↾ 𝑈) = 𝐹)
142	8, 140, 141	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝐹 ↾ 𝑈) = 𝐹)
143	142	rneqd 5882	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ran (𝐹 ↾ 𝑈) = ran 𝐹)
144	136, 139, 143	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) = ran 𝐹)
145	144	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) = ran 𝐹)
146	133, 145	neleqtrd 2857	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ ran 𝐹)
147	128, 146	fge0iccico 46786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐹:𝑈⟶(0[,)+∞))
148	147	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑈⟶(0[,)+∞))
149	119	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑈)
150		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
151	149, 150	sseldd 3918	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
152	151	adantll 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
153	148, 152	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
154	127, 153	sselid 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
155	154	recnd 11162	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
156	110, 124, 126, 155	fsumsplit 15692	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
157		infi 9169	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝐴) ∈ Fin)
158	125, 157	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ Fin)
159	158	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐴) ∈ Fin)
160		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)))
161		elinel1 4132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → 𝑦 ∈ 𝑥)
162	161	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → 𝑦 ∈ 𝑥)
163	160, 162, 154	syl2anc 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → (𝐹‘𝑦) ∈ ℝ)
164	159, 163	fsumrecl 15685	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ)
165		infi 9169	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝐵) ∈ Fin)
166	125, 165	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐵) ∈ Fin)
167	166	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ Fin)
168		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)))
169		elinel1 4132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → 𝑦 ∈ 𝑥)
170	169	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → 𝑦 ∈ 𝑥)
171	168, 170, 154	syl2anc 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (𝐹‘𝑦) ∈ ℝ)
172	167, 171	fsumrecl 15685	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ)
173		rexadd 13173	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) +𝑒 Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
174	164, 172, 173	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) +𝑒 Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
175	174	eqcomd 2741	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) + Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) +𝑒 Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
176	156, 175	eqtrd 2770	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) +𝑒 Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)))
177		ressxr 11178	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℝ*
178	177, 164	sselid 3915	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ*)
179	177, 172	sselid 3915	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ*)
180	1	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝐴 ∈ 𝑉)
181	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
182		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
183	181, 182	fge0iccico 46786	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,)+∞))
184	180, 183	sge0reval 46788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘(𝐹 ↾ 𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ))
185	184	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) = (Σ^‘(𝐹 ↾ 𝐴)))
186	34	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ*)
187	185, 186	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
188	187	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
189	3	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐵 ∈ 𝑊)
190	39	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
191		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
192	190, 191	fge0iccico 46786	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,)+∞))
193	189, 192	sge0reval 46788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))
194	193	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) = (Σ^‘(𝐹 ↾ 𝐵)))
195	41	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ*)
196	194, 195	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
197	196	adantlr 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
198	188, 197	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*))
199	198	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*))
200	178, 179, 199	jca31 514	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ* ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ*) ∧ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)))
201	180	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐴 ∈ 𝑉)
202	181	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
203	182	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
204	202, 203	fge0iccico 46786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,)+∞))
205	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐴) ⊆ 𝐴)
206	158	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐴) ∈ Fin)
207	201, 204, 205, 206	fsumlesge0 46793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) ≤ (Σ^‘(𝐹 ↾ 𝐴)))
208	100	sseli 3913	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → 𝑦 ∈ 𝐴)
209		fvres 6848	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
210	208, 209	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
211	210	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
212	211	sumeq2dv 15653	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦))
213	184	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ))
214	212, 213	breq12d 5087	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) ≤ (Σ^‘(𝐹 ↾ 𝐴)) ↔ Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < )))
215	207, 214	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ))
216	215	adantlr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ))
217	189	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐵 ∈ 𝑊)
218	190	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
219	191	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
220	218, 219	fge0iccico 46786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,)+∞))
221	103	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐵) ⊆ 𝐵)
222	166	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ Fin)
223	217, 220, 221, 222	fsumlesge0 46793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦) ≤ (Σ^‘(𝐹 ↾ 𝐵)))
224	103	sseli 3913	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → 𝑦 ∈ 𝐵)
225		fvres 6848	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦))
226	224, 225	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦))
227	226	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦))
228	227	sumeq2dv 15653	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦))
229	193	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))
230	228, 229	breq12d 5087	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦) ≤ (Σ^‘(𝐹 ↾ 𝐵)) ↔ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
231	223, 230	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))
232	231	adantllr 720	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))
233	216, 232	jca 511	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
234		xle2add 13200	. . . . . . . . . . . . . . . . . 18 ⊢ (((Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ* ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ*) ∧ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)) → ((Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∧ Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) +𝑒 Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))))
235	200, 233, 234	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) +𝑒 Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦)) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
236	176, 235	eqbrtrd 5096	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
237	236	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
238	98, 237	eqbrtrd 5096	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
239	238	3exp 1120	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))))
240	239	rexlimdv 3134	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))))
241	240	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → (∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))))
242	97, 241	mpd 15	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
243	242	ralrimiva 3127	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ∀𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
244	147	sge0rnre 46780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ)
245	177	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ℝ ⊆ ℝ*)
246	244, 245	sstrd 3927	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
247	188, 197	xaddcld 13242	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )) ∈ ℝ*)
248		supxrleub 13267	. . . . . . . . . 10 ⊢ ((ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ* ∧ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )) ∈ ℝ*) → (sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )) ↔ ∀𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))))
249	246, 247, 248	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )) ↔ ∀𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))))
250	243, 249	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
251	14	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝑈 ∈ V)
252	251, 147	sge0reval 46788	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
253	184	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ))
254	193	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))
255	253, 254	oveq12d 7374	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < )))
256	250, 252, 255	3brtr4d 5106	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
257	91, 256	pm2.61dan 813	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
258	69, 257	pm2.61dan 813	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
259	258	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
260		pnfge 13070	. . . . . . 7 ⊢ (((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ∈ ℝ* → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ≤ +∞)
261	42, 260	syl 17	. . . . . 6 ⊢ (𝜑 → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ≤ +∞)
262	261	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ≤ +∞)
263		id 22	. . . . . . 7 ⊢ ((Σ^‘𝐹) = +∞ → (Σ^‘𝐹) = +∞)
264	263	eqcomd 2741	. . . . . 6 ⊢ ((Σ^‘𝐹) = +∞ → +∞ = (Σ^‘𝐹))
265	264	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → +∞ = (Σ^‘𝐹))
266	262, 265	breqtrd 5100	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ≤ (Σ^‘𝐹))
267	29, 43, 259, 266	xrletrid 13095	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
268	22, 27, 267	syl2anc 585	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
269	21, 268	pm2.61dan 813	1 ⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∀wral 3049  ∃wrex 3059  Vcvv 3427   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885  ∅c0 4263  𝒫 cpw 4531   class class class wbr 5074   ↦ cmpt 5155  ran crn 5621   ↾ cres 5622   Fn wfn 6482  ⟶wf 6483  ‘cfv 6487  (class class class)co 7356  Fincfn 8882  supcsup 9342  ℝcr 11026  0cc0 11027   + caddc 11030  +∞cpnf 11165  -∞cmnf 11166  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169   +_𝑒 cxad 13050  [,)cico 13289  [,]cicc 13290  Σcsu 15637  Σ^{^}csumge0 46778

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-inf2 9551  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

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 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-sup 9344  df-oi 9414  df-card 9852  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 12164  df-2 12233  df-3 12234  df-n0 12427  df-z 12514  df-uz 12778  df-rp 12932  df-xadd 13053  df-ico 13293  df-icc 13294  df-fz 13451  df-fzo 13598  df-seq 13953  df-exp 14013  df-hash 14282  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15439  df-sum 15638  df-sumge0 46779

This theorem is referenced by:  sge0splitmpt  46827