Theorem sge0split 45204

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 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝐴 ∈ 𝑉)
3		sge0split.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝐵 ∈ 𝑊)
5		sge0split.u	. . . 4 ⊢ 𝑈 = (𝐴 ∪ 𝐵)
6		sge0split.in0	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (𝐴 ∩ 𝐵) = ∅)
8		sge0split.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝐹:𝑈⟶(0[,]+∞))
10		simpr 485	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) ∈ ℝ)
11	2, 4, 5, 7, 9, 10	sge0resplit 45201	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
12		unexg 7738	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
13	1, 3, 12	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
14	5, 13	eqeltrid 2837	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ V)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → 𝑈 ∈ V)
16	15, 9, 10	sge0ssre 45192	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ)
17	15, 9, 10	sge0ssre 45192	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ)
18		rexadd 13213	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
20	19	eqcomd 2738	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
21	11, 20	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
22		simpl 483	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → 𝜑)
23		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → ¬ (Σ^‘𝐹) ∈ ℝ)
24	14, 8	sge0repnf 45181	. . . . . 6 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞))
25	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞))
26	23, 25	mtbid 323	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → ¬ ¬ (Σ^‘𝐹) = +∞)
27	26	notnotrd 133	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = +∞)
28	14, 8	sge0xrcl 45180	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
29	28	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ ℝ*)
30		ssun1 4172	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31	30, 5	sseqtrri 4019	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑈
32	31	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
33	8, 32	fssresd 6758	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
34	1, 33	sge0xrcl 45180	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ*)
35		iccssxr 13409	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
36		ssun2 4173	. . . . . . . . . . 11 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
37	36, 5	sseqtrri 4019	. . . . . . . . . 10 ⊢ 𝐵 ⊆ 𝑈
38	37	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
39	8, 38	fssresd 6758	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
40	3, 39	sge0cl 45176	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐵)) ∈ (0[,]+∞))
41	35, 40	sselid 3980	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ*)
42	34, 41	xaddcld 13282	. . . . 5 ⊢ (𝜑 → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ∈ ℝ*)
43	42	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ∈ ℝ*)
44		pnfxr 11270	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
45		eqid 2732	. . . . . . . . 9 ⊢ +∞ = +∞
46		xreqle 44107	. . . . . . . . 9 ⊢ ((+∞ ∈ ℝ* ∧ +∞ = +∞) → +∞ ≤ +∞)
47	44, 45, 46	mp2an 690	. . . . . . . 8 ⊢ +∞ ≤ +∞
48	47	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ ≤ +∞)
49	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝑈 ∈ V)
50	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝐹:𝑈⟶(0[,]+∞))
51		rnresss 6017	. . . . . . . . . . 11 ⊢ ran (𝐹 ↾ 𝐴) ⊆ ran 𝐹
52	51	sseli 3978	. . . . . . . . . 10 ⊢ (+∞ ∈ ran (𝐹 ↾ 𝐴) → +∞ ∈ ran 𝐹)
53	52	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ ∈ ran 𝐹)
54	49, 50, 53	sge0pnfval 45168	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) = +∞)
55		xrge0neqmnf 13431	. . . . . . . . . . . . . 14 ⊢ ((Σ^‘(𝐹 ↾ 𝐵)) ∈ (0[,]+∞) → (Σ^‘(𝐹 ↾ 𝐵)) ≠ -∞)
56	40, 55	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐵)) ≠ -∞)
57		xaddpnf2 13208	. . . . . . . . . . . . 13 ⊢ (((Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ* ∧ (Σ^‘(𝐹 ↾ 𝐵)) ≠ -∞) → (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
58	41, 56, 57	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
59	58	eqcomd 2738	. . . . . . . . . . 11 ⊢ (𝜑 → +∞ = (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
60	59	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ = (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
61	1	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝐴 ∈ 𝑉)
62	33	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
63		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → +∞ ∈ ran (𝐹 ↾ 𝐴))
64	61, 62, 63	sge0pnfval 45168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘(𝐹 ↾ 𝐴)) = +∞)
65	64	oveq1d 7426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = (+∞ +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
66	60, 54, 65	3eqtr4d 2782	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
67	66, 54	eqtr3d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
68	54, 67	breq12d 5161	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ((Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ↔ +∞ ≤ +∞))
69	48, 68	mpbird 256	. . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝑈 ∈ V)
72	8	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
73		rnresss 6017	. . . . . . . . . . . . 13 ⊢ ran (𝐹 ↾ 𝐵) ⊆ ran 𝐹
74	73	sseli 3978	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ran (𝐹 ↾ 𝐵) → +∞ ∈ ran 𝐹)
75	74	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → +∞ ∈ ran 𝐹)
76	71, 72, 75	sge0pnfval 45168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) = +∞)
77	3	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐵 ∈ 𝑊)
78	39	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
79		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → +∞ ∈ ran (𝐹 ↾ 𝐵))
80	77, 78, 79	sge0pnfval 45168	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) = +∞)
81	80	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞))
82	1, 33	sge0cl 45176	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐴)) ∈ (0[,]+∞))
83		xrge0neqmnf 13431	. . . . . . . . . . . . . 14 ⊢ ((Σ^‘(𝐹 ↾ 𝐴)) ∈ (0[,]+∞) → (Σ^‘(𝐹 ↾ 𝐴)) ≠ -∞)
84	82, 83	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝐴)) ≠ -∞)
85		xaddpnf1 13207	. . . . . . . . . . . . 13 ⊢ (((Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ* ∧ (Σ^‘(𝐹 ↾ 𝐴)) ≠ -∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞) = +∞)
86	34, 84, 85	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞) = +∞)
87	86	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 +∞) = +∞)
88	81, 87	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) = +∞)
89	76, 88	breq12d 5161	. . . . . . . . 9 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ((Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ↔ +∞ ≤ +∞))
90	70, 89	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
91	90	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
92		simpr 485	. . . . . . . . . . . 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 3478	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
94		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
95	94	elrnmpt 5955	. . . . . . . . . . . . 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 217	. . . . . . . . . . 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 1138	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → 𝑧 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
99		inss1 4228	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝑥 ∩ 𝐴)
100		inss2 4229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
101	99, 100	sstri 3991	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ 𝐴
102		inss2 4229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝑥 ∩ 𝐵)
103		inss2 4229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
104	102, 103	sstri 3991	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ 𝐵
105	101, 104	ssini 4231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝐴 ∩ 𝐵)
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ (𝐴 ∩ 𝐵))
107	106, 6	sseqtrd 4022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ ∅)
108		ss0 4398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) ⊆ ∅ → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
110	109	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥 ∩ 𝐴) ∩ (𝑥 ∩ 𝐵)) = ∅)
111		indi 4273	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵))
112	111	eqcomi 2741	. . . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∪ 𝐵) = 𝑈
115	114	ineq2i 4209	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = (𝑥 ∩ 𝑈)
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ (𝐴 ∪ 𝐵)) = (𝑥 ∩ 𝑈))
117		elinel1 4195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
118		elpwi 4609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ 𝒫 𝑈 → 𝑥 ⊆ 𝑈)
119	117, 118	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ⊆ 𝑈)
120		df-ss 3965	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ⊆ 𝑈 ↔ (𝑥 ∩ 𝑈) = 𝑥)
121	120	biimpi 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ 𝑈 → (𝑥 ∩ 𝑈) = 𝑥)
122	119, 121	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝑈) = 𝑥)
123	113, 116, 122	3eqtrrd 2777	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
124	123	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ 𝐵)))
125		elinel2 4196	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
126	125	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
127		rge0ssre 13435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0[,)+∞) ⊆ ℝ
128	8	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
129		pm4.56 987	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((¬ +∞ ∈ ran (𝐹 ↾ 𝐴) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ↔ ¬ (+∞ ∈ ran (𝐹 ↾ 𝐴) ∨ +∞ ∈ ran (𝐹 ↾ 𝐵)))
130	129	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((¬ +∞ ∈ ran (𝐹 ↾ 𝐴) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ (+∞ ∈ ran (𝐹 ↾ 𝐴) ∨ +∞ ∈ ran (𝐹 ↾ 𝐵)))
131		elun 4148	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (+∞ ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) ↔ (+∞ ∈ ran (𝐹 ↾ 𝐴) ∨ +∞ ∈ ran (𝐹 ↾ 𝐵)))
132	130, 131	sylnibr 328	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ +∞ ∈ ran (𝐹 ↾ 𝐴) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)))
133	132	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)))
134		rnresun 43958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))
135	134	eqcomi 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = (𝐹 ↾ 𝑈)
138	137	rneqi 5936	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran (𝐹 ↾ 𝑈)
139	138	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran (𝐹 ↾ 𝑈))
140		ffn 6717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:𝑈⟶(0[,]+∞) → 𝐹 Fn 𝑈)
141		fnresdm 6669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 Fn 𝑈 → (𝐹 ↾ 𝑈) = 𝐹)
142	8, 140, 141	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝐹 ↾ 𝑈) = 𝐹)
143	142	rneqd 5937	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ran (𝐹 ↾ 𝑈) = ran 𝐹)
144	136, 139, 143	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) = ran 𝐹)
145	144	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) = ran 𝐹)
146	133, 145	neleqtrd 2855	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ ran 𝐹)
147	128, 146	fge0iccico 45165	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐹:𝑈⟶(0[,)+∞))
148	147	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑈⟶(0[,)+∞))
149	119	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑈)
150		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
151	149, 150	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
152	151	adantll 712	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑈)
153	148, 152	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
154	127, 153	sselid 3980	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
155	154	recnd 11244	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
156	110, 124, 126, 155	fsumsplit 15689	. . . . . . . . . . . . . . . . . 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 9270	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝐴) ∈ Fin)
158	125, 157	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐴) ∈ Fin)
159	158	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐴) ∈ Fin)
160		simpl 483	. . . . . . . . . . . . . . . . . . . . . 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 4195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → 𝑦 ∈ 𝑥)
162	161	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → 𝑦 ∈ 𝑥)
163	160, 162, 154	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → (𝐹‘𝑦) ∈ ℝ)
164	159, 163	fsumrecl 15682	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ)
165		infi 9270	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝐵) ∈ Fin)
166	125, 165	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ 𝐵) ∈ Fin)
167	166	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ Fin)
168		simpl 483	. . . . . . . . . . . . . . . . . . . . . 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 4195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → 𝑦 ∈ 𝑥)
170	169	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → 𝑦 ∈ 𝑥)
171	168, 170, 154	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (𝐹‘𝑦) ∈ ℝ)
172	167, 171	fsumrecl 15682	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ)
173		rexadd 13213	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . 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 11260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℝ*
178	177, 164	sselid 3980	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦) ∈ ℝ*)
179	177, 172	sselid 3980	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦) ∈ ℝ*)
180	1	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → 𝐴 ∈ 𝑉)
181	33	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
182		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
183	181, 182	fge0iccico 45165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,)+∞))
184	180, 183	sge0reval 45167	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘(𝐹 ↾ 𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ))
185	184	eqcomd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) = (Σ^‘(𝐹 ↾ 𝐴)))
186	34	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘(𝐹 ↾ 𝐴)) ∈ ℝ*)
187	185, 186	eqeltrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
188	187	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏 ∈ 𝑎 ((𝐹 ↾ 𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
189	3	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝐵 ∈ 𝑊)
190	39	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
191		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
192	190, 191	fge0iccico 45165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,)+∞))
193	189, 192	sge0reval 45167	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ))
194	193	eqcomd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) = (Σ^‘(𝐹 ↾ 𝐵)))
195	41	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘(𝐹 ↾ 𝐵)) ∈ ℝ*)
196	194, 195	eqeltrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑 ∈ 𝑐 ((𝐹 ↾ 𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
197	196	adantlr 713	. . . . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐴 ∈ 𝑉)
202	181	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹 ↾ 𝐴):𝐴⟶(0[,]+∞))
203	182	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐴))
204	202, 203	fge0iccico 45165	. . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐴) ∈ Fin)
207	201, 204, 205, 206	fsumlesge0 45172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) ≤ (Σ^‘(𝐹 ↾ 𝐴)))
208	100	sseli 3978	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐴) → 𝑦 ∈ 𝐴)
209		fvres 6910	. . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
212	211	sumeq2dv 15651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐴)((𝐹 ↾ 𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐴)(𝐹‘𝑦))
213	184	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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 5161	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐵 ∈ 𝑊)
218	190	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞))
219	191	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝐵))
220	218, 219	fge0iccico 45165	. . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ Fin)
223	217, 220, 221, 222	fsumlesge0 45172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦) ≤ (Σ^‘(𝐹 ↾ 𝐵)))
224	103	sseli 3978	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (𝑥 ∩ 𝐵) → 𝑦 ∈ 𝐵)
225		fvres 6910	. . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦))
228	227	sumeq2dv 15651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥 ∩ 𝐵)((𝐹 ↾ 𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥 ∩ 𝐵)(𝐹‘𝑦))
229	193	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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 5161	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . . 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 13240	. . . . . . . . . . . . . . . . . 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 5170	. . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 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 5170	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 3153	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 3146	. . . . . . . . 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 45159	. . . . . . . . . . 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 3992	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
247	188, 197	xaddcld 13282	. . . . . . . . . 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 13307	. . . . . . . . . 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 584	. . . . . . . . 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 256	. . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → 𝑈 ∈ V)
252	251, 147	sge0reval 45167	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
253	184	adantr 481	. . . . . . . . 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 713	. . . . . . . . 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 7429	. . . . . . . 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 5180	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐵)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
257	91, 256	pm2.61dan 811	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹 ↾ 𝐴)) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
258	69, 257	pm2.61dan 811	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
259	258	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ≤ ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
260		pnfge 13112	. . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ≤ +∞)
263		id 22	. . . . . . 7 ⊢ ((Σ^‘𝐹) = +∞ → (Σ^‘𝐹) = +∞)
264	263	eqcomd 2738	. . . . . 6 ⊢ ((Σ^‘𝐹) = +∞ → +∞ = (Σ^‘𝐹))
265	264	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → +∞ = (Σ^‘𝐹))
266	262, 265	breqtrd 5174	. . . 4 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))) ≤ (Σ^‘𝐹))
267	29, 43, 259, 266	xrletrid 13136	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
268	22, 27, 267	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) ∈ ℝ) → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
269	21, 268	pm2.61dan 811	1 ⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∪ cun 3946   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677   ↾ cres 5678   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411  Fincfn 8941  supcsup 9437  ℝcr 11111  0cc0 11112   + caddc 11115  +∞cpnf 11247  -∞cmnf 11248  ℝ^*cxr 11249   < clt 11250   ≤ cle 11251   +_𝑒 cxad 13092  [,)cico 13328  [,]cicc 13329  Σcsu 15634  Σ^{^}csumge0 45157

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-xadd 13095  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635  df-sumge0 45158

This theorem is referenced by:  sge0splitmpt  45206