Theorem sge0pr 47185

Description: Sum of a pair of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0pr.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sge0pr.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
sge0pr.d	⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))
sge0pr.e	⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
sge0pr.cd	⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)
sge0pr.ce	⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)
sge0pr.ab	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
sge0pr	⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝐷,𝑘   𝑘,𝐸   𝑘,𝑉   𝑘,𝑊   𝜑,𝑘

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem sge0pr

Step	Hyp	Ref	Expression
1		iccssxr 13477	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
2		sge0pr.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
3	1, 2	sselid 3936	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ*)
4		mnfxr 11285	. . . . . . . 8 ⊢ -∞ ∈ ℝ*
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → -∞ ∈ ℝ*)
6		0xr 11275	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ*)
8		mnflt0 13170	. . . . . . . . 9 ⊢ -∞ < 0
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -∞ < 0)
10		pnfxr 11282	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ*)
12		iccgelb 13449	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐸 ∈ (0[,]+∞)) → 0 ≤ 𝐸)
13	7, 11, 2, 12	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐸)
14	5, 7, 3, 9, 13	xrltletrd 13206	. . . . . . 7 ⊢ (𝜑 → -∞ < 𝐸)
15	5, 3, 14	xrgtned 13209	. . . . . 6 ⊢ (𝜑 → 𝐸 ≠ -∞)
16		xaddpnf2 13273	. . . . . 6 ⊢ ((𝐸 ∈ ℝ* ∧ 𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
17	3, 15, 16	syl2anc 596	. . . . 5 ⊢ (𝜑 → (+∞ +𝑒 𝐸) = +∞)
18	17	eqcomd 2771	. . . 4 ⊢ (𝜑 → +∞ = (+∞ +𝑒 𝐸))
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20		prex 5411	. . . . 5 ⊢ {𝐴, 𝐵} ∈ V
21	20	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → {𝐴, 𝐵} ∈ V)
22		sge0pr.cd	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)
23	22	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
24		sge0pr.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))
25	24	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,]+∞))
26	23, 25	eqeltrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
27	26	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
28		simpll 779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
29		simpl 488	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵})
30		neqne 2968	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐴 → 𝑘 ≠ 𝐴)
31	30	adantl 487	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ≠ 𝐴)
32		elprn1 4619	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘 ≠ 𝐴) → 𝑘 = 𝐵)
33	29, 31, 32	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
34	33	adantll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
35		sge0pr.ce	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)
36	35	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
37	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,]+∞))
38	36, 37	eqeltrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞))
39	28, 34, 38	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
40	27, 39	pm2.61dan 825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞))
41		eqid 2765	. . . . . 6 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
42	40, 41	fmptd 7113	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
43	42	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
44		id 23	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
45	44	eqcomd 2771	. . . . . 6 ⊢ (𝐷 = +∞ → +∞ = 𝐷)
46	45	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = 𝐷)
47		prid1g 4728	. . . . . . . 8 ⊢ (𝐷 ∈ (0[,]+∞) → 𝐷 ∈ {𝐷, 𝐸})
48	24, 47	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ {𝐷, 𝐸})
49		sge0pr.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
50		sge0pr.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
51	49, 50, 41, 22, 35	rnmptpr 45972	. . . . . . . 8 ⊢ (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
52	51	eqcomd 2771	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
53	48, 52	eleqtrd 2867	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
54	53	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
55	46, 54	eqeltrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
56	21, 43, 55	sge0pnfval 47164	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57		oveq1 7426	. . . 4 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
58	57	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
59	19, 56, 58	3eqtr4d 2810	. 2 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
60	1, 24	sselid 3936	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ*)
61		iccgelb 13449	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷)
62	7, 11, 24, 61	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
63	5, 7, 60, 9, 62	xrltletrd 13206	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝐷)
64	5, 60, 63	xrgtned 13209	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ≠ -∞)
65		xaddpnf1 13272	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
66	60, 64, 65	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐷 +𝑒 +∞) = +∞)
67	66	eqcomd 2771	. . . . . 6 ⊢ (𝜑 → +∞ = (𝐷 +𝑒 +∞))
68	67	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = (𝐷 +𝑒 +∞))
69	20	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → {𝐴, 𝐵} ∈ V)
70	42	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
71		id 23	. . . . . . . . 9 ⊢ (𝐸 = +∞ → 𝐸 = +∞)
72	71	eqcomd 2771	. . . . . . . 8 ⊢ (𝐸 = +∞ → +∞ = 𝐸)
73	72	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = 𝐸)
74		prid2g 4729	. . . . . . . . . 10 ⊢ (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
75	2, 74	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
76	75, 52	eleqtrd 2867	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
77	76	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
78	73, 77	eqeltrd 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
79	69, 70, 78	sge0pnfval 47164	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80		oveq2 7427	. . . . . 6 ⊢ (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
81	80	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
82	68, 79, 81	3eqtr4d 2810	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
83	82	adantlr 728	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
84		rge0ssre 13503	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
85		ax-resscn 11176	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
86	84, 85	sstri 3947	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
87	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ∈ ℝ*)
88	10	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → +∞ ∈ ℝ*)
89	60	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ ℝ*)
90	62	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ≤ 𝐷)
91		pnfge 13175	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℝ* → 𝐷 ≤ +∞)
92	60, 91	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≤ +∞)
93	92	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
94	44	necon3bi 2986	. . . . . . . . . . 11 ⊢ (¬ 𝐷 = +∞ → 𝐷 ≠ +∞)
95	94	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
96	89, 88, 93, 95	xrleneltd 46116	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
97	87, 88, 89, 90, 96	elicod 13442	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
98	97	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
99	86, 98	sselid 3936	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℂ)
100	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ∈ ℝ*)
101	10	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → +∞ ∈ ℝ*)
102	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ*)
103	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ≤ 𝐸)
104		pnfge 13175	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ* → 𝐸 ≤ +∞)
105	3, 104	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≤ +∞)
106	105	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
107	71	necon3bi 2986	. . . . . . . . . . 11 ⊢ (¬ 𝐸 = +∞ → 𝐸 ≠ +∞)
108	107	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109	102, 101, 106, 108	xrleneltd 46116	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110	100, 101, 102, 103, 109	elicod 13442	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
111	86, 110	sselid 3936	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
112	111	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
113	99, 112	jca 521	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
114	49, 50	jca 521	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
115	114	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
116		sge0pr.ab	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
117	116	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴 ≠ 𝐵)
118	22, 35, 113, 115, 117	sumpr 15826	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119		prfi 9290	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin
120	119	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → {𝐴, 𝐵} ∈ Fin)
121	22	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
122	97	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,)+∞))
123	121, 122	eqeltrd 2865	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
124	123	ad4ant14 765	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
125		simp-4l 795	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
126		simpllr 788	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞)
127	33	adantll 727	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
128	36	3adant2 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
129	110	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞))
130	128, 129	eqeltrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞))
131	125, 126, 127, 130	syl3anc 1398	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
132	124, 131	pm2.61dan 825	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞))
133	120, 132	sge0fsummpt 47181	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
134	84, 98	sselid 3936	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
135	84, 110	sselid 3936	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136	135	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137		rexadd 13278	. . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
138	134, 136, 137	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
139	118, 133, 138	3eqtr4d 2810	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
140	83, 139	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
141	59, 140	pm2.61dan 825	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  {cpr 4593   class class class wbr 5111   ↦ cmpt 5194  ran crn 5664  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  Fincfn 8949  ℂcc 11117  ℝcr 11118  0cc0 11119   + caddc 11122  +∞cpnf 11259  -∞cmnf 11260  ℝ^*cxr 11261   < clt 11262   ≤ cle 11263   +_𝑒 cxad 13155  [,)cico 13394  [,]cicc 13395  Σcsu 15765  Σ^{^}csumge0 47153

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196  ax-pre-sup 11197

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-div 11891  df-nn 12253  df-2 12322  df-3 12323  df-n0 12524  df-z 12611  df-uz 12883  df-rp 13037  df-xadd 13158  df-ico 13398  df-icc 13399  df-fz 13556  df-fzo 13704  df-seq 14060  df-exp 14120  df-hash 14389  df-cj 15178  df-re 15179  df-im 15180  df-sqrt 15314  df-abs 15315  df-clim 15567  df-sum 15766  df-sumge0 47154

This theorem is used by:  sge0prle  47192  meadjun  47253  ovnsubadd2lem  47436