Theorem sge0pr 47236

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 13505	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
2		sge0pr.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
3	1, 2	sselid 3929	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ*)
4		mnfxr 11312	. . . . . . . 8 ⊢ -∞ ∈ ℝ*
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → -∞ ∈ ℝ*)
6		0xr 11302	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ*)
8		mnflt0 13198	. . . . . . . . 9 ⊢ -∞ < 0
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -∞ < 0)
10		pnfxr 11309	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ*)
12		iccgelb 13477	. . . . . . . . 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 13234	. . . . . . 7 ⊢ (𝜑 → -∞ < 𝐸)
15	5, 3, 14	xrgtned 13237	. . . . . 6 ⊢ (𝜑 → 𝐸 ≠ -∞)
16		xaddpnf2 13301	. . . . . 6 ⊢ ((𝐸 ∈ ℝ* ∧ 𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
17	3, 15, 16	syl2anc 596	. . . . 5 ⊢ (𝜑 → (+∞ +𝑒 𝐸) = +∞)
18	17	eqcomd 2766	. . . 4 ⊢ (𝜑 → +∞ = (+∞ +𝑒 𝐸))
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20		prex 5403	. . . . 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 2860	. . . . . . . 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 2963	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐴 → 𝑘 ≠ 𝐴)
31	30	adantl 487	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ≠ 𝐴)
32		elprn1 4612	. . . . . . . . . 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 2860	. . . . . . . 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 2760	. . . . . 6 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
42	40, 41	fmptd 7109	. . . . 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 2766	. . . . . 6 ⊢ (𝐷 = +∞ → +∞ = 𝐷)
46	45	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = 𝐷)
47		prid1g 4721	. . . . . . . 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 46023	. . . . . . . 8 ⊢ (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
52	51	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
53	48, 52	eleqtrd 2862	. . . . . 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 2860	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
56	21, 43, 55	sge0pnfval 47215	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57		oveq1 7422	. . . 4 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
58	57	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
59	19, 56, 58	3eqtr4d 2805	. 2 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
60	1, 24	sselid 3929	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ*)
61		iccgelb 13477	. . . . . . . . . . 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 13234	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝐷)
64	5, 60, 63	xrgtned 13237	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ≠ -∞)
65		xaddpnf1 13300	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
66	60, 64, 65	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐷 +𝑒 +∞) = +∞)
67	66	eqcomd 2766	. . . . . 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 2766	. . . . . . . 8 ⊢ (𝐸 = +∞ → +∞ = 𝐸)
73	72	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = 𝐸)
74		prid2g 4722	. . . . . . . . . 10 ⊢ (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
75	2, 74	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
76	75, 52	eleqtrd 2862	. . . . . . . 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 2860	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
79	69, 70, 78	sge0pnfval 47215	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80		oveq2 7423	. . . . . 6 ⊢ (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
81	80	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
82	68, 79, 81	3eqtr4d 2805	. . . 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 13531	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
85		ax-resscn 11203	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
86	84, 85	sstri 3940	. . . . . . 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 13203	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℝ* → 𝐷 ≤ +∞)
92	60, 91	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≤ +∞)
93	92	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
94	44	necon3bi 2981	. . . . . . . . . . 11 ⊢ (¬ 𝐷 = +∞ → 𝐷 ≠ +∞)
95	94	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
96	89, 88, 93, 95	xrleneltd 46167	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
97	87, 88, 89, 90, 96	elicod 13470	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
98	97	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
99	86, 98	sselid 3929	. . . . . 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 13203	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ* → 𝐸 ≤ +∞)
105	3, 104	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≤ +∞)
106	105	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
107	71	necon3bi 2981	. . . . . . . . . . 11 ⊢ (¬ 𝐸 = +∞ → 𝐸 ≠ +∞)
108	107	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109	102, 101, 106, 108	xrleneltd 46167	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110	100, 101, 102, 103, 109	elicod 13470	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
111	86, 110	sselid 3929	. . . . . . 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 15856	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119		prfi 9297	. . . . . 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 2860	. . . . . . 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 2860	. . . . . . 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 47232	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
134	84, 98	sselid 3929	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
135	84, 110	sselid 3929	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136	135	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137		rexadd 13306	. . . . 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 2805	. . 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 2145   ≠ wne 2955  Vcvv 3450  {cpr 4586   class class class wbr 5103   ↦ cmpt 5186  ran crn 5656  ⟶wf 6530  ‘cfv 6534  (class class class)co 7415  Fincfn 8956  ℂcc 11144  ℝcr 11145  0cc0 11146   + caddc 11149  +∞cpnf 11286  -∞cmnf 11287  ℝ^*cxr 11288   < clt 11289   ≤ cle 11290   +_𝑒 cxad 13183  [,)cico 13422  [,]cicc 13423  Σcsu 15795  Σ^{^}csumge0 47204

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738  ax-inf2 9624  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7372  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-rdg 8401  df-1o 8459  df-2o 8460  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-oi 9486  df-card 9966  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11489  df-neg 11490  df-div 11918  df-nn 12280  df-2 12349  df-3 12350  df-n0 12551  df-z 12638  df-uz 12910  df-rp 13065  df-xadd 13186  df-ico 13426  df-icc 13427  df-fz 13584  df-fzo 13732  df-seq 14088  df-exp 14148  df-hash 14417  df-cj 15208  df-re 15209  df-im 15210  df-sqrt 15344  df-abs 15345  df-clim 15597  df-sum 15796  df-sumge0 47205

This theorem is used by:  sge0prle  47243  meadjun  47304  ovnsubadd2lem  47487