Theorem sge0pr 46845

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 13375	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
2		sge0pr.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
3	1, 2	sselid 3913	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ*)
4		mnfxr 11194	. . . . . . . 8 ⊢ -∞ ∈ ℝ*
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → -∞ ∈ ℝ*)
6		0xr 11184	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ*)
8		mnflt0 13068	. . . . . . . . 9 ⊢ -∞ < 0
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -∞ < 0)
10		pnfxr 11191	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ*)
12		iccgelb 13347	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐸 ∈ (0[,]+∞)) → 0 ≤ 𝐸)
13	7, 11, 2, 12	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐸)
14	5, 7, 3, 9, 13	xrltletrd 13104	. . . . . . 7 ⊢ (𝜑 → -∞ < 𝐸)
15	5, 3, 14	xrgtned 13107	. . . . . 6 ⊢ (𝜑 → 𝐸 ≠ -∞)
16		xaddpnf2 13171	. . . . . 6 ⊢ ((𝐸 ∈ ℝ* ∧ 𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
17	3, 15, 16	syl2anc 590	. . . . 5 ⊢ (𝜑 → (+∞ +𝑒 𝐸) = +∞)
18	17	eqcomd 2745	. . . 4 ⊢ (𝜑 → +∞ = (+∞ +𝑒 𝐸))
19	18	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20		prex 5368	. . . . 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 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
24		sge0pr.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))
25	24	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,]+∞))
26	23, 25	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
27	26	adantlr 721	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
28		simpll 772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
29		simpl 483	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵})
30		neqne 2942	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐴 → 𝑘 ≠ 𝐴)
31	30	adantl 482	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ≠ 𝐴)
32		elprn1 4584	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘 ≠ 𝐴) → 𝑘 = 𝐵)
33	29, 31, 32	syl2anc 590	. . . . . . . . 9 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
34	33	adantll 720	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
35		sge0pr.ce	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)
36	35	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
37	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,]+∞))
38	36, 37	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞))
39	28, 34, 38	syl2anc 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
40	27, 39	pm2.61dan 818	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞))
41		eqid 2739	. . . . . 6 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
42	40, 41	fmptd 7056	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
43	42	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
44		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
45	44	eqcomd 2745	. . . . . 6 ⊢ (𝐷 = +∞ → +∞ = 𝐷)
46	45	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = 𝐷)
47		prid1g 4693	. . . . . . . 8 ⊢ (𝐷 ∈ (0[,]+∞) → 𝐷 ∈ {𝐷, 𝐸})
48	24, 47	syl 17	. . . . . . 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 45632	. . . . . . . 8 ⊢ (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
52	51	eqcomd 2745	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
53	48, 52	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
54	53	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
55	46, 54	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
56	21, 43, 55	sge0pnfval 46824	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57		oveq1 7364	. . . 4 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
58	57	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
59	19, 56, 58	3eqtr4d 2784	. 2 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
60	1, 24	sselid 3913	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ*)
61		iccgelb 13347	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷)
62	7, 11, 24, 61	syl3anc 1379	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
63	5, 7, 60, 9, 62	xrltletrd 13104	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝐷)
64	5, 60, 63	xrgtned 13107	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ≠ -∞)
65		xaddpnf1 13170	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
66	60, 64, 65	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (𝐷 +𝑒 +∞) = +∞)
67	66	eqcomd 2745	. . . . . 6 ⊢ (𝜑 → +∞ = (𝐷 +𝑒 +∞))
68	67	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = (𝐷 +𝑒 +∞))
69	20	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → {𝐴, 𝐵} ∈ V)
70	42	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
71		id 22	. . . . . . . . 9 ⊢ (𝐸 = +∞ → 𝐸 = +∞)
72	71	eqcomd 2745	. . . . . . . 8 ⊢ (𝐸 = +∞ → +∞ = 𝐸)
73	72	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = 𝐸)
74		prid2g 4694	. . . . . . . . . 10 ⊢ (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
75	2, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
76	75, 52	eleqtrd 2841	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
77	76	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
78	73, 77	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
79	69, 70, 78	sge0pnfval 46824	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80		oveq2 7365	. . . . . 6 ⊢ (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
81	80	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
82	68, 79, 81	3eqtr4d 2784	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
83	82	adantlr 721	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
84		rge0ssre 13401	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
85		ax-resscn 11087	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
86	84, 85	sstri 3924	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
87	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ∈ ℝ*)
88	10	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → +∞ ∈ ℝ*)
89	60	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ ℝ*)
90	62	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ≤ 𝐷)
91		pnfge 13073	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℝ* → 𝐷 ≤ +∞)
92	60, 91	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≤ +∞)
93	92	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
94	44	necon3bi 2960	. . . . . . . . . . 11 ⊢ (¬ 𝐷 = +∞ → 𝐷 ≠ +∞)
95	94	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
96	89, 88, 93, 95	xrleneltd 45776	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
97	87, 88, 89, 90, 96	elicod 13340	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
98	97	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
99	86, 98	sselid 3913	. . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ*)
103	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ≤ 𝐸)
104		pnfge 13073	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ* → 𝐸 ≤ +∞)
105	3, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≤ +∞)
106	105	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
107	71	necon3bi 2960	. . . . . . . . . . 11 ⊢ (¬ 𝐸 = +∞ → 𝐸 ≠ +∞)
108	107	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109	102, 101, 106, 108	xrleneltd 45776	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110	100, 101, 102, 103, 109	elicod 13340	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
111	86, 110	sselid 3913	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
112	111	adantlr 721	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
113	99, 112	jca 516	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
114	49, 50	jca 516	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
115	114	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
116		sge0pr.ab	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
117	116	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴 ≠ 𝐵)
118	22, 35, 113, 115, 117	sumpr 15702	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119		prfi 9225	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin
120	119	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → {𝐴, 𝐵} ∈ Fin)
121	22	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
122	97	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,)+∞))
123	121, 122	eqeltrd 2839	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
124	123	ad4ant14 758	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
125		simp-4l 788	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
126		simpllr 781	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞)
127	33	adantll 720	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
128	36	3adant2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
129	110	3adant3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞))
130	128, 129	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞))
131	125, 126, 127, 130	syl3anc 1379	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
132	124, 131	pm2.61dan 818	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞))
133	120, 132	sge0fsummpt 46841	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
134	84, 98	sselid 3913	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
135	84, 110	sselid 3913	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136	135	adantlr 721	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137		rexadd 13176	. . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
138	134, 136, 137	syl2anc 590	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
139	118, 133, 138	3eqtr4d 2784	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
140	83, 139	pm2.61dan 818	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
141	59, 140	pm2.61dan 818	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431  {cpr 4558   class class class wbr 5073   ↦ cmpt 5154  ran crn 5620  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  Fincfn 8884  ℂcc 11028  ℝcr 11029  0cc0 11030   + caddc 11033  +∞cpnf 11168  -∞cmnf 11169  ℝ^*cxr 11170   < clt 11171   ≤ cle 11172   +_𝑒 cxad 13053  [,)cico 13292  [,]cicc 13293  Σcsu 15640  Σ^{^}csumge0 46813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-oi 9416  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-n0 12430  df-z 12517  df-uz 12781  df-rp 12935  df-xadd 13056  df-ico 13296  df-icc 13297  df-fz 13454  df-fzo 13601  df-seq 13956  df-exp 14016  df-hash 14285  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15442  df-sum 15641  df-sumge0 46814

This theorem is referenced by:  sge0prle  46852  meadjun  46913  ovnsubadd2lem  47096