Theorem sge0pr 46409

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 13470	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
2		sge0pr.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
3	1, 2	sselid 3981	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ*)
4		mnfxr 11318	. . . . . . . 8 ⊢ -∞ ∈ ℝ*
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → -∞ ∈ ℝ*)
6		0xr 11308	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ*)
8		mnflt0 13167	. . . . . . . . 9 ⊢ -∞ < 0
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -∞ < 0)
10		pnfxr 11315	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ*)
12		iccgelb 13443	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐸 ∈ (0[,]+∞)) → 0 ≤ 𝐸)
13	7, 11, 2, 12	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐸)
14	5, 7, 3, 9, 13	xrltletrd 13203	. . . . . . 7 ⊢ (𝜑 → -∞ < 𝐸)
15	5, 3, 14	xrgtned 45333	. . . . . 6 ⊢ (𝜑 → 𝐸 ≠ -∞)
16		xaddpnf2 13269	. . . . . 6 ⊢ ((𝐸 ∈ ℝ* ∧ 𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
17	3, 15, 16	syl2anc 584	. . . . 5 ⊢ (𝜑 → (+∞ +𝑒 𝐸) = +∞)
18	17	eqcomd 2743	. . . 4 ⊢ (𝜑 → +∞ = (+∞ +𝑒 𝐸))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20		prex 5437	. . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
24		sge0pr.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,]+∞))
26	23, 25	eqeltrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
27	26	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
28		simpll 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
29		simpl 482	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵})
30		neqne 2948	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐴 → 𝑘 ≠ 𝐴)
31	30	adantl 481	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ≠ 𝐴)
32		elprn1 45648	. . . . . . . . . 10 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘 ≠ 𝐴) → 𝑘 = 𝐵)
33	29, 31, 32	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
34	33	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
35		sge0pr.ce	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)
36	35	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
37	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,]+∞))
38	36, 37	eqeltrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞))
39	28, 34, 38	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
40	27, 39	pm2.61dan 813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞))
41		eqid 2737	. . . . . 6 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
42	40, 41	fmptd 7134	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
43	42	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
44		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
45	44	eqcomd 2743	. . . . . 6 ⊢ (𝐷 = +∞ → +∞ = 𝐷)
46	45	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = 𝐷)
47		prid1g 4760	. . . . . . . 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 45182	. . . . . . . 8 ⊢ (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
52	51	eqcomd 2743	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
53	48, 52	eleqtrd 2843	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
54	53	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
55	46, 54	eqeltrd 2841	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
56	21, 43, 55	sge0pnfval 46388	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57		oveq1 7438	. . . 4 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
58	57	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
59	19, 56, 58	3eqtr4d 2787	. 2 ⊢ ((𝜑 ∧ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
60	1, 24	sselid 3981	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ*)
61		iccgelb 13443	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷)
62	7, 11, 24, 61	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
63	5, 7, 60, 9, 62	xrltletrd 13203	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝐷)
64	5, 60, 63	xrgtned 45333	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ≠ -∞)
65		xaddpnf1 13268	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
66	60, 64, 65	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐷 +𝑒 +∞) = +∞)
67	66	eqcomd 2743	. . . . . 6 ⊢ (𝜑 → +∞ = (𝐷 +𝑒 +∞))
68	67	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = (𝐷 +𝑒 +∞))
69	20	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → {𝐴, 𝐵} ∈ V)
70	42	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
71		id 22	. . . . . . . . 9 ⊢ (𝐸 = +∞ → 𝐸 = +∞)
72	71	eqcomd 2743	. . . . . . . 8 ⊢ (𝐸 = +∞ → +∞ = 𝐸)
73	72	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = 𝐸)
74		prid2g 4761	. . . . . . . . . 10 ⊢ (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
75	2, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
76	75, 52	eleqtrd 2843	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
77	76	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
78	73, 77	eqeltrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
79	69, 70, 78	sge0pnfval 46388	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80		oveq2 7439	. . . . . 6 ⊢ (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
81	80	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
82	68, 79, 81	3eqtr4d 2787	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
83	82	adantlr 715	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
84		rge0ssre 13496	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
85		ax-resscn 11212	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
86	84, 85	sstri 3993	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
87	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ∈ ℝ*)
88	10	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → +∞ ∈ ℝ*)
89	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ ℝ*)
90	62	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ≤ 𝐷)
91		pnfge 13172	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℝ* → 𝐷 ≤ +∞)
92	60, 91	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≤ +∞)
93	92	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
94	44	necon3bi 2967	. . . . . . . . . . 11 ⊢ (¬ 𝐷 = +∞ → 𝐷 ≠ +∞)
95	94	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
96	89, 88, 93, 95	xrleneltd 45334	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
97	87, 88, 89, 90, 96	elicod 13437	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
98	97	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
99	86, 98	sselid 3981	. . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ*)
103	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ≤ 𝐸)
104		pnfge 13172	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ* → 𝐸 ≤ +∞)
105	3, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≤ +∞)
106	105	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
107	71	necon3bi 2967	. . . . . . . . . . 11 ⊢ (¬ 𝐸 = +∞ → 𝐸 ≠ +∞)
108	107	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109	102, 101, 106, 108	xrleneltd 45334	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110	100, 101, 102, 103, 109	elicod 13437	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
111	86, 110	sselid 3981	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
112	111	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
113	99, 112	jca 511	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
114	49, 50	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
115	114	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
116		sge0pr.ab	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
117	116	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴 ≠ 𝐵)
118	22, 35, 113, 115, 117	sumpr 15784	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119		prfi 9363	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin
120	119	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → {𝐴, 𝐵} ∈ Fin)
121	22	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
122	97	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,)+∞))
123	121, 122	eqeltrd 2841	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
124	123	ad4ant14 752	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
125		simp-4l 783	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
126		simpllr 776	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞)
127	33	adantll 714	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
128	36	3adant2 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
129	110	3adant3 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞))
130	128, 129	eqeltrd 2841	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞))
131	125, 126, 127, 130	syl3anc 1373	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
132	124, 131	pm2.61dan 813	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞))
133	120, 132	sge0fsummpt 46405	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
134	84, 98	sselid 3981	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
135	84, 110	sselid 3981	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136	135	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137		rexadd 13274	. . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
138	134, 136, 137	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
139	118, 133, 138	3eqtr4d 2787	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
140	83, 139	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
141	59, 140	pm2.61dan 813	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  {cpr 4628   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  ℂcc 11153  ℝcr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  -∞cmnf 11293  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296   +_𝑒 cxad 13152  [,)cico 13389  [,]cicc 13390  Σcsu 15722  Σ^{^}csumge0 46377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-sumge0 46378

This theorem is referenced by:  sge0prle  46416  meadjun  46477  ovnsubadd2lem  46660