esumrnmpt2 - Mathbox for Thierry Arnoux

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Theorem esumrnmpt2 31322

Description: Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 30-May-2020.)

**Hypotheses**
Ref	Expression
esumrnmpt2.1	⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
esumrnmpt2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumrnmpt2.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
esumrnmpt2.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
esumrnmpt2.5	⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
esumrnmpt2.6	⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)

**Assertion**
Ref	Expression
esumrnmpt2	⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐷)

Distinct variable groups: 𝐴,𝑘,𝑦 𝑦,𝐵 𝐶,𝑘 𝑦,𝐷 𝑘,𝑊 𝜑,𝑘,𝑦 Allowed substitution hints: 𝐵(𝑘) 𝐶(𝑦) 𝐷(𝑘) 𝑉(𝑦,𝑘) 𝑊(𝑦)

**Proof of Theorem esumrnmpt2**
Step	Hyp	Ref	Expression
1		nfrab1 3384	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}
2		esumrnmpt2.1	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
3		esumrnmpt2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssrab2 4055	. . . . . . 7 ⊢ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
6	3, 5	ssexd 5220	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
7	5	sselda 3966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
8		esumrnmpt2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
9	7, 8	syldan 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10		esumrnmpt2.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	7, 10	syldan 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ 𝑊)
12		rabid 3378	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝐵 = ∅))
13	12	simprbi 499	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
14	13	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15		elsng 4574	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
16	11, 15	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
17	14, 16	mtbird 327	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
18	11, 17	eldifd 3946	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19		esumrnmpt2.6	. . . . . 6 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)
20		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
21	1, 20	disjss1f 30316	. . . . . 6 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴 → (Disj 𝑘 ∈ 𝐴 𝐵 → Disj 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐵))
22	5, 19, 21	sylc 65	. . . . 5 ⊢ (𝜑 → Disj 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐵)
23	1, 2, 6, 9, 18, 22	esumrnmpt 31306	. . . 4 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
25		snex 5323	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → {∅} ∈ V)
27		velsn 4576	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
28	27	biimpi 218	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
29	28	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30		nfv 1911	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝜑
31		nfre1 3306	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝐵 = ∅
32	30, 31	nfan 1896	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
33		nfv 1911	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑦 = ∅
34	32, 33	nfan 1896	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35		nfv 1911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝐶 = 0
36		simpllr 774	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
38	36, 37	eqtr4d 2859	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = 𝐵)
39	38, 2	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 𝐷)
40		simp-4l 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝜑)
41		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑘 ∈ 𝐴)
42		esumrnmpt2.5	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
43	40, 41, 37, 42	syl21anc 835	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
44	39, 43	eqtrd 2856	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
46	34, 35, 44, 45	r19.29af2 3330	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
47	29, 46	syldan 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48		0e0iccpnf 12841	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
49	47, 48	eqeltrdi 2921	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50		nfcv 2977	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑦
51		nfmpt1 5156	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
52	51	nfrn 5818	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
53	50, 52	nfel 2992	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
54	30, 53	nfan 1896	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵))
55		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56		rabid 3378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = ∅))
57	56	simprbi 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} → 𝐵 = ∅)
58	57	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
59	55, 58	eqtrd 2856	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
60	59, 27	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61		vex 3497	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
62		eqid 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
63	62	elrnmpt 5822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵))
64	61, 63	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
65	64	biimpi 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
66	65	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
67	54, 60, 66	r19.29af 3331	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
68	67	ex 415	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
69	68	ssrdv 3972	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
70	69	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
71	24, 26, 49, 70	esummono 31308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ^𝑦 ∈ {∅}𝐶)
72		0ex 5203	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
73	72	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ∅ ∈ V)
74	48	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → 0 ∈ (0[,]+∞))
75	46, 73, 74	esumsn 31319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ {∅}𝐶 = 0)
76	71, 75	breqtrd 5084	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
78		nfv 1911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
79	31	nfn 1853	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
80		rabn0 4338	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
81	80	biimpi 218	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
82	81	necon1bi 3044	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} = ∅)
83		eqid 2821	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
85	79, 82, 84	mpteq12df 5140	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
86		mpt0 6484	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ∅ ↦ 𝐵) = ∅
87	85, 86	syl6eq 2872	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
88	87	rneqd 5802	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ran ∅)
89		rn0 5790	. . . . . . . . . . . . . 14 ⊢ ran ∅ = ∅
90	88, 89	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
91	78, 90	esumeq1d 31289	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑦 ∈ ∅𝐶)
92		esumnul 31302	. . . . . . . . . . . 12 ⊢ Σ^*𝑦 ∈ ∅𝐶 = 0
93	91, 92	syl6eq 2872	. . . . . . . . . . 11 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
94		0le0 11732	. . . . . . . . . . 11 ⊢ 0 ≤ 0
95	93, 94	eqbrtrdi 5097	. . . . . . . . . 10 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
96	77, 95	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
97	76, 96	pm2.61dan 811	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
98		ssrab2 4055	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴)
100	3, 99	ssexd 5220	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V)
101		nfrab1 3384	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}
102	101	mptexgf 6979	. . . . . . . . . . 11 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
103		rnexg 7608	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
104	100, 102, 103	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
105	2	adantl 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
106		simplll 773	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
107	99	sselda 3966	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
108	107	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
109	108	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
110	106, 109, 8	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
111	105, 110	eqeltrd 2913	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
112	54, 111, 66	r19.29af 3331	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
113	112	ralrimiva 3182	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
114		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
115	114	esumcl 31284	. . . . . . . . . 10 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V ∧ ∀𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
116	104, 113, 115	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
117		elxrge0 12839	. . . . . . . . . 10 ⊢ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) ↔ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^* ∧ 0 ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶))
118	117	simprbi 499	. . . . . . . . 9 ⊢ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) → 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)
119	116, 118	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)
120	97, 119	jca 514	. . . . . . 7 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶))
121		iccssxr 12813	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
122	121, 116	sseldi 3964	. . . . . . . 8 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
123	121, 48	sselii 3963	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
124	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
125		xrletri3 12541	. . . . . . . 8 ⊢ ((Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^ ∧ 0 ∈ ℝ^) → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0 ↔ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)))
126	122, 124, 125	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0 ↔ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)))
127	120, 126	mpbird 259	. . . . . 6 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
128	127	oveq1d 7165	. . . . 5 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = (0 +_𝑒 Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶))
129	9	ralrimiva 3182	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
130	1	esumcl 31284	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
131	6, 129, 130	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
132	121, 131	sseldi 3964	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^)
133	23, 132	eqeltrd 2913	. . . . . 6 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
134		xaddid2 12629	. . . . . 6 ⊢ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^ → (0 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
135	133, 134	syl 17	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
136	128, 135	eqtrd 2856	. . . 4 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
137		simpl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝜑)
138	57	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
139	137, 107, 138, 42	syl21anc 835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 = 0)
140	139	ralrimiva 3182	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
141	30, 140	esumeq2d 31291	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0)
142	101	esum0 31303	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0 = 0)
143	100, 142	syl 17	. . . . . . 7 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0 = 0)
144	141, 143	eqtrd 2856	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
145	144	oveq1d 7165	. . . . 5 ⊢ (𝜑 → (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +_𝑒 Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
146		xaddid2 12629	. . . . . 6 ⊢ (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^ → (0 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
147	132, 146	syl 17	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
148	145, 147	eqtrd 2856	. . . 4 ⊢ (𝜑 → (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
149	23, 136, 148	3eqtr4d 2866	. . 3 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
150		nfv 1911	. . . 4 ⊢ Ⅎ𝑦𝜑
151		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
152	1	mptexgf 6979	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
153		rnexg 7608	. . . . 5 ⊢ ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
154	6, 152, 153	3syl 18	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
155	69	ssrind 4211	. . . . . 6 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
156		incom 4177	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
157	13	neqned 3023	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
158	157	necomd 3071	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
159	158	neneqd 3021	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
160	159	nrex 3269	. . . . . . . . 9 ⊢ ¬ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
161		eqid 2821	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
162	161	elrnmpt 5822	. . . . . . . . . 10 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵))
163	72, 162	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵)
164	160, 163	mtbir 325	. . . . . . . 8 ⊢ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
165		disjsn 4640	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
166	164, 165	mpbir 233	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
167	156, 166	eqtr3i 2846	. . . . . 6 ⊢ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
168	155, 167	sseqtrdi 4016	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
169		ss0 4351	. . . . 5 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅ → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
170	168, 169	syl 17	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
171		nfmpt1 5156	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
172	171	nfrn 5818	. . . . . . 7 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
173	50, 172	nfel 2992	. . . . . 6 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
174	30, 173	nfan 1896	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
175	2	adantl 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
176		simplll 773	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
177	7	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
178	177	adantr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
179	176, 178, 8	syl2anc 586	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
180	175, 179	eqeltrd 2913	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
181	161	elrnmpt 5822	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵))
182	61, 181	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
183	182	biimpi 218	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
184	183	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185	174, 180, 184	r19.29af 3331	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
186	150, 114, 151, 104, 154, 170, 112, 185	esumsplit 31307	. . 3 ⊢ (𝜑 → Σ^𝑦 ∈ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶 = (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 +_𝑒 Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶))
187		rabnc 4340	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
188	187	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
189	107, 8	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
190	30, 101, 1, 100, 6, 188, 189, 9	esumsplit 31307	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷 = (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
191	149, 186, 190	3eqtr4d 2866	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶 = Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
192		rabxm 4339	. . . . . . . 8 ⊢ 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})
193	192, 83	mpteq12i 5151	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
194		mptun 6488	. . . . . . 7 ⊢ (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
195	193, 194	eqtri 2844	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
196	195	rneqi 5801	. . . . 5 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197		rnun 5998	. . . . 5 ⊢ ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
198	196, 197	eqtri 2844	. . . 4 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
199	198	a1i 11	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ 𝐵) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
200	150, 199	esumeq1d 31289	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ^𝑦 ∈ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶)
201	192	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}))
202	30, 201	esumeq1d 31289	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐷 = Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
203	191, 200, 202	3eqtr4d 2866	1 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐷)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 Disj wdisj 5023 class class class wbr 5058 ↦ cmpt 5138 ran crn 5550 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 ℝ^*cxr 10668 ≤ cle 10670 +_𝑒 cxad 12499 [,]cicc 12735 Σ^*cesum 31281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-ordt 16768 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-ps 17804 df-tsr 17805 df-plusf 17845 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-abv 19582 df-lmod 19630 df-scaf 19631 df-sra 19938 df-rgmod 19939 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-tmd 22674 df-tgp 22675 df-tsms 22729 df-trg 22762 df-xms 22924 df-ms 22925 df-tms 22926 df-nm 23186 df-ngp 23187 df-nrg 23189 df-nlm 23190 df-ii 23479 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-esum 31282

This theorem is referenced by: carsggect 31571 carsgclctunlem2 31572 pmeasadd 31578

W3C validator