esumrnmpt2 - Mathbox for Thierry Arnoux

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Theorem esumrnmpt2 29457

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 3099	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}
2		esumrnmpt2.1	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
3		esumrnmpt2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssrab2 3650	. . . . . . 7 ⊢ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
6	3, 5	ssexd 4733	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
7	5	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
8		esumrnmpt2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
9	7, 8	syldan 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10		esumrnmpt2.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	7, 10	syldan 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ 𝑊)
12		rabid 3095	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝐵 = ∅))
13	12	simprbi 479	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
14	13	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15		elsng 4139	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
16	11, 15	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
17	14, 16	mtbird 314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
18	11, 17	eldifd 3551	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19		esumrnmpt2.6	. . . . . 6 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)
20		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
21	1, 20	disjss1f 28768	. . . . . 6 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴 → (Disj 𝑘 ∈ 𝐴 𝐵 → Disj 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐵))
22	5, 19, 21	sylc 63	. . . . 5 ⊢ (𝜑 → Disj 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐵)
23	1, 2, 6, 9, 18, 22	esumrnmpt 29441	. . . 4 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
25		snex 4835	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → {∅} ∈ V)
27		velsn 4141	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
28	27	biimpi 205	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
29	28	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30		nfv 1830	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝜑
31		nfre1 2988	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝐵 = ∅
32	30, 31	nfan 1816	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
33		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑦 = ∅
34	32, 33	nfan 1816	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝐶 = 0
36		simpllr 795	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
38	36, 37	eqtr4d 2647	. . . . . . . . . . . . . . . 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 802	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝜑)
41		simplr 788	. . . . . . . . . . . . . . . 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 1317	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
44	39, 43	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
46	34, 35, 44, 45	r19.29af2 3057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
47	29, 46	syldan 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48		0e0iccpnf 12154	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
49	47, 48	syl6eqel 2696	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50		nfcv 2751	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑦
51		nfmpt1 4675	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
52	51	nfrn 5289	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
53	50, 52	nfel 2763	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
54	30, 53	nfan 1816	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵))
55		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56		rabid 3095	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = ∅))
57	56	simprbi 479	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} → 𝐵 = ∅)
58	57	ad2antlr 759	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
59	55, 58	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
60	59, 27	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
62		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
63	62	elrnmpt 5293	. . . . . . . . . . . . . . . . . 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 205	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
66	65	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
67	54, 60, 66	r19.29af 3058	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
68	67	ex 449	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
69	68	ssrdv 3574	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
70	69	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
71	24, 26, 49, 70	esummono 29443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ^𝑦 ∈ {∅}𝐶)
72		0ex 4718	. . . . . . . . . . . 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 29454	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ {∅}𝐶 = 0)
76	71, 75	breqtrd 4609	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
78		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
79	31	nfn 1768	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
80		nfrab1 3099	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}
81		nfcv 2751	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘∅
82		rabn0 3912	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
83	82	biimpi 205	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
84	83	necon1bi 2810	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} = ∅)
85		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
86	85	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
87	79, 80, 81, 84, 86	mpteq12df 28837	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
88		mpt0 5934	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ∅ ↦ 𝐵) = ∅
89	87, 88	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
90	89	rneqd 5274	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ran ∅)
91		rn0 5298	. . . . . . . . . . . . . 14 ⊢ ran ∅ = ∅
92	90, 91	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
93	78, 92	esumeq1d 29424	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑦 ∈ ∅𝐶)
94		esumnul 29437	. . . . . . . . . . . 12 ⊢ Σ^*𝑦 ∈ ∅𝐶 = 0
95	93, 94	syl6eq 2660	. . . . . . . . . . 11 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
96		0le0 10987	. . . . . . . . . . 11 ⊢ 0 ≤ 0
97	95, 96	syl6eqbr 4622	. . . . . . . . . 10 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
98	77, 97	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
99	76, 98	pm2.61dan 828	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
100		ssrab2 3650	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴)
102	3, 101	ssexd 4733	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V)
103	80	mptexgf 28809	. . . . . . . . . . 11 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
104		rnexg 6990	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
105	102, 103, 104	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
106	2	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
107		simplll 794	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
108	101	sselda 3568	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
109	108	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
110	109	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
111	107, 110, 8	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
112	106, 111	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
113	54, 112, 66	r19.29af 3058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
114	113	ralrimiva 2949	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
115		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
116	115	esumcl 29419	. . . . . . . . . 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[,]+∞))
117	105, 114, 116	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
118		elxrge0 12152	. . . . . . . . . 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))
119	118	simprbi 479	. . . . . . . . 9 ⊢ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) → 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)
120	117, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)
121	99, 120	jca 553	. . . . . . 7 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶))
122		iccssxr 12127	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
123	122, 117	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
124	122, 48	sselii 3565	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
125	124	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
126		xrletri3 11861	. . . . . . . 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)))
127	123, 125, 126	syl2anc 691	. . . . . . 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)))
128	121, 127	mpbird 246	. . . . . 6 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
129	128	oveq1d 6564	. . . . 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))
130	9	ralrimiva 2949	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
131	1	esumcl 29419	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
132	6, 130, 131	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
133	122, 132	sseldi 3566	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^)
134	23, 133	eqeltrd 2688	. . . . . 6 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
135		xaddid2 11947	. . . . . 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)
136	134, 135	syl 17	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
137	129, 136	eqtrd 2644	. . . 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)
138		simpl 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝜑)
139	57	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
140	138, 108, 139, 42	syl21anc 1317	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 = 0)
141	140	ralrimiva 2949	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
142	30, 141	esumeq2d 29426	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0)
143	80	esum0 29438	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0 = 0)
144	102, 143	syl 17	. . . . . . 7 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0 = 0)
145	142, 144	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
146	145	oveq1d 6564	. . . . 5 ⊢ (𝜑 → (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +_𝑒 Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
147		xaddid2 11947	. . . . . 6 ⊢ (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^ → (0 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
148	133, 147	syl 17	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
149	146, 148	eqtrd 2644	. . . 4 ⊢ (𝜑 → (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
150	23, 137, 149	3eqtr4d 2654	. . 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))
151		nfv 1830	. . . 4 ⊢ Ⅎ𝑦𝜑
152		nfcv 2751	. . . 4 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
153	1	mptexgf 28809	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
154		rnexg 6990	. . . . 5 ⊢ ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
155	6, 153, 154	3syl 18	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
156		ssrin 3800	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅} → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
157	69, 156	syl 17	. . . . . 6 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
158		incom 3767	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
159	13	neqned 2789	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
160	159	necomd 2837	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
161	160	neneqd 2787	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
162	161	nrex 2983	. . . . . . . . 9 ⊢ ¬ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
163		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
164	163	elrnmpt 5293	. . . . . . . . . 10 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵))
165	72, 164	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵)
166	162, 165	mtbir 312	. . . . . . . 8 ⊢ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
167		disjsn 4192	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
168	166, 167	mpbir 220	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
169	158, 168	eqtr3i 2634	. . . . . 6 ⊢ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
170	157, 169	syl6sseq 3614	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
171		ss0 3926	. . . . 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)) = ∅)
172	170, 171	syl 17	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
173		nfmpt1 4675	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
174	173	nfrn 5289	. . . . . . 7 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
175	50, 174	nfel 2763	. . . . . 6 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
176	30, 175	nfan 1816	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
177	2	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
178		simplll 794	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
179	7	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
180	179	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
181	178, 180, 8	syl2anc 691	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
182	177, 181	eqeltrd 2688	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
183	163	elrnmpt 5293	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵))
184	61, 183	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185	184	biimpi 205	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
186	185	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
187	176, 182, 186	r19.29af 3058	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
188	151, 115, 152, 105, 155, 172, 113, 187	esumsplit 29442	. . 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))
189		rabnc 3916	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
190	189	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
191	108, 8	syldan 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
192	30, 80, 1, 102, 6, 190, 191, 9	esumsplit 29442	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷 = (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
193	150, 188, 192	3eqtr4d 2654	. 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)
194		rabxm 3915	. . . . . . . 8 ⊢ 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})
195	194, 85	mpteq12i 4670	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
196		mptun 5938	. . . . . . 7 ⊢ (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197	195, 196	eqtri 2632	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
198	197	rneqi 5273	. . . . 5 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
199		rnun 5460	. . . . 5 ⊢ ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
200	198, 199	eqtri 2632	. . . 4 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
201	200	a1i 11	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ 𝐵) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
202	151, 201	esumeq1d 29424	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ^𝑦 ∈ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶)
203	194	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}))
204	30, 203	esumeq1d 29424	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐷 = Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
205	193, 202, 204	3eqtr4d 2654	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 Disj wdisj 4553 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 ℝ^*cxr 9952 ≤ cle 9954 +_𝑒 cxad 11820 [,]cicc 12049 Σ^*cesum 29416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417

This theorem is referenced by: carsggect 29707 carsgclctunlem2 29708 pmeasadd 29714

W3C validator