esumrnmpt2 - Mathbox for Thierry Arnoux

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Theorem esumrnmpt2 32045

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 3318	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}
2		esumrnmpt2.1	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
3		esumrnmpt2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssrab2 4014	. . . . . . 7 ⊢ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
6	3, 5	ssexd 5249	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
7	5	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
8		esumrnmpt2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
9	7, 8	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10		esumrnmpt2.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	7, 10	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ 𝑊)
12		rabid 3311	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝐵 = ∅))
13	12	simprbi 497	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
14	13	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15		elsng 4576	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
16	11, 15	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
17	14, 16	mtbird 325	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
18	11, 17	eldifd 3899	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19		esumrnmpt2.6	. . . . . 6 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)
20		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
21	1, 20	disjss1f 30920	. . . . . 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 32029	. . . 4 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
25		snex 5355	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → {∅} ∈ V)
27		velsn 4578	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
28	27	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
29	28	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30		nfv 1918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝜑
31		nfre1 3240	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝐵 = ∅
32	30, 31	nfan 1903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
33		nfv 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑦 = ∅
34	32, 33	nfan 1903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35		nfv 1918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝐶 = 0
36		simpllr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
38	36, 37	eqtr4d 2782	. . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝜑)
41		simplr 766	. . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
46	34, 35, 44, 45	r19.29af2 3262	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
47	29, 46	syldan 591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48		0e0iccpnf 13200	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
49	47, 48	eqeltrdi 2848	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50		nfcv 2908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑦
51		nfmpt1 5183	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
52	51	nfrn 5864	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
53	50, 52	nfel 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
54	30, 53	nfan 1903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵))
55		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56		rabid 3311	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = ∅))
57	56	simprbi 497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} → 𝐵 = ∅)
58	57	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
59	55, 58	eqtrd 2779	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
60	59, 27	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61		vex 3437	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
62		eqid 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
63	62	elrnmpt 5868	. . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
66	65	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
67	54, 60, 66	r19.29af 3263	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
68	67	ex 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
69	68	ssrdv 3928	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
70	69	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
71	24, 26, 49, 70	esummono 32031	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ^𝑦 ∈ {∅}𝐶)
72		0ex 5232	. . . . . . . . . . . 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 32042	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ {∅}𝐶 = 0)
76	71, 75	breqtrd 5101	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
78		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
79	31	nfn 1861	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
80		rabn0 4320	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
81	80	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
82	81	necon1bi 2973	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} = ∅)
83		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
85	79, 82, 84	mpteq12df 5161	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
86		mpt0 6584	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ∅ ↦ 𝐵) = ∅
87	85, 86	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
88	87	rneqd 5850	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ran ∅)
89		rn0 5838	. . . . . . . . . . . . . 14 ⊢ ran ∅ = ∅
90	88, 89	eqtrdi 2795	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
91	78, 90	esumeq1d 32012	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑦 ∈ ∅𝐶)
92		esumnul 32025	. . . . . . . . . . . 12 ⊢ Σ^*𝑦 ∈ ∅𝐶 = 0
93	91, 92	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
94		0le0 12083	. . . . . . . . . . 11 ⊢ 0 ≤ 0
95	93, 94	eqbrtrdi 5114	. . . . . . . . . 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 810	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
98		ssrab2 4014	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴)
100	3, 99	ssexd 5249	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V)
101		nfrab1 3318	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}
102	101	mptexgf 7107	. . . . . . . . . . 11 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
103		rnexg 7760	. . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
106		simplll 772	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
107	99	sselda 3922	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
108	107	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
109	108	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
110	106, 109, 8	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
111	105, 110	eqeltrd 2840	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
112	54, 111, 66	r19.29af 3263	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
113	112	ralrimiva 3104	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
114		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
115	114	esumcl 32007	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
117		elxrge0 13198	. . . . . . . . . 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 497	. . . . . . . . 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 512	. . . . . . 7 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶))
121		iccssxr 13171	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
122	121, 116	sselid 3920	. . . . . . . 8 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
123	121, 48	sselii 3919	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
124	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
125		xrletri3 12897	. . . . . . . 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 584	. . . . . . 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 256	. . . . . 6 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
128	127	oveq1d 7299	. . . . 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 3104	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
130	1	esumcl 32007	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
131	6, 129, 130	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
132	121, 131	sselid 3920	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^)
133	23, 132	eqeltrd 2840	. . . . . 6 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
134		xaddid2 12985	. . . . . 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 2779	. . . 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 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝜑)
138	57	adantl 482	. . . . . . . . . 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 3104	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
141	30, 140	esumeq2d 32014	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0)
142	101	esum0 32026	. . . . . . . 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 2779	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
145	144	oveq1d 7299	. . . . 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 12985	. . . . . 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 2779	. . . 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 2789	. . 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 1918	. . . 4 ⊢ Ⅎ𝑦𝜑
151		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
152	1	mptexgf 7107	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
153		rnexg 7760	. . . . 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 4170	. . . . . 6 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
156		incom 4136	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
157	13	neqned 2951	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
158	157	necomd 3000	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
159	158	neneqd 2949	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
160	159	nrex 3198	. . . . . . . . 9 ⊢ ¬ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
161		eqid 2739	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
162	161	elrnmpt 5868	. . . . . . . . . 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 323	. . . . . . . 8 ⊢ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
165		disjsn 4648	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
166	164, 165	mpbir 230	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
167	156, 166	eqtr3i 2769	. . . . . 6 ⊢ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
168	155, 167	sseqtrdi 3972	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
169		ss0 4333	. . . . 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 5183	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
172	171	nfrn 5864	. . . . . . 7 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
173	50, 172	nfel 2922	. . . . . 6 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
174	30, 173	nfan 1903	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
175	2	adantl 482	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
176		simplll 772	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
177	7	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
178	177	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
179	176, 178, 8	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
180	175, 179	eqeltrd 2840	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
181	161	elrnmpt 5868	. . . . . . . 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 215	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
184	183	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185	174, 180, 184	r19.29af 3263	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
186	150, 114, 151, 104, 154, 170, 112, 185	esumsplit 32030	. . 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 4322	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
188	187	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
189	107, 8	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
190	30, 101, 1, 100, 6, 188, 189, 9	esumsplit 32030	. . 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 2789	. 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 4321	. . . . . . . 8 ⊢ 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})
193	192, 83	mpteq12i 5181	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
194		mptun 6588	. . . . . . 7 ⊢ (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
195	193, 194	eqtri 2767	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
196	195	rneqi 5849	. . . . 5 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197		rnun 6054	. . . . 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 2767	. . . 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 32012	. 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 32012	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐷 = Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
203	191, 200, 202	3eqtr4d 2789	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 ∃wrex 3066 {crab 3069 Vcvv 3433 ∪ cun 3886 ∩ cin 3887 ⊆ wss 3888 ∅c0 4257 {csn 4562 Disj wdisj 5040 class class class wbr 5075 ↦ cmpt 5158 ran crn 5591 (class class class)co 7284 0cc0 10880 +∞cpnf 11015 ℝ^*cxr 11017 ≤ cle 11019 +_𝑒 cxad 12855 [,]cicc 13091 Σ^*cesum 32004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-inf2 9408 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958 ax-addf 10959 ax-mulf 10960

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-disj 5041 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-isom 6446 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-of 7542 df-om 7722 df-1st 7840 df-2nd 7841 df-supp 7987 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-2o 8307 df-er 8507 df-map 8626 df-pm 8627 df-ixp 8695 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-fsupp 9138 df-fi 9179 df-sup 9210 df-inf 9211 df-oi 9278 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-z 12329 df-dec 12447 df-uz 12592 df-q 12698 df-rp 12740 df-xneg 12857 df-xadd 12858 df-xmul 12859 df-ioo 13092 df-ioc 13093 df-ico 13094 df-icc 13095 df-fz 13249 df-fzo 13392 df-fl 13521 df-mod 13599 df-seq 13731 df-exp 13792 df-fac 13997 df-bc 14026 df-hash 14054 df-shft 14787 df-cj 14819 df-re 14820 df-im 14821 df-sqrt 14955 df-abs 14956 df-limsup 15189 df-clim 15206 df-rlim 15207 df-sum 15407 df-ef 15786 df-sin 15788 df-cos 15789 df-pi 15791 df-struct 16857 df-sets 16874 df-slot 16892 df-ndx 16904 df-base 16922 df-ress 16951 df-plusg 16984 df-mulr 16985 df-starv 16986 df-sca 16987 df-vsca 16988 df-ip 16989 df-tset 16990 df-ple 16991 df-ds 16993 df-unif 16994 df-hom 16995 df-cco 16996 df-rest 17142 df-topn 17143 df-0g 17161 df-gsum 17162 df-topgen 17163 df-pt 17164 df-prds 17167 df-ordt 17221 df-xrs 17222 df-qtop 17227 df-imas 17228 df-xps 17230 df-mre 17304 df-mrc 17305 df-acs 17307 df-ps 18293 df-tsr 18294 df-plusf 18334 df-mgm 18335 df-sgrp 18384 df-mnd 18395 df-mhm 18439 df-submnd 18440 df-grp 18589 df-minusg 18590 df-sbg 18591 df-mulg 18710 df-subg 18761 df-cntz 18932 df-cmn 19397 df-abl 19398 df-mgp 19730 df-ur 19747 df-ring 19794 df-cring 19795 df-subrg 20031 df-abv 20086 df-lmod 20134 df-scaf 20135 df-sra 20443 df-rgmod 20444 df-psmet 20598 df-xmet 20599 df-met 20600 df-bl 20601 df-mopn 20602 df-fbas 20603 df-fg 20604 df-cnfld 20607 df-top 22052 df-topon 22069 df-topsp 22091 df-bases 22105 df-cld 22179 df-ntr 22180 df-cls 22181 df-nei 22258 df-lp 22296 df-perf 22297 df-cn 22387 df-cnp 22388 df-haus 22475 df-tx 22722 df-hmeo 22915 df-fil 23006 df-fm 23098 df-flim 23099 df-flf 23100 df-tmd 23232 df-tgp 23233 df-tsms 23287 df-trg 23320 df-xms 23482 df-ms 23483 df-tms 23484 df-nm 23747 df-ngp 23748 df-nrg 23750 df-nlm 23751 df-ii 24049 df-cncf 24050 df-limc 25039 df-dv 25040 df-log 25721 df-esum 32005

This theorem is referenced by: carsggect 32294 carsgclctunlem2 32295 pmeasadd 32301

W3C validator