esumrnmpt2 - Mathbox for Thierry Arnoux

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Theorem esumrnmpt2 34581

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 3431	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}
2		esumrnmpt2.1	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
3		esumrnmpt2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssrab2 4028	. . . . . . 7 ⊢ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
6	3, 5	ssexd 5289	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V)
7	5	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
8		esumrnmpt2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
9	7, 8	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10		esumrnmpt2.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	7, 10	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ 𝑊)
12		rabid 3432	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝐵 = ∅))
13	12	simprbi 503	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
14	13	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15		elsng 4598	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
16	11, 15	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → (𝐵 ∈ {∅} ↔ 𝐵 = ∅))
17	14, 16	mtbird 328	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 ∈ {∅})
18	11, 17	eldifd 3910	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19		esumrnmpt2.6	. . . . . 6 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)
20		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
21	1, 20	disjss1f 33048	. . . . . 6 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴 → (Disj 𝑘 ∈ 𝐴 𝐵 → Disj 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐵))
22	5, 19, 21	sylc 66	. . . . 5 ⊢ (𝜑 → Disj 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐵)
23	1, 2, 6, 9, 18, 22	esumrnmpt 34565	. . . 4 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
25		snex 5404	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → {∅} ∈ V)
27		velsn 4600	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
28	27	bilani 510	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
29		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝜑
30		nfre1 3287	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝐵 = ∅
31	29, 30	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
32		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑦 = ∅
33	31, 32	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
34		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝐶 = 0
35		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
36		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
37	35, 36	eqtr4d 2798	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = 𝐵)
38	37, 2	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 𝐷)
39		simp-4l 795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝜑)
40		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑘 ∈ 𝐴)
41		esumrnmpt2.5	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
42	39, 40, 36, 41	syl21anc 851	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
43	38, 42	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
44		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
45	33, 34, 43, 44	r19.29af2 3270	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
46	28, 45	syldan 603	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
47		0e0iccpnf 13515	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
48	46, 47	eqeltrdi 2868	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
49		nfcv 2922	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑦
50		nfmpt1 5204	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
51	50	nfrn 5936	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
52	49, 51	nfel 2936	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
53	29, 52	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵))
54		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
55		rabid 3432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = ∅))
56	55	simprbi 503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} → 𝐵 = ∅)
57	56	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
58	54, 57	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
59	58, 27	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
60		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
61		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
62	61	elrnmpt 5942	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵))
63	60, 62	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
64	63	bilani 510	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝑦 = 𝐵)
65	53, 59, 64	r19.29af 3271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
66	65	ex 418	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) → 𝑦 ∈ {∅}))
67	66	ssrdv 3937	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
68	67	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
69	24, 26, 48, 68	esummono 34567	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ^𝑦 ∈ {∅}𝐶)
70		0ex 5264	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ∅ ∈ V)
72	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → 0 ∈ (0[,]+∞))
73	45, 71, 72	esumsn 34578	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ {∅}𝐶 = 0)
74	69, 73	breqtrd 5131	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
75		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
76		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
77	30	nfn 1890	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
78		rabn0 4339	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
79	78	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
80	79	necon1bi 2983	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} = ∅)
81		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
83	77, 80, 82	mpteq12df 5189	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
84		mpt0 6675	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ∅ ↦ 𝐵) = ∅
85	83, 84	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
86	85	rneqd 5922	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ran ∅)
87		rn0 5910	. . . . . . . . . . . . . 14 ⊢ ran ∅ = ∅
88	86, 87	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
89	76, 88	esumeq1d 34548	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ^𝑦 ∈ ∅𝐶)
90		esumnul 34561	. . . . . . . . . . . 12 ⊢ Σ^*𝑦 ∈ ∅𝐶 = 0
91	89, 90	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
92		0le0 12369	. . . . . . . . . . 11 ⊢ 0 ≤ 0
93	91, 92	eqbrtrdi 5144	. . . . . . . . . 10 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
94	75, 93	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
95	74, 94	pm2.61dan 825	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
96		ssrab2 4028	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴)
98	3, 97	ssexd 5289	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V)
99		nfrab1 3431	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}
100	99	mptexgf 7222	. . . . . . . . . . 11 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
101		rnexg 7900	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
102	98, 100, 101	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
103	2	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
104		simplll 787	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
105	97	sselda 3931	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
106	105	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
107	106	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
108	104, 107, 8	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
109	103, 108	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
110	53, 109, 64	r19.29af 3271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
111	110	ralrimiva 3154	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
112		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
113	112	esumcl 34543	. . . . . . . . . 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[,]+∞))
114	102, 111, 113	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
115		elxrge0 13513	. . . . . . . . . 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))
116	115	simprbi 503	. . . . . . . . 9 ⊢ (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞) → 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)
117	114, 116	syl 18	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶)
118	95, 117	jca 521	. . . . . . 7 ⊢ (𝜑 → (Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶))
119		iccssxr 13486	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
120	119, 114	sselid 3929	. . . . . . . 8 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
121	119, 47	sselii 3928	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
122	121	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
123		xrletri3 13208	. . . . . . . 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)))
124	120, 122, 123	syl2anc 596	. . . . . . 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)))
125	118, 124	mpbird 260	. . . . . 6 ⊢ (𝜑 → Σ^*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
126	125	oveq1d 7429	. . . . 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))
127	9	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
128	1	esumcl 34543	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
129	6, 127, 128	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
130	119, 129	sselid 3929	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^)
131	23, 130	eqeltrd 2860	. . . . . 6 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ^)
132		xaddlid 13297	. . . . . 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)
133	131, 132	syl 18	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶) = Σ^𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶)
134	126, 133	eqtrd 2795	. . . 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)
135		simpl 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝜑)
136	56	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
137	135, 105, 136, 41	syl21anc 851	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 = 0)
138	137	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
139	29, 138	esumeq2d 34550	. . . . . . 7 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0)
140	99	esum0 34562	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0 = 0)
141	98, 140	syl 18	. . . . . . 7 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0 = 0)
142	139, 141	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
143	142	oveq1d 7429	. . . . 5 ⊢ (𝜑 → (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +_𝑒 Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
144		xaddlid 13297	. . . . . 6 ⊢ (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ ℝ^ → (0 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
145	130, 144	syl 18	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
146	143, 145	eqtrd 2795	. . . 4 ⊢ (𝜑 → (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
147	23, 134, 146	3eqtr4d 2805	. . 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))
148		nfv 1947	. . . 4 ⊢ Ⅎ𝑦𝜑
149		nfcv 2922	. . . 4 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
150	1	mptexgf 7222	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
151		rnexg 7900	. . . . 5 ⊢ ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
152	6, 150, 151	3syl 19	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
153	67	ssrind 4189	. . . . . 6 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
154		incom 4155	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
155	13	neqned 2962	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
156	155	necomd 3010	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
157	156	neneqd 2960	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
158	157	nrex 3090	. . . . . . . . 9 ⊢ ¬ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
159		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
160	159	elrnmpt 5942	. . . . . . . . . 10 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵))
161	70, 160	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵)
162	158, 161	mtbir 326	. . . . . . . 8 ⊢ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
163		disjsn 4672	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
164	162, 163	mpbir 234	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
165	154, 164	eqtr3i 2785	. . . . . 6 ⊢ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
166	153, 165	sseqtrdi 3971	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
167		ss0 4352	. . . . 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)) = ∅)
168	166, 167	syl 18	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅)
169		nfmpt1 5204	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
170	169	nfrn 5936	. . . . . . 7 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
171	49, 170	nfel 2936	. . . . . 6 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
172	29, 171	nfan 1932	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
173	2	adantl 487	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
174		simplll 787	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
175	7	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
176	175	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
177	174, 176, 8	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
178	173, 177	eqeltrd 2860	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
179	159	elrnmpt 5942	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵))
180	60, 179	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ↔ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
181	180	bilani 510	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
182	172, 178, 181	r19.29af 3271	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
183	148, 112, 149, 102, 152, 168, 110, 182	esumsplit 34566	. . 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))
184		rabnc 4341	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
185	184	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
186	105, 8	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
187	29, 99, 1, 98, 6, 185, 186, 9	esumsplit 34566	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷 = (Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +_𝑒 Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
188	147, 183, 187	3eqtr4d 2805	. 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)
189		rabxm 4340	. . . . . . . 8 ⊢ 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})
190	189, 81	mpteq12i 5202	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
191		mptun 6679	. . . . . . 7 ⊢ (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
192	190, 191	eqtri 2783	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
193	192	rneqi 5921	. . . . 5 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
194		rnun 6136	. . . . 5 ⊢ ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
195	193, 194	eqtri 2783	. . . 4 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
196	195	a1i 11	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ 𝐵) = (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
197	148, 196	esumeq1d 34548	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ^𝑦 ∈ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))𝐶)
198	189	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}))
199	29, 198	esumeq1d 34548	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐷 = Σ^𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
200	188, 197, 199	3eqtr4d 2805	1 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐷)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 Disj wdisj 5070 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 (class class class)co 7414 0cc0 11127 +∞cpnf 11267 ℝ^*cxr 11269 ≤ cle 11271 +_𝑒 cxad 13164 [,]cicc 13404 Σ^*cesum 34540

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-xneg 13166 df-xadd 13167 df-xmul 13168 df-ioo 13405 df-ioc 13406 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-fl 13856 df-mod 13934 df-seq 14069 df-exp 14129 df-fac 14341 df-bc 14370 df-hash 14398 df-shft 15143 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-limsup 15561 df-clim 15578 df-rlim 15579 df-sum 15777 df-ef 16156 df-sin 16158 df-cos 16159 df-pi 16161 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-rest 17510 df-topn 17511 df-0g 17529 df-gsum 17530 df-topgen 17531 df-pt 17532 df-prds 17535 df-ordt 17590 df-xrs 17591 df-qtop 17596 df-imas 17597 df-xps 17599 df-mre 17673 df-mrc 17674 df-acs 17676 df-ps 18657 df-tsr 18658 df-plusf 18732 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-mhm 18894 df-submnd 18895 df-grp 19063 df-minusg 19064 df-sbg 19065 df-mulg 19194 df-subg 19249 df-cntz 19447 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-cring 20378 df-subrng 20711 df-subrg 20735 df-abv 20978 df-lmod 21049 df-scaf 21050 df-sra 21360 df-rgmod 21361 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-mopn 21584 df-fbas 21585 df-fg 21586 df-cnfld 21589 df-top 23122 df-topon 23139 df-topsp 23161 df-bases 23174 df-cld 23247 df-ntr 23248 df-cls 23249 df-nei 23326 df-lp 23364 df-perf 23365 df-cn 23455 df-cnp 23456 df-haus 23543 df-tx 23791 df-hmeo 23984 df-fil 24075 df-fm 24167 df-flim 24168 df-flf 24169 df-tmd 24301 df-tgp 24302 df-tsms 24356 df-trg 24389 df-xms 24549 df-ms 24550 df-tms 24551 df-nm 24811 df-ngp 24812 df-nrg 24814 df-nlm 24815 df-ii 25108 df-cncf 25109 df-limc 26096 df-dv 26097 df-log 26796 df-esum 34541

This theorem is used by: carsggect 34832 carsgclctunlem2 34833 pmeasadd 34839

W3C validator