Theorem esumrnmpt2 34231

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 3410	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}
2		esumrnmpt2.1	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
3		esumrnmpt2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssrab2 4021	. . . . . . 7 ⊢ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ⊆ 𝐴)
6	3, 5	ssexd 5262	. . . . 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 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
10		esumrnmpt2.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	7, 10	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ 𝑊)
12		rabid 3411	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝐵 = ∅))
13	12	simprbi 497	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ 𝐵 = ∅)
14	13	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → ¬ 𝐵 = ∅)
15		elsng 4582	. . . . . . . 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 3901	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝐵 ∈ (𝑊 ∖ {∅}))
19		esumrnmpt2.6	. . . . . 6 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)
20		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
21	1, 20	disjss1f 32660	. . . . . 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 34215	. . . 4 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷)
24		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
25		snex 5377	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → {∅} ∈ V)
27		velsn 4584	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
28	27	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
29	28	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝑦 = ∅)
30		nfv 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝜑
31		nfre1 3263	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝐵 = ∅
32	30, 31	nfan 1901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
33		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑦 = ∅
34	32, 33	nfan 1901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅)
35		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝐶 = 0
36		simpllr 776	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝑦 = ∅)
37		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐵 = ∅)
38	36, 37	eqtr4d 2775	. . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝜑)
41		simplr 769	. . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)
44	39, 43	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → 𝐶 = 0)
45		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
46	34, 35, 44, 45	r19.29af2 3246	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 = ∅) → 𝐶 = 0)
47	29, 46	syldan 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 = 0)
48		0e0iccpnf 13406	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
49	47, 48	eqeltrdi 2845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) ∧ 𝑦 ∈ {∅}) → 𝐶 ∈ (0[,]+∞))
50		nfcv 2899	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑦
51		nfmpt1 5185	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
52	51	nfrn 5902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
53	50, 52	nfel 2914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
54	30, 53	nfan 1901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵))
55		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
56		rabid 3411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = ∅))
57	56	simprbi 497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} → 𝐵 = ∅)
58	57	ad2antlr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐵 = ∅)
59	55, 58	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 = ∅)
60	59, 27	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑦 ∈ {∅})
61		vex 3434	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
62		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
63	62	elrnmpt 5908	. . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . 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 3247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝑦 ∈ {∅})
68	67	ex 412	. . . . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ⊆ {∅})
71	24, 26, 49, 70	esummono 34217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ Σ*𝑦 ∈ {∅}𝐶)
72		0ex 5243	. . . . . . . . . . . 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 34228	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ*𝑦 ∈ {∅}𝐶 = 0)
76	71, 75	breqtrd 5112	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
77		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅) → ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
78		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
79	31	nfn 1859	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅
80		rabn0 4330	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = ∅)
81	80	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ≠ ∅ → ∃𝑘 ∈ 𝐴 𝐵 = ∅)
82	81	necon1bi 2961	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} = ∅)
83		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → 𝐵 = 𝐵)
85	79, 82, 84	mpteq12df 5170	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ ∅ ↦ 𝐵))
86		mpt0 6635	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ∅ ↦ 𝐵) = ∅
87	85, 86	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
88	87	rneqd 5888	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ran ∅)
89		rn0 5876	. . . . . . . . . . . . . 14 ⊢ ran ∅ = ∅
90	88, 89	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) = ∅)
91	78, 90	esumeq1d 34198	. . . . . . . . . . . 12 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = Σ*𝑦 ∈ ∅𝐶)
92		esumnul 34211	. . . . . . . . . . . 12 ⊢ Σ*𝑦 ∈ ∅𝐶 = 0
93	91, 92	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (¬ ∃𝑘 ∈ 𝐴 𝐵 = ∅ → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
94		0le0 12276	. . . . . . . . . . 11 ⊢ 0 ≤ 0
95	93, 94	eqbrtrdi 5125	. . . . . . . . . 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 813	. . . . . . . 8 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0)
98		ssrab2 4021	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ⊆ 𝐴)
100	3, 99	ssexd 5262	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V)
101		nfrab1 3410	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}
102	101	mptexgf 7171	. . . . . . . . . . 11 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∈ V)
103		rnexg 7847	. . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
106		simplll 775	. . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
109	108	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
110	106, 109, 8	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
111	105, 110	eqeltrd 2837	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
112	54, 111, 66	r19.29af 3247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
113	112	ralrimiva 3130	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
114		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)
115	114	esumcl 34193	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ (0[,]+∞))
117		elxrge0 13404	. . . . . . . . . 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 511	. . . . . . 7 ⊢ (𝜑 → (Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 ≤ 0 ∧ 0 ≤ Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶))
121		iccssxr 13377	. . . . . . . . 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 13099	. . . . . . . 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 585	. . . . . . 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 257	. . . . . 6 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵)𝐶 = 0)
128	127	oveq1d 7376	. . . . 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 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
130	1	esumcl 34193	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V ∧ ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷 ∈ (0[,]+∞))
131	6, 129, 130	syl2anc 585	. . . . . . . 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 2837	. . . . . 6 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)𝐶 ∈ ℝ*)
134		xaddlid 13188	. . . . . 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 2772	. . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝜑)
138	57	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
139	137, 107, 138, 42	syl21anc 838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 = 0)
140	139	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
141	30, 140	esumeq2d 34200	. . . . . . 7 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}0)
142	101	esum0 34212	. . . . . . . 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 2772	. . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 = 0)
145	144	oveq1d 7376	. . . . 5 ⊢ (𝜑 → (Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}𝐷 +𝑒 Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷) = (0 +𝑒 Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝐷))
146		xaddlid 13188	. . . . . 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 2772	. . . 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 2782	. . 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 1916	. . . 4 ⊢ Ⅎ𝑦𝜑
151		nfcv 2899	. . . 4 ⊢ Ⅎ𝑦ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
152	1	mptexgf 7171	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ∈ V → (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∈ V)
153		rnexg 7847	. . . . 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 4185	. . . . . 6 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)))
156		incom 4150	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
157	13	neqned 2940	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → 𝐵 ≠ ∅)
158	157	necomd 2988	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ∅ ≠ 𝐵)
159	158	neneqd 2938	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} → ¬ ∅ = 𝐵)
160	159	nrex 3066	. . . . . . . . 9 ⊢ ¬ ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}∅ = 𝐵
161		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) = (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
162	161	elrnmpt 5908	. . . . . . . . . 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 4656	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
166	164, 165	mpbir 231	. . . . . . 7 ⊢ (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) ∩ {∅}) = ∅
167	156, 166	eqtr3i 2762	. . . . . 6 ⊢ ({∅} ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) = ∅
168	155, 167	sseqtrdi 3963	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∩ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ⊆ ∅)
169		ss0 4343	. . . . 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 5185	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
172	171	nfrn 5902	. . . . . . 7 ⊢ Ⅎ𝑘ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
173	50, 172	nfel 2914	. . . . . 6 ⊢ Ⅎ𝑘 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)
174	30, 173	nfan 1901	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
175	2	adantl 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
176		simplll 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝜑)
177	7	adantlr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) → 𝑘 ∈ 𝐴)
178	177	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝑘 ∈ 𝐴)
179	176, 178, 8	syl2anc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐷 ∈ (0[,]+∞))
180	175, 179	eqeltrd 2837	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ∧ 𝑦 = 𝐵) → 𝐶 ∈ (0[,]+∞))
181	161	elrnmpt 5908	. . . . . . . 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 216	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
184	183	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → ∃𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}𝑦 = 𝐵)
185	174, 180, 184	r19.29af 3247	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵)) → 𝐶 ∈ (0[,]+∞))
186	150, 114, 151, 104, 154, 170, 112, 185	esumsplit 34216	. . 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 4332	. . . . 5 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅
188	187	a1i 11	. . . 4 ⊢ (𝜑 → ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∩ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) = ∅)
189	107, 8	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐷 ∈ (0[,]+∞))
190	30, 101, 1, 100, 6, 188, 189, 9	esumsplit 34216	. . 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 2782	. 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 4331	. . . . . . . 8 ⊢ 𝐴 = ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})
193	192, 83	mpteq12i 5183	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵)
194		mptun 6639	. . . . . . 7 ⊢ (𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅}) ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
195	193, 194	eqtri 2760	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
196	195	rneqi 5887	. . . . 5 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = ran ((𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ↦ 𝐵) ∪ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅} ↦ 𝐵))
197		rnun 6104	. . . . 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 2760	. . . 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 34198	. 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 34198	. 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐷 = Σ*𝑘 ∈ ({𝑘 ∈ 𝐴 ∣ 𝐵 = ∅} ∪ {𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅})𝐷)
203	191, 200, 202	3eqtr4d 2782	1 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑘 ∈ 𝐴𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  Disj wdisj 5053   class class class wbr 5086   ↦ cmpt 5167  ran crn 5626  (class class class)co 7361  0cc0 11032  +∞cpnf 11170  ℝ^*cxr 11172   ≤ cle 11174   +_𝑒 cxad 13055  [,]cicc 13295  Σ^*cesum 34190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110  ax-addf 11111  ax-mulf 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-q 12893  df-rp 12937  df-xneg 13057  df-xadd 13058  df-xmul 13059  df-ioo 13296  df-ioc 13297  df-ico 13298  df-icc 13299  df-fz 13456  df-fzo 13603  df-fl 13745  df-mod 13823  df-seq 13958  df-exp 14018  df-fac 14230  df-bc 14259  df-hash 14287  df-shft 15023  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-limsup 15427  df-clim 15444  df-rlim 15445  df-sum 15643  df-ef 16026  df-sin 16028  df-cos 16029  df-pi 16031  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-hom 17238  df-cco 17239  df-rest 17379  df-topn 17380  df-0g 17398  df-gsum 17399  df-topgen 17400  df-pt 17401  df-prds 17404  df-ordt 17459  df-xrs 17460  df-qtop 17465  df-imas 17466  df-xps 17468  df-mre 17542  df-mrc 17543  df-acs 17545  df-ps 18526  df-tsr 18527  df-plusf 18601  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-subrng 20517  df-subrg 20541  df-abv 20780  df-lmod 20851  df-scaf 20852  df-sra 21163  df-rgmod 21164  df-psmet 21339  df-xmet 21340  df-met 21341  df-bl 21342  df-mopn 21343  df-fbas 21344  df-fg 21345  df-cnfld 21348  df-top 22872  df-topon 22889  df-topsp 22911  df-bases 22924  df-cld 22997  df-ntr 22998  df-cls 22999  df-nei 23076  df-lp 23114  df-perf 23115  df-cn 23205  df-cnp 23206  df-haus 23293  df-tx 23540  df-hmeo 23733  df-fil 23824  df-fm 23916  df-flim 23917  df-flf 23918  df-tmd 24050  df-tgp 24051  df-tsms 24105  df-trg 24138  df-xms 24298  df-ms 24299  df-tms 24300  df-nm 24560  df-ngp 24561  df-nrg 24563  df-nlm 24564  df-ii 24857  df-cncf 24858  df-limc 25846  df-dv 25847  df-log 26536  df-esum 34191

This theorem is referenced by:  carsggect  34481  carsgclctunlem2  34482  pmeasadd  34488