Theorem sge0fodjrnlem 47059

Description: Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned 0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0fodjrnlem.k	⊢ Ⅎ𝑘𝜑
sge0fodjrnlem.n	⊢ Ⅎ𝑛𝜑
sge0fodjrnlem.bd	⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
sge0fodjrnlem.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
sge0fodjrnlem.f	⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)
sge0fodjrnlem.dj	⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))
sge0fodjrnlem.fng	⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
sge0fodjrnlem.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
sge0fodjrnlem.b0	⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)
sge0fodjrnlem.z	⊢ 𝑍 = (◡𝐹 “ {∅})

Assertion

Ref	Expression
sge0fodjrnlem	⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑘,𝑛   𝐷,𝑘   𝑘,𝐹,𝑛   𝑘,𝐺   𝑘,𝑍,𝑛

Allowed substitution hints:   𝜑(𝑘,𝑛)   𝐵(𝑘)   𝐷(𝑛)   𝐺(𝑛)   𝑉(𝑘,𝑛)

Proof of Theorem sge0fodjrnlem

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0fodjrnlem.k	. . . 4 ⊢ Ⅎ𝑘𝜑
2		sge0fodjrnlem.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
3		sge0fodjrnlem.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)
4		focdmex 7955	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝐹:𝐶–onto→𝐴 → 𝐴 ∈ V))
5	2, 3, 4	sylc 66	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
6		difssd 4099	. . . 4 ⊢ (𝜑 → (𝐴 ∖ {∅}) ⊆ 𝐴)
7		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝜑)
8	6	sselda 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝑘 ∈ 𝐴)
9		sge0fodjrnlem.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
10	7, 8, 9	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝐵 ∈ (0[,]+∞))
11		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝜑)
12		dfin4 4239	. . . . . . . . . 10 ⊢ (𝐴 ∩ {∅}) = (𝐴 ∖ (𝐴 ∖ {∅}))
13	12	eqcomi 2778	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐴 ∖ {∅})) = (𝐴 ∩ {∅})
14		inss2 4198	. . . . . . . . 9 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
15	13, 14	eqsstri 3991	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∖ {∅})) ⊆ {∅}
16		id 23	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})))
17	15, 16	sselid 3943	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ {∅})
18		elsni 4611	. . . . . . 7 ⊢ (𝑘 ∈ {∅} → 𝑘 = ∅)
19	17, 18	syl 18	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 = ∅)
20	19	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝑘 = ∅)
21		sge0fodjrnlem.b0	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)
22	11, 20, 21	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝐵 = 0)
23	1, 5, 6, 10, 22	sge0ss 47055	. . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))
24	23	eqcomd 2775	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)))
25		sge0fodjrnlem.n	. . 3 ⊢ Ⅎ𝑛𝜑
26		sge0fodjrnlem.bd	. . 3 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
27	2	difexd 5304	. . 3 ⊢ (𝜑 → (𝐶 ∖ 𝑍) ∈ V)
28		eqid 2769	. . . . 5 ⊢ (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
29		fof 6795	. . . . . . 7 ⊢ (𝐹:𝐶–onto→𝐴 → 𝐹:𝐶⟶𝐴)
30	3, 29	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
31	30	ffvelcdmda 7082	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
32		sge0fodjrnlem.dj	. . . . 5 ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))
33		fveq2 6884	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
34	33	neeq1d 3023	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ≠ ∅ ↔ (𝐹‘𝑛) ≠ ∅))
35	34	cbvrabv 3433	. . . . 5 ⊢ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} = {𝑛 ∈ 𝐶 ∣ (𝐹‘𝑛) ≠ ∅}
36	33	cbvmptv 5219	. . . . . . 7 ⊢ (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
37	36	rneqi 5930	. . . . . 6 ⊢ ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = ran (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
38	37	difeq1i 4085	. . . . 5 ⊢ (ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}) = (ran (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ∖ {∅})
39	25, 28, 31, 32, 35, 38	disjf1o 45838	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ↾ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}):{𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}–1-1-onto→(ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}))
40	30	feqmptd 6952	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)))
41		difssd 4099	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ∖ 𝑍) ⊆ 𝐶)
42	41	sselda 3945	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝐶)
43		eldifi 4093	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝑛 ∈ 𝐶)
44	43	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ 𝐶)
45		fvex 6897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹‘𝑛) ∈ V
46	45	elsn 4609	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑛) ∈ {∅} ↔ (𝐹‘𝑛) = ∅)
47	46	bilanri 511	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → (𝐹‘𝑛) ∈ {∅})
48	44, 47	jca 520	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
49	48	adantll 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
50	30	ffnd 6709	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝐶)
51		elpreima 7056	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 Fn 𝐶 → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
52	50, 51	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
53	52	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
54	49, 53	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ (◡𝐹 “ {∅}))
55		sge0fodjrnlem.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (◡𝐹 “ {∅})
56	54, 55	eleqtrrdi 2880	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ 𝑍)
57		eldifn 4094	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → ¬ 𝑛 ∈ 𝑍)
58	57	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → ¬ 𝑛 ∈ 𝑍)
59	56, 58	pm2.65da 828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ¬ (𝐹‘𝑛) = ∅)
60	59	neqned 2971	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝐹‘𝑛) ≠ ∅)
61	42, 60	jca 520	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ≠ ∅))
62	34	elrab 3659	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ≠ ∅))
63	61, 62	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅})
64	63	ex 417	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
65	62	simplbi 501	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ 𝐶)
66	65	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → 𝑛 ∈ 𝐶)
67	55	eleq2i 2861	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (◡𝐹 “ {∅}))
68	67	bilani 509	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (◡𝐹 “ {∅}))
69	52	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
70	68, 69	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
71	70	simprd 500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ {∅})
72		elsni 4611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) ∈ {∅} → (𝐹‘𝑛) = ∅)
73	71, 72	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ∅)
74	73	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ∅)
75	62	simprbi 502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → (𝐹‘𝑛) ≠ ∅)
76	75	ad2antlr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≠ ∅)
77	76	neneqd 2969	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → ¬ (𝐹‘𝑛) = ∅)
78	74, 77	pm2.65da 828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → ¬ 𝑛 ∈ 𝑍)
79	66, 78	eldifd 3924	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → 𝑛 ∈ (𝐶 ∖ 𝑍))
80	79	ex 417	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ (𝐶 ∖ 𝑍)))
81	25, 80	ralrimi 3269	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}𝑛 ∈ (𝐶 ∖ 𝑍))
82		dfss3 3934	. . . . . . . . . . 11 ⊢ ({𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ⊆ (𝐶 ∖ 𝑍) ↔ ∀𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}𝑛 ∈ (𝐶 ∖ 𝑍))
83	81, 82	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ⊆ (𝐶 ∖ 𝑍))
84	83	sseld 3944	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ (𝐶 ∖ 𝑍)))
85	64, 84	impbid 215	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
86	25, 85	alrimi 2255	. . . . . . 7 ⊢ (𝜑 → ∀𝑛(𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
87		dfcleq 2762	. . . . . . 7 ⊢ ((𝐶 ∖ 𝑍) = {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ↔ ∀𝑛(𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
88	86, 87	sylibr 237	. . . . . 6 ⊢ (𝜑 → (𝐶 ∖ 𝑍) = {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅})
89	40, 88	reseq12d 5982	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐶 ∖ 𝑍)) = ((𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ↾ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
90	40, 36	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)))
91	90	eqcomd 2775	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = 𝐹)
92	91	rneqd 5931	. . . . . . 7 ⊢ (𝜑 → ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = ran 𝐹)
93		forn 6798	. . . . . . . 8 ⊢ (𝐹:𝐶–onto→𝐴 → ran 𝐹 = 𝐴)
94	3, 93	syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐴)
95	92, 94	eqtr2d 2805	. . . . . 6 ⊢ (𝜑 → 𝐴 = ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)))
96	95	difeq1d 4088	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {∅}) = (ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}))
97	89, 88, 96	f1oeq123d 6817	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ 𝑍)):(𝐶 ∖ 𝑍)–1-1-onto→(𝐴 ∖ {∅}) ↔ ((𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ↾ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}):{𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}–1-1-onto→(ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅})))
98	39, 97	mpbird 260	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶 ∖ 𝑍)):(𝐶 ∖ 𝑍)–1-1-onto→(𝐴 ∖ {∅}))
99		fvres 6903	. . . . 5 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = (𝐹‘𝑛))
100	99	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = (𝐹‘𝑛))
101		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝜑)
102		sge0fodjrnlem.fng	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
103	101, 42, 102	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝐹‘𝑛) = 𝐺)
104	100, 103	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = 𝐺)
105	1, 25, 26, 27, 98, 104, 10	sge0f1o 47025	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑛 ∈ (𝐶 ∖ 𝑍) ↦ 𝐷)))
106	102	eqcomd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 = (𝐹‘𝑛))
107	106, 31	eqeltrd 2869	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
108	101, 42, 107	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝐺 ∈ 𝐴)
109	108	ex 417	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝐺 ∈ 𝐴))
110	109	imdistani 578	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝜑 ∧ 𝐺 ∈ 𝐴))
111		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑘𝐺
112		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑘 𝐺 ∈ 𝐴
113	1, 112	nfan 1926	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐺 ∈ 𝐴)
114		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
115	113, 114	nfim 1923	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
116		eleq1 2857	. . . . . . 7 ⊢ (𝑘 = 𝐺 → (𝑘 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
117	116	anbi2d 641	. . . . . 6 ⊢ (𝑘 = 𝐺 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
118	26	eleq1d 2854	. . . . . 6 ⊢ (𝑘 = 𝐺 → (𝐵 ∈ (0[,]+∞) ↔ 𝐷 ∈ (0[,]+∞)))
119	117, 118	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝐺 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))))
120	111, 115, 119, 9	vtoclgf 3543	. . . 4 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞)))
121	108, 110, 120	sylc 66	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝐷 ∈ (0[,]+∞))
122		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝜑)
123		eldifi 4093	. . . . . 6 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝐶)
124	123	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝑛 ∈ 𝐶)
125	122, 124, 107	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐺 ∈ 𝐴)
126		dfin4 4239	. . . . . . . . 9 ⊢ (𝑍 ∩ 𝐶) = (𝑍 ∖ (𝑍 ∖ 𝐶))
127		difss 4098	. . . . . . . . 9 ⊢ (𝑍 ∖ (𝑍 ∖ 𝐶)) ⊆ 𝑍
128	126, 127	eqsstri 3991	. . . . . . . 8 ⊢ (𝑍 ∩ 𝐶) ⊆ 𝑍
129		inss2 4198	. . . . . . . . . 10 ⊢ (𝐶 ∩ 𝑍) ⊆ 𝑍
130		id 23	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)))
131		dfin4 4239	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝑍) = (𝐶 ∖ (𝐶 ∖ 𝑍))
132	131	eqcomi 2778	. . . . . . . . . . 11 ⊢ (𝐶 ∖ (𝐶 ∖ 𝑍)) = (𝐶 ∩ 𝑍)
133	130, 132	eleqtrdi 2879	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝐶 ∩ 𝑍))
134	129, 133	sselid 3943	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝑍)
135	134, 123	elind 4161	. . . . . . . 8 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝑍 ∩ 𝐶))
136	128, 135	sselid 3943	. . . . . . 7 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝑍)
137	136	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝑛 ∈ 𝑍)
138	73	eqcomd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∅ = (𝐹‘𝑛))
139		simpl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝜑)
140	70	simpld 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝐶)
141	139, 140, 102	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = 𝐺)
142	138, 141	eqtr2d 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐺 = ∅)
143	122, 137, 142	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐺 = ∅)
144	122, 143	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → (𝜑 ∧ 𝐺 = ∅))
145		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑘 𝐺 = ∅
146	1, 145	nfan 1926	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐺 = ∅)
147		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 = 0
148	146, 147	nfim 1923	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0)
149		eqeq1 2773	. . . . . . 7 ⊢ (𝑘 = 𝐺 → (𝑘 = ∅ ↔ 𝐺 = ∅))
150	149	anbi2d 641	. . . . . 6 ⊢ (𝑘 = 𝐺 → ((𝜑 ∧ 𝑘 = ∅) ↔ (𝜑 ∧ 𝐺 = ∅)))
151	26	eqeq1d 2771	. . . . . 6 ⊢ (𝑘 = 𝐺 → (𝐵 = 0 ↔ 𝐷 = 0))
152	150, 151	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝐺 → (((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0) ↔ ((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0)))
153	111, 148, 152, 21	vtoclgf 3543	. . . 4 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0))
154	125, 144, 153	sylc 66	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐷 = 0)
155	25, 2, 41, 121, 154	sge0ss 47055	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (𝐶 ∖ 𝑍) ↦ 𝐷)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
156	24, 105, 155	3eqtrd 2808	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594  Disj wdisj 5080   ↦ cmpt 5196  ^◡ccnv 5663  ran crn 5665   ↾ cres 5666   “ cima 5667   Fn wfn 6534  ⟶wf 6535  –onto→wfo 6537  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7413  0cc0 11102  +∞cpnf 11242  [,]cicc 13377  Σ^{^}csumge0 47005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-inf2 9612  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179  ax-pre-sup 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9404  df-oi 9474  df-card 9927  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-div 11874  df-nn 12236  df-2 12305  df-3 12306  df-n0 12507  df-z 12594  df-uz 12865  df-rp 13019  df-xadd 13140  df-ico 13380  df-icc 13381  df-fz 13538  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-clim 15541  df-sum 15740  df-sumge0 47006

This theorem is referenced by:  sge0fodjrn  47060