Theorem sge0fodjrnlem 42705

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		fornex 7659	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝐹:𝐶–onto→𝐴 → 𝐴 ∈ V))
5	2, 3, 4	sylc 65	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
6		difssd 4111	. . . 4 ⊢ (𝜑 → (𝐴 ∖ {∅}) ⊆ 𝐴)
7		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝜑)
8	6	sselda 3969	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝑘 ∈ 𝐴)
9		sge0fodjrnlem.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
10	7, 8, 9	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝐵 ∈ (0[,]+∞))
11		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝜑)
12		dfin4 4246	. . . . . . . . . 10 ⊢ (𝐴 ∩ {∅}) = (𝐴 ∖ (𝐴 ∖ {∅}))
13	12	eqcomi 2832	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐴 ∖ {∅})) = (𝐴 ∩ {∅})
14		inss2 4208	. . . . . . . . 9 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
15	13, 14	eqsstri 4003	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∖ {∅})) ⊆ {∅}
16		id 22	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})))
17	15, 16	sseldi 3967	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ {∅})
18		elsni 4586	. . . . . . 7 ⊢ (𝑘 ∈ {∅} → 𝑘 = ∅)
19	17, 18	syl 17	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 = ∅)
20	19	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝑘 = ∅)
21		sge0fodjrnlem.b0	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)
22	11, 20, 21	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝐵 = 0)
23	1, 5, 6, 10, 22	sge0ss 42701	. . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))
24	23	eqcomd 2829	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)))
25		sge0fodjrnlem.n	. . 3 ⊢ Ⅎ𝑛𝜑
26		sge0fodjrnlem.bd	. . 3 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
27		difexg 5233	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∖ 𝑍) ∈ V)
28	2, 27	syl 17	. . 3 ⊢ (𝜑 → (𝐶 ∖ 𝑍) ∈ V)
29		eqid 2823	. . . . 5 ⊢ (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
30		fof 6592	. . . . . . 7 ⊢ (𝐹:𝐶–onto→𝐴 → 𝐹:𝐶⟶𝐴)
31	3, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
32	31	ffvelrnda 6853	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
33		sge0fodjrnlem.dj	. . . . 5 ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))
34		fveq2 6672	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
35	34	neeq1d 3077	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ≠ ∅ ↔ (𝐹‘𝑛) ≠ ∅))
36	35	cbvrabv 3493	. . . . 5 ⊢ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} = {𝑛 ∈ 𝐶 ∣ (𝐹‘𝑛) ≠ ∅}
37	34	cbvmptv 5171	. . . . . . 7 ⊢ (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
38	37	rneqi 5809	. . . . . 6 ⊢ ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = ran (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
39	38	difeq1i 4097	. . . . 5 ⊢ (ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}) = (ran (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ∖ {∅})
40	25, 29, 32, 33, 36, 39	disjf1o 41459	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ↾ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}):{𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}–1-1-onto→(ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}))
41	31	feqmptd 6735	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)))
42		difssd 4111	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ∖ 𝑍) ⊆ 𝐶)
43	42	sselda 3969	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝐶)
44		eldifi 4105	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝑛 ∈ 𝐶)
45	44	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ 𝐶)
46		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑛) = ∅ → (𝐹‘𝑛) = ∅)
47		fvex 6685	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑛) ∈ V
48	47	elsn 4584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑛) ∈ {∅} ↔ (𝐹‘𝑛) = ∅)
49	46, 48	sylibr 236	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑛) = ∅ → (𝐹‘𝑛) ∈ {∅})
50	49	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → (𝐹‘𝑛) ∈ {∅})
51	45, 50	jca 514	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
52	51	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
53	31	ffnd 6517	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝐶)
54		elpreima 6830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 Fn 𝐶 → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
56	55	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
57	52, 56	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ (◡𝐹 “ {∅}))
58		sge0fodjrnlem.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (◡𝐹 “ {∅})
59	57, 58	eleqtrrdi 2926	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ 𝑍)
60		eldifn 4106	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → ¬ 𝑛 ∈ 𝑍)
61	60	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → ¬ 𝑛 ∈ 𝑍)
62	59, 61	pm2.65da 815	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ¬ (𝐹‘𝑛) = ∅)
63	62	neqned 3025	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝐹‘𝑛) ≠ ∅)
64	43, 63	jca 514	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ≠ ∅))
65	35	elrab 3682	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ≠ ∅))
66	64, 65	sylibr 236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅})
67	66	ex 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
68	65	simplbi 500	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ 𝐶)
69	68	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → 𝑛 ∈ 𝐶)
70	58	eleq2i 2906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (◡𝐹 “ {∅}))
71	70	biimpi 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (◡𝐹 “ {∅}))
72	71	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (◡𝐹 “ {∅}))
73	55	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
74	72, 73	mpbid 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
75	74	simprd 498	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ {∅})
76		elsni 4586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) ∈ {∅} → (𝐹‘𝑛) = ∅)
77	75, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ∅)
78	77	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ∅)
79	65	simprbi 499	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → (𝐹‘𝑛) ≠ ∅)
80	79	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≠ ∅)
81	80	neneqd 3023	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → ¬ (𝐹‘𝑛) = ∅)
82	78, 81	pm2.65da 815	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → ¬ 𝑛 ∈ 𝑍)
83	69, 82	eldifd 3949	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → 𝑛 ∈ (𝐶 ∖ 𝑍))
84	83	ex 415	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ (𝐶 ∖ 𝑍)))
85	25, 84	ralrimi 3218	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}𝑛 ∈ (𝐶 ∖ 𝑍))
86		dfss3 3958	. . . . . . . . . . 11 ⊢ ({𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ⊆ (𝐶 ∖ 𝑍) ↔ ∀𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}𝑛 ∈ (𝐶 ∖ 𝑍))
87	85, 86	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ⊆ (𝐶 ∖ 𝑍))
88	87	sseld 3968	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ (𝐶 ∖ 𝑍)))
89	67, 88	impbid 214	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
90	25, 89	alrimi 2213	. . . . . . 7 ⊢ (𝜑 → ∀𝑛(𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
91		dfcleq 2817	. . . . . . 7 ⊢ ((𝐶 ∖ 𝑍) = {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ↔ ∀𝑛(𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
92	90, 91	sylibr 236	. . . . . 6 ⊢ (𝜑 → (𝐶 ∖ 𝑍) = {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅})
93	41, 92	reseq12d 5856	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐶 ∖ 𝑍)) = ((𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ↾ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
94	41, 37	syl6eqr 2876	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)))
95	94	eqcomd 2829	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = 𝐹)
96	95	rneqd 5810	. . . . . . 7 ⊢ (𝜑 → ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = ran 𝐹)
97		forn 6595	. . . . . . . 8 ⊢ (𝐹:𝐶–onto→𝐴 → ran 𝐹 = 𝐴)
98	3, 97	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐴)
99	96, 98	eqtr2d 2859	. . . . . 6 ⊢ (𝜑 → 𝐴 = ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)))
100	99	difeq1d 4100	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {∅}) = (ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}))
101	93, 92, 100	f1oeq123d 6612	. . . 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)) ∖ {∅})))
102	40, 101	mpbird 259	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶 ∖ 𝑍)):(𝐶 ∖ 𝑍)–1-1-onto→(𝐴 ∖ {∅}))
103		fvres 6691	. . . . 5 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = (𝐹‘𝑛))
104	103	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = (𝐹‘𝑛))
105		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝜑)
106		sge0fodjrnlem.fng	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
107	105, 43, 106	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝐹‘𝑛) = 𝐺)
108	104, 107	eqtrd 2858	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = 𝐺)
109	1, 25, 26, 28, 102, 108, 10	sge0f1o 42671	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑛 ∈ (𝐶 ∖ 𝑍) ↦ 𝐷)))
110	106	eqcomd 2829	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 = (𝐹‘𝑛))
111	110, 32	eqeltrd 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
112	105, 43, 111	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝐺 ∈ 𝐴)
113	112	ex 415	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝐺 ∈ 𝐴))
114	113	imdistani 571	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝜑 ∧ 𝐺 ∈ 𝐴))
115		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑘𝐺
116		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑘 𝐺 ∈ 𝐴
117	1, 116	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐺 ∈ 𝐴)
118		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
119	117, 118	nfim 1897	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
120		eleq1 2902	. . . . . . 7 ⊢ (𝑘 = 𝐺 → (𝑘 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
121	120	anbi2d 630	. . . . . 6 ⊢ (𝑘 = 𝐺 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
122	26	eleq1d 2899	. . . . . 6 ⊢ (𝑘 = 𝐺 → (𝐵 ∈ (0[,]+∞) ↔ 𝐷 ∈ (0[,]+∞)))
123	121, 122	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝐺 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))))
124	115, 119, 123, 9	vtoclgf 3567	. . . 4 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞)))
125	112, 114, 124	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝐷 ∈ (0[,]+∞))
126		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝜑)
127		eldifi 4105	. . . . . 6 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝐶)
128	127	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝑛 ∈ 𝐶)
129	126, 128, 111	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐺 ∈ 𝐴)
130		dfin4 4246	. . . . . . . . 9 ⊢ (𝑍 ∩ 𝐶) = (𝑍 ∖ (𝑍 ∖ 𝐶))
131		difss 4110	. . . . . . . . 9 ⊢ (𝑍 ∖ (𝑍 ∖ 𝐶)) ⊆ 𝑍
132	130, 131	eqsstri 4003	. . . . . . . 8 ⊢ (𝑍 ∩ 𝐶) ⊆ 𝑍
133		inss2 4208	. . . . . . . . . 10 ⊢ (𝐶 ∩ 𝑍) ⊆ 𝑍
134		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)))
135		dfin4 4246	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝑍) = (𝐶 ∖ (𝐶 ∖ 𝑍))
136	135	eqcomi 2832	. . . . . . . . . . 11 ⊢ (𝐶 ∖ (𝐶 ∖ 𝑍)) = (𝐶 ∩ 𝑍)
137	134, 136	eleqtrdi 2925	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝐶 ∩ 𝑍))
138	133, 137	sseldi 3967	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝑍)
139	138, 127	elind 4173	. . . . . . . 8 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝑍 ∩ 𝐶))
140	132, 139	sseldi 3967	. . . . . . 7 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝑍)
141	140	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝑛 ∈ 𝑍)
142	77	eqcomd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∅ = (𝐹‘𝑛))
143		simpl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝜑)
144	74	simpld 497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝐶)
145	143, 144, 106	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = 𝐺)
146	142, 145	eqtr2d 2859	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐺 = ∅)
147	126, 141, 146	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐺 = ∅)
148	126, 147	jca 514	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → (𝜑 ∧ 𝐺 = ∅))
149		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑘 𝐺 = ∅
150	1, 149	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐺 = ∅)
151		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 = 0
152	150, 151	nfim 1897	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0)
153		eqeq1 2827	. . . . . . 7 ⊢ (𝑘 = 𝐺 → (𝑘 = ∅ ↔ 𝐺 = ∅))
154	153	anbi2d 630	. . . . . 6 ⊢ (𝑘 = 𝐺 → ((𝜑 ∧ 𝑘 = ∅) ↔ (𝜑 ∧ 𝐺 = ∅)))
155	26	eqeq1d 2825	. . . . . 6 ⊢ (𝑘 = 𝐺 → (𝐵 = 0 ↔ 𝐷 = 0))
156	154, 155	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝐺 → (((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0) ↔ ((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0)))
157	115, 152, 156, 21	vtoclgf 3567	. . . 4 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0))
158	129, 148, 157	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐷 = 0)
159	25, 2, 42, 125, 158	sge0ss 42701	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (𝐶 ∖ 𝑍) ↦ 𝐷)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
160	24, 109, 159	3eqtrd 2862	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  {crab 3144  Vcvv 3496   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  {csn 4569  Disj wdisj 5033   ↦ cmpt 5148  ^◡ccnv 5556  ran crn 5558   ↾ cres 5559   “ cima 5560   Fn wfn 6352  ⟶wf 6353  –onto→wfo 6355  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  0cc0 10539  +∞cpnf 10674  [,]cicc 12744  Σ^{^}csumge0 42651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-disj 5034  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-xadd 12511  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-sumge0 42652

This theorem is referenced by:  sge0fodjrn  42706