Theorem sge0fodjrnlem 46372

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 7979	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝐹:𝐶–onto→𝐴 → 𝐴 ∈ V))
5	2, 3, 4	sylc 65	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
6		difssd 4147	. . . 4 ⊢ (𝜑 → (𝐴 ∖ {∅}) ⊆ 𝐴)
7		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝜑)
8	6	sselda 3995	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝑘 ∈ 𝐴)
9		sge0fodjrnlem.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
10	7, 8, 9	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {∅})) → 𝐵 ∈ (0[,]+∞))
11		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝜑)
12		dfin4 4284	. . . . . . . . . 10 ⊢ (𝐴 ∩ {∅}) = (𝐴 ∖ (𝐴 ∖ {∅}))
13	12	eqcomi 2744	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐴 ∖ {∅})) = (𝐴 ∩ {∅})
14		inss2 4246	. . . . . . . . 9 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
15	13, 14	eqsstri 4030	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∖ {∅})) ⊆ {∅}
16		id 22	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})))
17	15, 16	sselid 3993	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ {∅})
18		elsni 4648	. . . . . . 7 ⊢ (𝑘 ∈ {∅} → 𝑘 = ∅)
19	17, 18	syl 17	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 = ∅)
20	19	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝑘 = ∅)
21		sge0fodjrnlem.b0	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)
22	11, 20, 21	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝐵 = 0)
23	1, 5, 6, 10, 22	sge0ss 46368	. . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))
24	23	eqcomd 2741	. 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 5337	. . 3 ⊢ (𝜑 → (𝐶 ∖ 𝑍) ∈ V)
28		eqid 2735	. . . . 5 ⊢ (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
29		fof 6821	. . . . . . 7 ⊢ (𝐹:𝐶–onto→𝐴 → 𝐹:𝐶⟶𝐴)
30	3, 29	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
31	30	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
32		sge0fodjrnlem.dj	. . . . 5 ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))
33		fveq2 6907	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
34	33	neeq1d 2998	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ≠ ∅ ↔ (𝐹‘𝑛) ≠ ∅))
35	34	cbvrabv 3444	. . . . 5 ⊢ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} = {𝑛 ∈ 𝐶 ∣ (𝐹‘𝑛) ≠ ∅}
36	33	cbvmptv 5261	. . . . . . 7 ⊢ (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
37	36	rneqi 5951	. . . . . 6 ⊢ ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = ran (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))
38	37	difeq1i 4132	. . . . 5 ⊢ (ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}) = (ran (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ∖ {∅})
39	25, 28, 31, 32, 35, 38	disjf1o 45134	. . . 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 6977	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)))
41		difssd 4147	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ∖ 𝑍) ⊆ 𝐶)
42	41	sselda 3995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝐶)
43		eldifi 4141	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝑛 ∈ 𝐶)
44	43	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ 𝐶)
45		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑛) = ∅ → (𝐹‘𝑛) = ∅)
46		fvex 6920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑛) ∈ V
47	46	elsn 4646	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑛) ∈ {∅} ↔ (𝐹‘𝑛) = ∅)
48	45, 47	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑛) = ∅ → (𝐹‘𝑛) ∈ {∅})
49	48	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → (𝐹‘𝑛) ∈ {∅})
50	44, 49	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝐶 ∖ 𝑍) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
51	50	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
52	30	ffnd 6738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝐶)
53		elpreima 7078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 Fn 𝐶 → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
55	54	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
56	51, 55	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ (◡𝐹 “ {∅}))
57		sge0fodjrnlem.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (◡𝐹 “ {∅})
58	56, 57	eleqtrrdi 2850	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → 𝑛 ∈ 𝑍)
59		eldifn 4142	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → ¬ 𝑛 ∈ 𝑍)
60	59	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) ∧ (𝐹‘𝑛) = ∅) → ¬ 𝑛 ∈ 𝑍)
61	58, 60	pm2.65da 817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ¬ (𝐹‘𝑛) = ∅)
62	61	neqned 2945	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝐹‘𝑛) ≠ ∅)
63	42, 62	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ≠ ∅))
64	34	elrab 3695	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ≠ ∅))
65	63, 64	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅})
66	65	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
67	64	simplbi 497	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ 𝐶)
68	67	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → 𝑛 ∈ 𝐶)
69	57	eleq2i 2831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (◡𝐹 “ {∅}))
70	69	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (◡𝐹 “ {∅}))
71	70	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (◡𝐹 “ {∅}))
72	54	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 ∈ (◡𝐹 “ {∅}) ↔ (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅})))
73	71, 72	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 ∈ 𝐶 ∧ (𝐹‘𝑛) ∈ {∅}))
74	73	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ {∅})
75		elsni 4648	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) ∈ {∅} → (𝐹‘𝑛) = ∅)
76	74, 75	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ∅)
77	76	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ∅)
78	64	simprbi 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → (𝐹‘𝑛) ≠ ∅)
79	78	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≠ ∅)
80	79	neneqd 2943	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) ∧ 𝑛 ∈ 𝑍) → ¬ (𝐹‘𝑛) = ∅)
81	77, 80	pm2.65da 817	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → ¬ 𝑛 ∈ 𝑍)
82	68, 81	eldifd 3974	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}) → 𝑛 ∈ (𝐶 ∖ 𝑍))
83	82	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ (𝐶 ∖ 𝑍)))
84	25, 83	ralrimi 3255	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}𝑛 ∈ (𝐶 ∖ 𝑍))
85		dfss3 3984	. . . . . . . . . . 11 ⊢ ({𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ⊆ (𝐶 ∖ 𝑍) ↔ ∀𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}𝑛 ∈ (𝐶 ∖ 𝑍))
86	84, 85	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ⊆ (𝐶 ∖ 𝑍))
87	86	sseld 3994	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} → 𝑛 ∈ (𝐶 ∖ 𝑍)))
88	66, 87	impbid 212	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
89	25, 88	alrimi 2211	. . . . . . 7 ⊢ (𝜑 → ∀𝑛(𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
90		dfcleq 2728	. . . . . . 7 ⊢ ((𝐶 ∖ 𝑍) = {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅} ↔ ∀𝑛(𝑛 ∈ (𝐶 ∖ 𝑍) ↔ 𝑛 ∈ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
91	89, 90	sylibr 234	. . . . . 6 ⊢ (𝜑 → (𝐶 ∖ 𝑍) = {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅})
92	40, 91	reseq12d 6001	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐶 ∖ 𝑍)) = ((𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) ↾ {𝑚 ∈ 𝐶 ∣ (𝐹‘𝑚) ≠ ∅}))
93	40, 36	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)))
94	93	eqcomd 2741	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = 𝐹)
95	94	rneqd 5952	. . . . . . 7 ⊢ (𝜑 → ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) = ran 𝐹)
96		forn 6824	. . . . . . . 8 ⊢ (𝐹:𝐶–onto→𝐴 → ran 𝐹 = 𝐴)
97	3, 96	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐴)
98	95, 97	eqtr2d 2776	. . . . . 6 ⊢ (𝜑 → 𝐴 = ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)))
99	98	difeq1d 4135	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {∅}) = (ran (𝑚 ∈ 𝐶 ↦ (𝐹‘𝑚)) ∖ {∅}))
100	92, 91, 99	f1oeq123d 6843	. . . 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)) ∖ {∅})))
101	39, 100	mpbird 257	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶 ∖ 𝑍)):(𝐶 ∖ 𝑍)–1-1-onto→(𝐴 ∖ {∅}))
102		fvres 6926	. . . . 5 ⊢ (𝑛 ∈ (𝐶 ∖ 𝑍) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = (𝐹‘𝑛))
103	102	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = (𝐹‘𝑛))
104		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝜑)
105		sge0fodjrnlem.fng	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
106	104, 42, 105	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝐹‘𝑛) = 𝐺)
107	103, 106	eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → ((𝐹 ↾ (𝐶 ∖ 𝑍))‘𝑛) = 𝐺)
108	1, 25, 26, 27, 101, 107, 10	sge0f1o 46338	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑛 ∈ (𝐶 ∖ 𝑍) ↦ 𝐷)))
109	105	eqcomd 2741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 = (𝐹‘𝑛))
110	109, 31	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
111	104, 42, 110	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝐺 ∈ 𝐴)
112	111	ex 412	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝐶 ∖ 𝑍) → 𝐺 ∈ 𝐴))
113	112	imdistani 568	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → (𝜑 ∧ 𝐺 ∈ 𝐴))
114		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑘𝐺
115		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑘 𝐺 ∈ 𝐴
116	1, 115	nfan 1897	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐺 ∈ 𝐴)
117		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
118	116, 117	nfim 1894	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))
119		eleq1 2827	. . . . . . 7 ⊢ (𝑘 = 𝐺 → (𝑘 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
120	119	anbi2d 630	. . . . . 6 ⊢ (𝑘 = 𝐺 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
121	26	eleq1d 2824	. . . . . 6 ⊢ (𝑘 = 𝐺 → (𝐵 ∈ (0[,]+∞) ↔ 𝐷 ∈ (0[,]+∞)))
122	120, 121	imbi12d 344	. . . . 5 ⊢ (𝑘 = 𝐺 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞))))
123	114, 118, 122, 9	vtoclgf 3569	. . . 4 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞)))
124	111, 113, 123	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ 𝑍)) → 𝐷 ∈ (0[,]+∞))
125		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝜑)
126		eldifi 4141	. . . . . 6 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝐶)
127	126	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝑛 ∈ 𝐶)
128	125, 127, 110	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐺 ∈ 𝐴)
129		dfin4 4284	. . . . . . . . 9 ⊢ (𝑍 ∩ 𝐶) = (𝑍 ∖ (𝑍 ∖ 𝐶))
130		difss 4146	. . . . . . . . 9 ⊢ (𝑍 ∖ (𝑍 ∖ 𝐶)) ⊆ 𝑍
131	129, 130	eqsstri 4030	. . . . . . . 8 ⊢ (𝑍 ∩ 𝐶) ⊆ 𝑍
132		inss2 4246	. . . . . . . . . 10 ⊢ (𝐶 ∩ 𝑍) ⊆ 𝑍
133		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)))
134		dfin4 4284	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝑍) = (𝐶 ∖ (𝐶 ∖ 𝑍))
135	134	eqcomi 2744	. . . . . . . . . . 11 ⊢ (𝐶 ∖ (𝐶 ∖ 𝑍)) = (𝐶 ∩ 𝑍)
136	133, 135	eleqtrdi 2849	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝐶 ∩ 𝑍))
137	132, 136	sselid 3993	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝑍)
138	137, 126	elind 4210	. . . . . . . 8 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ (𝑍 ∩ 𝐶))
139	131, 138	sselid 3993	. . . . . . 7 ⊢ (𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍)) → 𝑛 ∈ 𝑍)
140	139	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝑛 ∈ 𝑍)
141	76	eqcomd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∅ = (𝐹‘𝑛))
142		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝜑)
143	73	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝐶)
144	142, 143, 105	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = 𝐺)
145	141, 144	eqtr2d 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐺 = ∅)
146	125, 140, 145	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐺 = ∅)
147	125, 146	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → (𝜑 ∧ 𝐺 = ∅))
148		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑘 𝐺 = ∅
149	1, 148	nfan 1897	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐺 = ∅)
150		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 = 0
151	149, 150	nfim 1894	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0)
152		eqeq1 2739	. . . . . . 7 ⊢ (𝑘 = 𝐺 → (𝑘 = ∅ ↔ 𝐺 = ∅))
153	152	anbi2d 630	. . . . . 6 ⊢ (𝑘 = 𝐺 → ((𝜑 ∧ 𝑘 = ∅) ↔ (𝜑 ∧ 𝐺 = ∅)))
154	26	eqeq1d 2737	. . . . . 6 ⊢ (𝑘 = 𝐺 → (𝐵 = 0 ↔ 𝐷 = 0))
155	153, 154	imbi12d 344	. . . . 5 ⊢ (𝑘 = 𝐺 → (((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0) ↔ ((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0)))
156	114, 151, 155, 21	vtoclgf 3569	. . . 4 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 = ∅) → 𝐷 = 0))
157	128, 147, 156	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐶 ∖ (𝐶 ∖ 𝑍))) → 𝐷 = 0)
158	25, 2, 41, 124, 157	sge0ss 46368	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (𝐶 ∖ 𝑍) ↦ 𝐷)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
159	24, 108, 158	3eqtrd 2779	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {crab 3433  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631  Disj wdisj 5115   ↦ cmpt 5231  ^◡ccnv 5688  ran crn 5690   ↾ cres 5691   “ cima 5692   Fn wfn 6558  ⟶wf 6559  –onto→wfo 6561  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  0cc0 11153  +∞cpnf 11290  [,]cicc 13387  Σ^{^}csumge0 46318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-xadd 13153  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-sumge0 46319

This theorem is referenced by:  sge0fodjrn  46373