Theorem gsumzaddlem 19698

Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)

Hypotheses

Ref	Expression
gsumzadd.b	⊢ 𝐵 = (Base‘𝐺)
gsumzadd.0	⊢ 0 = (0g‘𝐺)
gsumzadd.p	⊢ + = (+g‘𝐺)
gsumzadd.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzadd.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzadd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzadd.fn	⊢ (𝜑 → 𝐹 finSupp 0 )
gsumzadd.hn	⊢ (𝜑 → 𝐻 finSupp 0 )
gsumzaddlem.w	⊢ 𝑊 = ((𝐹 ∪ 𝐻) supp 0 )
gsumzaddlem.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzaddlem.h	⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
gsumzaddlem.1	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzaddlem.2	⊢ (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
gsumzaddlem.3	⊢ (𝜑 → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻)))
gsumzaddlem.4	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑘 ∈ (𝐴 ∖ 𝑥))) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))

Assertion

Ref	Expression
gsumzaddlem	⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))

Distinct variable groups:   𝑥,𝑘, +   0 ,𝑘,𝑥   𝑘,𝐹,𝑥   𝑘,𝐺,𝑥   𝐴,𝑘,𝑥   𝐵,𝑘,𝑥   𝑘,𝐻,𝑥   𝜑,𝑘,𝑥   𝑥,𝑉   𝑘,𝑊,𝑥   𝑘,𝑍,𝑥

Allowed substitution hint:   𝑉(𝑘)

Proof of Theorem gsumzaddlem

Dummy variables 𝑓 𝑛 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzadd.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		gsumzadd.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
3		gsumzadd.0	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
4	2, 3	mndidcl 18571	. . . . . . 7 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
5	1, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐵)
6		gsumzadd.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	2, 6, 3	mndlid 18576	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
8	1, 5, 7	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ( 0 + 0 ) = 0 )
10		gsumzaddlem.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
11		gsumzadd.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12	3	fvexi 6856	. . . . . . . . 9 ⊢ 0 ∈ V
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
14		gsumzaddlem.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
15	14, 11	fexd 7177	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ V)
16	15	suppun 8115	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))
17		gsumzaddlem.w	. . . . . . . . 9 ⊢ 𝑊 = ((𝐹 ∪ 𝐻) supp 0 )
18	16, 17	sseqtrrdi 3995	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
19	10, 11, 13, 18	gsumcllem 19685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 0 ))
20	19	oveq2d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )))
21	3	gsumz 18646	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
22	1, 11, 21	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
23	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
24	20, 23	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐹) = 0 )
25	10, 11	fexd 7177	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
26	25	suppun 8115	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐻 ∪ 𝐹) supp 0 ))
27		uncom 4113	. . . . . . . . . . 11 ⊢ (𝐹 ∪ 𝐻) = (𝐻 ∪ 𝐹)
28	27	oveq1i 7367	. . . . . . . . . 10 ⊢ ((𝐹 ∪ 𝐻) supp 0 ) = ((𝐻 ∪ 𝐹) supp 0 )
29	26, 28	sseqtrrdi 3995	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))
30	29, 17	sseqtrrdi 3995	. . . . . . . 8 ⊢ (𝜑 → (𝐻 supp 0 ) ⊆ 𝑊)
31	14, 11, 13, 30	gsumcllem 19685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐻 = (𝑥 ∈ 𝐴 ↦ 0 ))
32	31	oveq2d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐻) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )))
33	32, 23	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐻) = 0 )
34	24, 33	oveq12d 7375	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ( 0 + 0 ))
35	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐴 ∈ 𝑉)
36	5	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) → 0 ∈ 𝐵)
37	35, 36, 36, 19, 31	offval2 7637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 ∘f + 𝐻) = (𝑥 ∈ 𝐴 ↦ ( 0 + 0 )))
38	9	mpteq2dv 5207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ 𝐴 ↦ ( 0 + 0 )) = (𝑥 ∈ 𝐴 ↦ 0 ))
39	37, 38	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 ∘f + 𝐻) = (𝑥 ∈ 𝐴 ↦ 0 ))
40	39	oveq2d 7373	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )))
41	40, 23	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = 0 )
42	9, 34, 41	3eqtr4rd 2787	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
43	42	ex 413	. 2 ⊢ (𝜑 → (𝑊 = ∅ → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
44	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐺 ∈ Mnd)
45	2, 6	mndcl 18564	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
46	45	3expb 1120	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 + 𝑤) ∈ 𝐵)
47	44, 46	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 + 𝑤) ∈ 𝐵)
48	47	caovclg 7546	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
49		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈ ℕ)
50		nnuz 12806	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
51	49, 50	eleqtrdi 2848	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘1))
52	10	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹:𝐴⟶𝐵)
53		f1of1 6783	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))–1-1→𝑊)
54	53	ad2antll 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1→𝑊)
55		suppssdm 8108	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∪ 𝐻) supp 0 ) ⊆ dom (𝐹 ∪ 𝐻)
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ∪ 𝐻) supp 0 ) ⊆ dom (𝐹 ∪ 𝐻))
57	17	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 = ((𝐹 ∪ 𝐻) supp 0 ))
58		dmun 5866	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ∪ 𝐻) = (dom 𝐹 ∪ dom 𝐻)
59	10	fdmd 6679	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 = 𝐴)
60	14	fdmd 6679	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐻 = 𝐴)
61	59, 60	uneq12d 4124	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (dom 𝐹 ∪ dom 𝐻) = (𝐴 ∪ 𝐴))
62		unidm 4112	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∪ 𝐴) = 𝐴
63	61, 62	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom 𝐹 ∪ dom 𝐻) = 𝐴)
64	58, 63	eqtr2id 2789	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = dom (𝐹 ∪ 𝐻))
65	56, 57, 64	3sstr4d 3991	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ⊆ 𝐴)
66	65	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴)
67		f1ss 6744	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))–1-1→𝑊 ∧ 𝑊 ⊆ 𝐴) → 𝑓:(1...(♯‘𝑊))–1-1→𝐴)
68	54, 66, 67	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1→𝐴)
69		f1f 6738	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝑊))–1-1→𝐴 → 𝑓:(1...(♯‘𝑊))⟶𝐴)
70	68, 69	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝐴)
71		fco 6692	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵)
72	52, 70, 71	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵)
73	72	ffvelcdmda 7035	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵)
74	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐻:𝐴⟶𝐵)
75		fco 6692	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵)
76	74, 70, 75	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵)
77	76	ffvelcdmda 7035	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘𝑘) ∈ 𝐵)
78	52	ffnd 6669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹 Fn 𝐴)
79	74	ffnd 6669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐻 Fn 𝐴)
80	11	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐴 ∈ 𝑉)
81		ovexd 7392	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (1...(♯‘𝑊)) ∈ V)
82		inidm 4178	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐴) = 𝐴
83	78, 79, 70, 80, 80, 81, 82	ofco 7640	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐹 ∘f + 𝐻) ∘ 𝑓) = ((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓)))
84	83	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (((𝐹 ∘f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))‘𝑘))
85	84	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))‘𝑘))
86		fnfco 6707	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹 ∘ 𝑓) Fn (1...(♯‘𝑊)))
87	78, 70, 86	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘ 𝑓) Fn (1...(♯‘𝑊)))
88		fnfco 6707	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻 ∘ 𝑓) Fn (1...(♯‘𝑊)))
89	79, 70, 88	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 ∘ 𝑓) Fn (1...(♯‘𝑊)))
90		inidm 4178	. . . . . . . . 9 ⊢ ((1...(♯‘𝑊)) ∩ (1...(♯‘𝑊))) = (1...(♯‘𝑊))
91		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = ((𝐹 ∘ 𝑓)‘𝑘))
92		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘))
93	87, 89, 81, 81, 90, 91, 92	ofval 7628	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))‘𝑘) = (((𝐹 ∘ 𝑓)‘𝑘) + ((𝐻 ∘ 𝑓)‘𝑘)))
94	85, 93	eqtrd 2776	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹 ∘ 𝑓)‘𝑘) + ((𝐻 ∘ 𝑓)‘𝑘)))
95	1	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐺 ∈ Mnd)
96		elfzouz 13576	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1..^(♯‘𝑊)) → 𝑛 ∈ (ℤ≥‘1))
97	96	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ (ℤ≥‘1))
98		elfzouz2 13587	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1..^(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝑛))
99	98	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (♯‘𝑊) ∈ (ℤ≥‘𝑛))
100		fzss2 13481	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝑛) → (1...𝑛) ⊆ (1...(♯‘𝑊)))
101	99, 100	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (1...𝑛) ⊆ (1...(♯‘𝑊)))
102	101	sselda 3944	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...(♯‘𝑊)))
103	73	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵)
104	102, 103	syldan 591	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵)
105	2, 6	mndcl 18564	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑘 + 𝑥) ∈ 𝐵)
106	105	3expb 1120	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
107	95, 106	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
108	97, 104, 107	seqcl 13928	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐹 ∘ 𝑓))‘𝑛) ∈ 𝐵)
109	77	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘𝑘) ∈ 𝐵)
110	102, 109	syldan 591	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐻 ∘ 𝑓)‘𝑘) ∈ 𝐵)
111	97, 110, 107	seqcl 13928	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ 𝐵)
112		fzofzp1 13669	. . . . . . . . 9 ⊢ (𝑛 ∈ (1..^(♯‘𝑊)) → (𝑛 + 1) ∈ (1...(♯‘𝑊)))
113		ffvelcdm 7032	. . . . . . . . 9 ⊢ (((𝐹 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵)
114	72, 112, 113	syl2an 596	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵)
115		ffvelcdm 7032	. . . . . . . . 9 ⊢ (((𝐻 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵)
116	76, 112, 115	syl2an 596	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵)
117		fvco3 6940	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1))))
118	70, 112, 117	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1))))
119		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘(𝑛 + 1)) → (𝐹‘𝑘) = (𝐹‘(𝑓‘(𝑛 + 1))))
120	119	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘(𝑛 + 1)) → ((𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ↔ (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
121		gsumzaddlem.4	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑘 ∈ (𝐴 ∖ 𝑥))) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))
122	121	expr 457	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (𝑘 ∈ (𝐴 ∖ 𝑥) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})))
123	122	ralrimiv 3142	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))
124	123	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})))
125	124	alrimiv 1930	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})))
126	125	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑥(𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})))
127		imassrn 6024	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ (1...𝑛)) ⊆ ran 𝑓
128	70	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))⟶𝐴)
129	128	frnd 6676	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝑓 ⊆ 𝐴)
130	127, 129	sstrid 3955	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ⊆ 𝐴)
131		vex 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
132	131	imaex 7853	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ (1...𝑛)) ∈ V
133		sseq1 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ (1...𝑛)) ⊆ 𝐴))
134		difeq2 4076	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑓 “ (1...𝑛))))
135		reseq2 5932	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝐻 ↾ 𝑥) = (𝐻 ↾ (𝑓 “ (1...𝑛))))
136	135	oveq2d 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝐺 Σg (𝐻 ↾ 𝑥)) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
137	136	sneqd 4598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → {(𝐺 Σg (𝐻 ↾ 𝑥))} = {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})
138	137	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}) = (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
139	138	eleq2d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → ((𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}) ↔ (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
140	134, 139	raleqbidv 3319	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}) ↔ ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
141	133, 140	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓 “ (1...𝑛)) → ((𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) ↔ ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))))
142	132, 141	spcv 3564	. . . . . . . . . . . . 13 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) → ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
143	126, 130, 142	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
144		ffvelcdm 7032	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴)
145	70, 112, 144	syl2an 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴)
146		fzp1nel 13525	. . . . . . . . . . . . . 14 ⊢ ¬ (𝑛 + 1) ∈ (1...𝑛)
147	68	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))–1-1→𝐴)
148	112	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑛 + 1) ∈ (1...(♯‘𝑊)))
149		f1elima 7210	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(1...(♯‘𝑊))–1-1→𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊)) ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛)))
150	147, 148, 101, 149	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛)))
151	146, 150	mtbiri 326	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ¬ (𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)))
152	145, 151	eldifd 3921	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ (𝐴 ∖ (𝑓 “ (1...𝑛))))
153	120, 143, 152	rspcdva 3582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
154	118, 153	eqeltrd 2838	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
155		gsumzadd.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (Cntz‘𝐺)
156	132	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ∈ V)
157	14	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐻:𝐴⟶𝐵)
158	157, 130	fssresd 6709	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐻 ↾ (𝑓 “ (1...𝑛))):(𝑓 “ (1...𝑛))⟶𝐵)
159		gsumzaddlem.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
160	159	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
161		resss 5962	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ 𝐻
162	161	rnssi 5895	. . . . . . . . . . . . . 14 ⊢ ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻
163	155	cntzidss 19118	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐻 ⊆ (𝑍‘ran 𝐻) ∧ ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛)))))
164	160, 162, 163	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛)))))
165	97, 50	eleqtrrdi 2849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ ℕ)
166		f1ores 6798	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(1...(♯‘𝑊))–1-1→𝐴 ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)))
167	147, 101, 166	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)))
168		f1of1 6783	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛)))
169	167, 168	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛)))
170		suppssdm 8108	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ dom (𝐻 ↾ (𝑓 “ (1...𝑛)))
171		dmres 5959	. . . . . . . . . . . . . . . 16 ⊢ dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻)
172	171	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻))
173	170, 172	sseqtrid 3996	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ((𝑓 “ (1...𝑛)) ∩ dom 𝐻))
174		inss1 4188	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ (𝑓 “ (1...𝑛))
175		df-ima 5646	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛))
176	175	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛)))
177	174, 176	sseqtrid 3996	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ ran (𝑓 ↾ (1...𝑛)))
178	173, 177	sstrd 3954	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ran (𝑓 ↾ (1...𝑛)))
179		eqid 2736	. . . . . . . . . . . . 13 ⊢ (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 ) = (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 )
180	2, 3, 6, 155, 95, 156, 158, 164, 165, 169, 178, 179	gsumval3 19684	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))) = (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛))
181	175	eqimss2i 4003	. . . . . . . . . . . . . . . . . 18 ⊢ ran (𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛))
182		cores 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛)) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛))))
183	181, 182	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))
184		resco 6202	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 ∘ 𝑓) ↾ (1...𝑛)) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))
185	183, 184	eqtr4i 2767	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = ((𝐻 ∘ 𝑓) ↾ (1...𝑛))
186	185	fveq1i 6843	. . . . . . . . . . . . . . 15 ⊢ (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = (((𝐻 ∘ 𝑓) ↾ (1...𝑛))‘𝑘)
187		fvres 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑛) → (((𝐻 ∘ 𝑓) ↾ (1...𝑛))‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘))
188	186, 187	eqtrid 2788	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑛) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘))
189	188	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘))
190	97, 189	seqfveq 13932	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛) = (seq1( + , (𝐻 ∘ 𝑓))‘𝑛))
191	180, 190	eqtr2d 2777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
192		fvex 6855	. . . . . . . . . . . 12 ⊢ (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ V
193	192	elsn 4601	. . . . . . . . . . 11 ⊢ ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))} ↔ (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
194	191, 193	sylibr 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})
195	6, 155	cntzi 19109	. . . . . . . . . 10 ⊢ ((((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ∧ (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) → (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) = ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1))))
196	154, 194, 195	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) = ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1))))
197	196	eqcomd 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1))) = (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)))
198	2, 6, 95, 108, 111, 114, 116, 197	mnd4g 18570	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((seq1( + , (𝐹 ∘ 𝑓))‘𝑛) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) + (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + ((𝐻 ∘ 𝑓)‘(𝑛 + 1)))) = (((seq1( + , (𝐹 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1))) + ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐻 ∘ 𝑓)‘(𝑛 + 1)))))
199	48, 48, 51, 73, 77, 94, 198	seqcaopr3 13943	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (seq1( + , ((𝐹 ∘f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)) = ((seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻 ∘ 𝑓))‘(♯‘𝑊))))
200	47, 52, 74, 80, 80, 82	off 7635	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘f + 𝐻):𝐴⟶𝐵)
201		gsumzaddlem.3	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻)))
202	201	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻)))
203	44, 106	sylan 580	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
204	203, 52, 74, 80, 80, 82	off 7635	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘f + 𝐻):𝐴⟶𝐵)
205		eldifi 4086	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ ran 𝑓) → 𝑥 ∈ 𝐴)
206		eqidd 2737	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
207		eqidd 2737	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = (𝐻‘𝑥))
208	78, 79, 80, 80, 82, 206, 207	ofval 7628	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f + 𝐻)‘𝑥) = ((𝐹‘𝑥) + (𝐻‘𝑥)))
209	205, 208	sylan2 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹 ∘f + 𝐻)‘𝑥) = ((𝐹‘𝑥) + (𝐻‘𝑥)))
210	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))
211		f1ofo 6791	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))–onto→𝑊)
212		forn 6759	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(1...(♯‘𝑊))–onto→𝑊 → ran 𝑓 = 𝑊)
213	211, 212	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ran 𝑓 = 𝑊)
214	213, 17	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ran 𝑓 = ((𝐹 ∪ 𝐻) supp 0 ))
215	214	sseq2d 3976	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )))
216	215	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )))
217	210, 216	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 supp 0 ) ⊆ ran 𝑓)
218	12	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 0 ∈ V)
219	52, 217, 80, 218	suppssr 8127	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐹‘𝑥) = 0 )
220	26	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐻 ∪ 𝐹) supp 0 ))
221	220, 28	sseqtrrdi 3995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))
222	214	sseq2d 3976	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )))
223	222	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )))
224	221, 223	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 supp 0 ) ⊆ ran 𝑓)
225	74, 224, 80, 218	suppssr 8127	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐻‘𝑥) = 0 )
226	219, 225	oveq12d 7375	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹‘𝑥) + (𝐻‘𝑥)) = ( 0 + 0 ))
227	8	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ( 0 + 0 ) = 0 )
228	209, 226, 227	3eqtrd 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹 ∘f + 𝐻)‘𝑥) = 0 )
229	204, 228	suppss 8125	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐹 ∘f + 𝐻) supp 0 ) ⊆ ran 𝑓)
230		ovex 7390	. . . . . . . . 9 ⊢ (𝐹 ∘f + 𝐻) ∈ V
231	230, 131	coex 7867	. . . . . . . 8 ⊢ ((𝐹 ∘f + 𝐻) ∘ 𝑓) ∈ V
232		suppimacnv 8105	. . . . . . . . 9 ⊢ ((((𝐹 ∘f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (((𝐹 ∘f + 𝐻) ∘ 𝑓) supp 0 ) = (◡((𝐹 ∘f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })))
233	232	eqcomd 2742	. . . . . . . 8 ⊢ ((((𝐹 ∘f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (◡((𝐹 ∘f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹 ∘f + 𝐻) ∘ 𝑓) supp 0 ))
234	231, 12, 233	mp2an 690	. . . . . . 7 ⊢ (◡((𝐹 ∘f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹 ∘f + 𝐻) ∘ 𝑓) supp 0 )
235	2, 3, 6, 155, 44, 80, 200, 202, 49, 68, 229, 234	gsumval3 19684	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = (seq1( + , ((𝐹 ∘f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)))
236		gsumzaddlem.1	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
237	236	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
238		eqid 2736	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
239	2, 3, 6, 155, 44, 80, 52, 237, 49, 68, 217, 238	gsumval3 19684	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))
240	159	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
241		eqid 2736	. . . . . . . 8 ⊢ ((𝐻 ∘ 𝑓) supp 0 ) = ((𝐻 ∘ 𝑓) supp 0 )
242	2, 3, 6, 155, 44, 80, 74, 240, 49, 68, 224, 241	gsumval3 19684	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg 𝐻) = (seq1( + , (𝐻 ∘ 𝑓))‘(♯‘𝑊)))
243	239, 242	oveq12d 7375	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ((seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻 ∘ 𝑓))‘(♯‘𝑊))))
244	199, 235, 243	3eqtr4d 2786	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
245	244	expr 457	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
246	245	exlimdv 1936	. . 3 ⊢ ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
247	246	expimpd 454	. 2 ⊢ (𝜑 → (((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
248		gsumzadd.fn	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 )
249		gsumzadd.hn	. . . . 5 ⊢ (𝜑 → 𝐻 finSupp 0 )
250	248, 249	fsuppun 9324	. . . 4 ⊢ (𝜑 → ((𝐹 ∪ 𝐻) supp 0 ) ∈ Fin)
251	17, 250	eqeltrid 2842	. . 3 ⊢ (𝜑 → 𝑊 ∈ Fin)
252		fz1f1o 15595	. . 3 ⊢ (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
253	251, 252	syl 17	. 2 ⊢ (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
254	43, 247, 253	mpjaod 858	1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {csn 4586   class class class wbr 5105   ↦ cmpt 5188  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   ↾ cres 5635   “ cima 5636   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  –1-1→wf1 6493  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615   supp csupp 8092  Fincfn 8883   finSupp cfsupp 9305  1c1 11052   + caddc 11054  ℕcn 12153  ℤ_≥cuz 12763  ...cfz 13424  ..^cfzo 13567  seqcseq 13906  ♯chash 14230  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321   Σ_g cgsu 17322  Mndcmnd 18556  Cntzccntz 19095

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-cntz 19097

This theorem is referenced by:  gsumzadd  19699  dprdfadd  19799