Theorem gsumval3 18793

Description: Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 31-May-2019.)

Hypotheses

Ref	Expression
gsumval3.b	⊢ 𝐵 = (Base‘𝐺)
gsumval3.0	⊢ 0 = (0g‘𝐺)
gsumval3.p	⊢ + = (+g‘𝐺)
gsumval3.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumval3.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumval3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumval3.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumval3.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
gsumval3.h	⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
gsumval3.n	⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
gsumval3.w	⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )

Assertion

Ref	Expression
gsumval3	⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))

Proof of Theorem gsumval3

Dummy variables 𝑓 𝑘 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		gsumval3.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		gsumval3.0	. . . . . 6 ⊢ 0 = (0g‘𝐺)
4	3	gsumz 17854	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
5	1, 2, 4	syl2anc 576	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
6	5	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
7		gsumval3.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8	7	feqmptd 6560	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
10		gsumval3.h	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
11		f1f 6401	. . . . . . . . . . . . . 14 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
12	10, 11	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
13	12	ad2antrr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → 𝐻:(1...𝑀)⟶𝐴)
14		f1f1orn 6452	. . . . . . . . . . . . . . . 16 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
15	10, 14	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
16	15	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
17		f1ocnv 6453	. . . . . . . . . . . . . 14 ⊢ (𝐻:(1...𝑀)–1-1-onto→ran 𝐻 → ◡𝐻:ran 𝐻–1-1-onto→(1...𝑀))
18		f1of 6441	. . . . . . . . . . . . . 14 ⊢ (◡𝐻:ran 𝐻–1-1-onto→(1...𝑀) → ◡𝐻:ran 𝐻⟶(1...𝑀))
19	16, 17, 18	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ◡𝐻:ran 𝐻⟶(1...𝑀))
20	19	ffvelrnda 6674	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (◡𝐻‘𝑥) ∈ (1...𝑀))
21		fvco3 6586	. . . . . . . . . . . 12 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (◡𝐻‘𝑥) ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘(◡𝐻‘𝑥)) = (𝐹‘(𝐻‘(◡𝐻‘𝑥))))
22	13, 20, 21	syl2anc 576	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → ((𝐹 ∘ 𝐻)‘(◡𝐻‘𝑥)) = (𝐹‘(𝐻‘(◡𝐻‘𝑥))))
23		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝑊 = ∅)
24	23	difeq2d 3982	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((1...𝑀) ∖ 𝑊) = ((1...𝑀) ∖ ∅))
25		dif0 4212	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∖ ∅) = (1...𝑀)
26	24, 25	syl6eq 2823	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((1...𝑀) ∖ 𝑊) = (1...𝑀))
27	26	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → ((1...𝑀) ∖ 𝑊) = (1...𝑀))
28	20, 27	eleqtrrd 2862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (◡𝐻‘𝑥) ∈ ((1...𝑀) ∖ 𝑊))
29		fco 6358	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
30	7, 12, 29	syl2anc 576	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
31	30	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
32		gsumval3.w	. . . . . . . . . . . . . . 15 ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )
33	32	eqimss2i 3909	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ 𝑊
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ 𝑊)
35		ovexd 7008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (1...𝑀) ∈ V)
36	3	fvexi 6510	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 0 ∈ V)
38	31, 34, 35, 37	suppssr 7662	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (◡𝐻‘𝑥) ∈ ((1...𝑀) ∖ 𝑊)) → ((𝐹 ∘ 𝐻)‘(◡𝐻‘𝑥)) = 0 )
39	28, 38	syldan 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → ((𝐹 ∘ 𝐻)‘(◡𝐻‘𝑥)) = 0 )
40		f1ocnvfv2 6857	. . . . . . . . . . . . 13 ⊢ ((𝐻:(1...𝑀)–1-1-onto→ran 𝐻 ∧ 𝑥 ∈ ran 𝐻) → (𝐻‘(◡𝐻‘𝑥)) = 𝑥)
41	16, 40	sylan 572	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐻‘(◡𝐻‘𝑥)) = 𝑥)
42	41	fveq2d 6500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘(𝐻‘(◡𝐻‘𝑥))) = (𝐹‘𝑥))
43	22, 39, 42	3eqtr3rd 2816	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) = 0 )
44		fvex 6509	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
45	44	elsn 4450	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 )
46	43, 45	sylibr 226	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) ∈ { 0 })
47	46	adantlr 703	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) ∈ { 0 })
48		eldif 3832	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ ran 𝐻) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻))
49		gsumval3.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
50	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ V)
51	7, 49, 2, 50	suppssr 7662	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran 𝐻)) → (𝐹‘𝑥) = 0 )
52	51, 45	sylibr 226	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran 𝐻)) → (𝐹‘𝑥) ∈ { 0 })
53	48, 52	sylan2br 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻)) → (𝐹‘𝑥) ∈ { 0 })
54	53	adantlr 703	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻)) → (𝐹‘𝑥) ∈ { 0 })
55	54	anassrs 460	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) ∈ { 0 })
56	47, 55	pm2.61dan 801	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ { 0 })
57	56, 45	sylib 210	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 0 )
58	57	mpteq2dva 5018	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 0 ))
59	9, 58	eqtrd 2807	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 0 ))
60	59	oveq2d 6990	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )))
61		gsumval3.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
62	61, 3	mndidcl 17788	. . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
63		gsumval3.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
64	61, 63, 3	mndlid 17791	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
65	1, 62, 64	syl2anc2 577	. . . . 5 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
66	65	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ( 0 + 0 ) = 0 )
67		gsumval3.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
68		nnuz 12093	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
69	67, 68	syl6eleq 2869	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
70	69	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝑀 ∈ (ℤ≥‘1))
71	26	eleq2d 2844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ ((1...𝑀) ∖ 𝑊) ↔ 𝑥 ∈ (1...𝑀)))
72	71	biimpar 470	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (1...𝑀)) → 𝑥 ∈ ((1...𝑀) ∖ 𝑊))
73	31, 34, 35, 37	suppssr 7662	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ((1...𝑀) ∖ 𝑊)) → ((𝐹 ∘ 𝐻)‘𝑥) = 0 )
74	72, 73	syldan 583	. . . 4 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑥) = 0 )
75	66, 70, 74	seqid3 13227	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = 0 )
76	6, 60, 75	3eqtr4d 2817	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
77		fzf 12710	. . . . 5 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
78		ffn 6341	. . . . 5 ⊢ (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
79		ovelrn 7138	. . . . 5 ⊢ (... Fn (ℤ × ℤ) → (𝐴 ∈ ran ... ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝐴 = (𝑚...𝑛)))
80	77, 78, 79	mp2b 10	. . . 4 ⊢ (𝐴 ∈ ran ... ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝐴 = (𝑚...𝑛))
81	1	ad2antrr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐺 ∈ Mnd)
82		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐴 = (𝑚...𝑛))
83		frel 6346	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
84		reldm0 5638	. . . . . . . . . . . . . . . . 17 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
85	7, 83, 84	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
86	7	fdmd 6350	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐹 = 𝐴)
87	86	eqeq1d 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (dom 𝐹 = ∅ ↔ 𝐴 = ∅))
88	85, 87	bitrd 271	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 = ∅ ↔ 𝐴 = ∅))
89		coeq1 5574	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 = ∅ → (𝐹 ∘ 𝐻) = (∅ ∘ 𝐻))
90		co01 5950	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∘ 𝐻) = ∅
91	89, 90	syl6eq 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = ∅ → (𝐹 ∘ 𝐻) = ∅)
92	91	oveq1d 6989	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = (∅ supp 0 ))
93		supp0 7636	. . . . . . . . . . . . . . . . . 18 ⊢ ( 0 ∈ V → (∅ supp 0 ) = ∅)
94	36, 93	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ supp 0 ) = ∅
95	92, 94	syl6eq 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅)
96	32, 95	syl5eq 2819	. . . . . . . . . . . . . . 15 ⊢ (𝐹 = ∅ → 𝑊 = ∅)
97	88, 96	syl6bir 246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 = ∅ → 𝑊 = ∅))
98	97	necon3d 2981	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ≠ ∅ → 𝐴 ≠ ∅))
99	98	imp 398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → 𝐴 ≠ ∅)
100	99	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐴 ≠ ∅)
101	82, 100	eqnetrrd 3028	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝑚...𝑛) ≠ ∅)
102		fzn0 12735	. . . . . . . . . 10 ⊢ ((𝑚...𝑛) ≠ ∅ ↔ 𝑛 ∈ (ℤ≥‘𝑚))
103	101, 102	sylib 210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝑛 ∈ (ℤ≥‘𝑚))
104	7	ad2antrr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐹:𝐴⟶𝐵)
105	82	feq2d 6327	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:(𝑚...𝑛)⟶𝐵))
106	104, 105	mpbid 224	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐹:(𝑚...𝑛)⟶𝐵)
107	61, 63, 81, 103, 106	gsumval2 17760	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝐺 Σg 𝐹) = (seq𝑚( + , 𝐹)‘𝑛))
108		frn 6347	. . . . . . . . . . . . . . 15 ⊢ (𝐻:(1...𝑀)⟶𝐴 → ran 𝐻 ⊆ 𝐴)
109	10, 11, 108	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
110	109	ad2antrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ⊆ 𝐴)
111	110, 82	sseqtrd 3890	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ⊆ (𝑚...𝑛))
112		fzssuz 12762	. . . . . . . . . . . . 13 ⊢ (𝑚...𝑛) ⊆ (ℤ≥‘𝑚)
113		uzssz 12076	. . . . . . . . . . . . . 14 ⊢ (ℤ≥‘𝑚) ⊆ ℤ
114		zssre 11798	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
115	113, 114	sstri 3860	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑚) ⊆ ℝ
116	112, 115	sstri 3860	. . . . . . . . . . . 12 ⊢ (𝑚...𝑛) ⊆ ℝ
117	111, 116	syl6ss 3863	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ⊆ ℝ)
118		ltso 10519	. . . . . . . . . . 11 ⊢ < Or ℝ
119		soss 5341	. . . . . . . . . . 11 ⊢ (ran 𝐻 ⊆ ℝ → ( < Or ℝ → < Or ran 𝐻))
120	117, 118, 119	mpisyl 21	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → < Or ran 𝐻)
121		fzfi 13153	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ Fin
122	121	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
123		fex2 7451	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
124	12, 122, 2, 123	syl3anc 1352	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 ∈ V)
125		f1oen3g 8320	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
126	124, 15, 125	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
127		enfi 8527	. . . . . . . . . . . . 13 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
128	126, 127	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
129	121, 128	mpbii 225	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
130	129	ad2antrr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ∈ Fin)
131		fz1iso 13631	. . . . . . . . . 10 ⊢ (( < Or ran 𝐻 ∧ ran 𝐻 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))
132	120, 130, 131	syl2anc 576	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))
133	67	nnnn0d 11765	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
134		hashfz1 13519	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
135	133, 134	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
136	122, 15	hasheqf1od 13527	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(1...𝑀)) = (♯‘ran 𝐻))
137	135, 136	eqtr3d 2809	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 = (♯‘ran 𝐻))
138	137	ad2antrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑀 = (♯‘ran 𝐻))
139	138	fveq2d 6500	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq1( + , (𝐹 ∘ 𝑓))‘𝑀) = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘ran 𝐻)))
140	1	ad2antrr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐺 ∈ Mnd)
141	61, 63	mndcl 17781	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
142	141	3expb 1101	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
143	140, 142	sylan 572	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
144		gsumval3.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
145	144	ad2antrr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
146	145	sselda 3851	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
147		gsumval3.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (Cntz‘𝐺)
148	63, 147	cntzi 18242	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
149	146, 148	sylan 572	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
150	149	anasss 459	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
151	61, 63	mndass 17782	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
152	140, 151	sylan 572	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
153	69	ad2antrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑀 ∈ (ℤ≥‘1))
154	7	ad2antrr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐹:𝐴⟶𝐵)
155	154	frnd 6348	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐹 ⊆ 𝐵)
156		simprr 761	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))
157		isof1o 6897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻) → 𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻)
158	156, 157	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻)
159	138	oveq2d 6990	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (1...𝑀) = (1...(♯‘ran 𝐻)))
160	159	f1oeq2d 6437	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 ↔ 𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻))
161	158, 160	mpbird 249	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...𝑀)–1-1-onto→ran 𝐻)
162		f1ocnv 6453	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 → ◡𝑓:ran 𝐻–1-1-onto→(1...𝑀))
163	161, 162	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ◡𝑓:ran 𝐻–1-1-onto→(1...𝑀))
164	15	ad2antrr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
165		f1oco 6463	. . . . . . . . . . . . . 14 ⊢ ((◡𝑓:ran 𝐻–1-1-onto→(1...𝑀) ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (◡𝑓 ∘ 𝐻):(1...𝑀)–1-1-onto→(1...𝑀))
166	163, 164, 165	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (◡𝑓 ∘ 𝐻):(1...𝑀)–1-1-onto→(1...𝑀))
167		ffn 6341	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
168		dffn4 6422	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
169	167, 168	sylib 210	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
170		fof 6416	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐴–onto→ran 𝐹 → 𝐹:𝐴⟶ran 𝐹)
171	154, 169, 170	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐹:𝐴⟶ran 𝐹)
172		f1of 6441	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 → 𝑓:(1...𝑀)⟶ran 𝐻)
173	161, 172	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...𝑀)⟶ran 𝐻)
174	109	ad2antrr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐻 ⊆ 𝐴)
175	173, 174	fssd 6355	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...𝑀)⟶𝐴)
176		fco 6358	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ 𝑓:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝑓):(1...𝑀)⟶ran 𝐹)
177	171, 175, 176	syl2anc 576	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐹 ∘ 𝑓):(1...𝑀)⟶ran 𝐹)
178	177	ffvelrnda 6674	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (1...𝑀)) → ((𝐹 ∘ 𝑓)‘𝑥) ∈ ran 𝐹)
179		f1ococnv2 6467	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 → (𝑓 ∘ ◡𝑓) = ( I ↾ ran 𝐻))
180	161, 179	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑓 ∘ ◡𝑓) = ( I ↾ ran 𝐻))
181	180	coeq1d 5578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ((𝑓 ∘ ◡𝑓) ∘ 𝐻) = (( I ↾ ran 𝐻) ∘ 𝐻))
182		f1of 6441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐻:(1...𝑀)–1-1-onto→ran 𝐻 → 𝐻:(1...𝑀)⟶ran 𝐻)
183		fcoi2 6379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐻:(1...𝑀)⟶ran 𝐻 → (( I ↾ ran 𝐻) ∘ 𝐻) = 𝐻)
184	164, 182, 183	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (( I ↾ ran 𝐻) ∘ 𝐻) = 𝐻)
185	181, 184	eqtr2d 2808	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻 = ((𝑓 ∘ ◡𝑓) ∘ 𝐻))
186		coass 5954	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∘ ◡𝑓) ∘ 𝐻) = (𝑓 ∘ (◡𝑓 ∘ 𝐻))
187	185, 186	syl6eq 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻 = (𝑓 ∘ (◡𝑓 ∘ 𝐻)))
188	187	coeq2d 5579	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐹 ∘ 𝐻) = (𝐹 ∘ (𝑓 ∘ (◡𝑓 ∘ 𝐻))))
189		coass 5954	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ 𝐻)) = (𝐹 ∘ (𝑓 ∘ (◡𝑓 ∘ 𝐻)))
190	188, 189	syl6eqr 2825	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐹 ∘ 𝐻) = ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ 𝐻)))
191	190	fveq1d 6498	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ((𝐹 ∘ 𝐻)‘𝑘) = (((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ 𝐻))‘𝑘))
192	191	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑘) = (((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ 𝐻))‘𝑘))
193		f1of 6441	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑓:ran 𝐻–1-1-onto→(1...𝑀) → ◡𝑓:ran 𝐻⟶(1...𝑀))
194	161, 162, 193	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ◡𝑓:ran 𝐻⟶(1...𝑀))
195	164, 182	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻:(1...𝑀)⟶ran 𝐻)
196		fco 6358	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑓:ran 𝐻⟶(1...𝑀) ∧ 𝐻:(1...𝑀)⟶ran 𝐻) → (◡𝑓 ∘ 𝐻):(1...𝑀)⟶(1...𝑀))
197	194, 195, 196	syl2anc 576	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (◡𝑓 ∘ 𝐻):(1...𝑀)⟶(1...𝑀))
198		fvco3 6586	. . . . . . . . . . . . . . 15 ⊢ (((◡𝑓 ∘ 𝐻):(1...𝑀)⟶(1...𝑀) ∧ 𝑘 ∈ (1...𝑀)) → (((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ 𝐻))‘𝑘) = ((𝐹 ∘ 𝑓)‘((◡𝑓 ∘ 𝐻)‘𝑘)))
199	197, 198	sylan 572	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑘 ∈ (1...𝑀)) → (((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ 𝐻))‘𝑘) = ((𝐹 ∘ 𝑓)‘((◡𝑓 ∘ 𝐻)‘𝑘)))
200	192, 199	eqtrd 2807	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑘) = ((𝐹 ∘ 𝑓)‘((◡𝑓 ∘ 𝐻)‘𝑘)))
201	143, 150, 152, 153, 155, 166, 178, 200	seqf1o 13224	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq1( + , (𝐹 ∘ 𝑓))‘𝑀))
202	61, 63, 3	mndlid 17791	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
203	140, 202	sylan 572	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
204	61, 63, 3	mndrid 17792	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
205	140, 204	sylan 572	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
206	140, 62	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 0 ∈ 𝐵)
207		fdm 6349	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:(1...𝑀)⟶𝐴 → dom 𝐻 = (1...𝑀))
208	10, 11, 207	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐻 = (1...𝑀))
209		eluzfz1 12728	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑀))
210		ne0i 4180	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ (1...𝑀) → (1...𝑀) ≠ ∅)
211	69, 209, 210	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑀) ≠ ∅)
212	208, 211	eqnetrd 3027	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐻 ≠ ∅)
213		dm0rn0 5637	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝐻 = ∅ ↔ ran 𝐻 = ∅)
214	213	necon3bii 3012	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐻 ≠ ∅ ↔ ran 𝐻 ≠ ∅)
215	212, 214	sylib 210	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐻 ≠ ∅)
216	215	ad2antrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐻 ≠ ∅)
217	111	adantrr 705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐻 ⊆ (𝑚...𝑛))
218		simprl 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐴 = (𝑚...𝑛))
219	218	eleq2d 2844	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝑚...𝑛)))
220	219	biimpar 470	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (𝑚...𝑛)) → 𝑥 ∈ 𝐴)
221	154	ffvelrnda 6674	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
222	220, 221	syldan 583	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (𝑚...𝑛)) → (𝐹‘𝑥) ∈ 𝐵)
223	218	difeq1d 3981	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐴 ∖ ran 𝐻) = ((𝑚...𝑛) ∖ ran 𝐻))
224	223	eleq2d 2844	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑥 ∈ (𝐴 ∖ ran 𝐻) ↔ 𝑥 ∈ ((𝑚...𝑛) ∖ ran 𝐻)))
225	224	biimpar 470	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ((𝑚...𝑛) ∖ ran 𝐻)) → 𝑥 ∈ (𝐴 ∖ ran 𝐻))
226	51	ad4ant14 740	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝐻)) → (𝐹‘𝑥) = 0 )
227	225, 226	syldan 583	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ((𝑚...𝑛) ∖ ran 𝐻)) → (𝐹‘𝑥) = 0 )
228		f1of 6441	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻 → 𝑓:(1...(♯‘ran 𝐻))⟶ran 𝐻)
229	156, 157, 228	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...(♯‘ran 𝐻))⟶ran 𝐻)
230		fvco3 6586	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘ran 𝐻))⟶ran 𝐻 ∧ 𝑦 ∈ (1...(♯‘ran 𝐻))) → ((𝐹 ∘ 𝑓)‘𝑦) = (𝐹‘(𝑓‘𝑦)))
231	229, 230	sylan 572	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑦 ∈ (1...(♯‘ran 𝐻))) → ((𝐹 ∘ 𝑓)‘𝑦) = (𝐹‘(𝑓‘𝑦)))
232	203, 205, 143, 206, 156, 216, 217, 222, 227, 231	seqcoll2 13634	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘ran 𝐻)))
233	139, 201, 232	3eqtr4d 2817	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛))
234	233	expr 449	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛)))
235	234	exlimdv 1893	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛)))
236	132, 235	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛))
237	107, 236	eqtr4d 2810	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
238	237	ex 405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐴 = (𝑚...𝑛) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
239	238	rexlimdvw 3228	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (∃𝑛 ∈ ℤ 𝐴 = (𝑚...𝑛) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
240	239	rexlimdvw 3228	. . . 4 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝐴 = (𝑚...𝑛) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
241	80, 240	syl5bi 234	. . 3 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐴 ∈ ran ... → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
242		suppssdm 7644	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
243	32, 242	eqsstri 3884	. . . . . . . . . 10 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
244	243, 30	fssdm 6357	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
245		fz1ssnn 12752	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
246		nnssre 11441	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ
247	245, 246	sstri 3860	. . . . . . . . 9 ⊢ (1...𝑀) ⊆ ℝ
248	244, 247	syl6ss 3863	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ ℝ)
249		soss 5341	. . . . . . . 8 ⊢ (𝑊 ⊆ ℝ → ( < Or ℝ → < Or 𝑊))
250	248, 118, 249	mpisyl 21	. . . . . . 7 ⊢ (𝜑 → < Or 𝑊)
251		ssfi 8531	. . . . . . . 8 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
252	121, 244, 251	sylancr 579	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
253		fz1iso 13631	. . . . . . 7 ⊢ (( < Or 𝑊 ∧ 𝑊 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
254	250, 252, 253	syl2anc 576	. . . . . 6 ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
255	254	ad2antrr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
256	61, 3, 63, 147, 1, 2, 7, 144, 67, 10, 49, 32	gsumval3lem2 18792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))
257	1	ad2antrr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
258	257, 202	sylan 572	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
259	257, 204	sylan 572	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
260	257, 142	sylan 572	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
261	257, 62	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 0 ∈ 𝐵)
262		simprr 761	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
263		simplr 757	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ≠ ∅)
264	244	ad2antrr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
265	30	ad2antrr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
266	265	ffvelrnda 6674	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑥) ∈ 𝐵)
267	33	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ 𝑊)
268		ovexd 7008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (1...𝑀) ∈ V)
269	36	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 0 ∈ V)
270	265, 267, 268, 269	suppssr 7662	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ ((1...𝑀) ∖ 𝑊)) → ((𝐹 ∘ 𝐻)‘𝑥) = 0 )
271		coass 5954	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐻) ∘ 𝑓) = (𝐹 ∘ (𝐻 ∘ 𝑓))
272	271	fveq1i 6497	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝐻) ∘ 𝑓)‘𝑦) = ((𝐹 ∘ (𝐻 ∘ 𝑓))‘𝑦)
273		isof1o 6897	. . . . . . . . . . . 12 ⊢ (𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
274		f1of 6441	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))⟶𝑊)
275	262, 273, 274	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
276		fvco3 6586	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑦 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘ 𝐻) ∘ 𝑓)‘𝑦) = ((𝐹 ∘ 𝐻)‘(𝑓‘𝑦)))
277	275, 276	sylan 572	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑦 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘ 𝐻) ∘ 𝑓)‘𝑦) = ((𝐹 ∘ 𝐻)‘(𝑓‘𝑦)))
278	272, 277	syl5eqr 2821	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑦 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ (𝐻 ∘ 𝑓))‘𝑦) = ((𝐹 ∘ 𝐻)‘(𝑓‘𝑦)))
279	258, 259, 260, 261, 262, 263, 264, 266, 270, 278	seqcoll2 13634	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))
280	256, 279	eqtr4d 2810	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
281	280	expr 449	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → (𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
282	281	exlimdv 1893	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
283	255, 282	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
284	283	ex 405	. . 3 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (¬ 𝐴 ∈ ran ... → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
285	241, 284	pm2.61d 172	. 2 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
286	76, 285	pm2.61dane 3048	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508  ∃wex 1743   ∈ wcel 2051   ≠ wne 2960  ∃wrex 3082  Vcvv 3408   ∖ cdif 3819   ⊆ wss 3822  ∅c0 4172  𝒫 cpw 4416  {csn 4435   class class class wbr 4925   ↦ cmpt 5004   I cid 5307   Or wor 5321   × cxp 5401  ^◡ccnv 5402  dom cdm 5403  ran crn 5404   ↾ cres 5405   ∘ ccom 5407  Rel wrel 5408   Fn wfn 6180  ⟶wf 6181  –1-1→wf1 6182  –onto→wfo 6183  –1-1-onto→wf1o 6184  ‘cfv 6185   Isom wiso 6186  (class class class)co 6974   supp csupp 7631   ≈ cen 8301  Fincfn 8304  ℝcr 10332  1c1 10334   < clt 10472  ℕcn 11437  ℕ₀cn0 11705  ℤcz 11791  ℤ_≥cuz 12056  ...cfz 12706  seqcseq 13182  ♯chash 13503  Basecbs 16337  +_gcplusg 16419  0_gc0g 16567   Σ_g cgsu 16568  Mndcmnd 17774  Cntzccntz 18228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-supp 7632  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-oi 8767  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-n0 11706  df-z 11792  df-uz 12057  df-fz 12707  df-fzo 12848  df-seq 13183  df-hash 13504  df-0g 16569  df-gsum 16570  df-mgm 17722  df-sgrp 17764  df-mnd 17775  df-cntz 18230

This theorem is referenced by:  gsumzres  18795  gsumzcl2  18796  gsumzf1o  18798  gsumzaddlem  18806  gsumconst  18819  gsumzmhm  18822  gsumzoppg  18829  gsumfsum  20329  wilthlem3  25364