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Theorem gsumval3 19769

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 = (0_g‘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 Σ_g 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 = (0_g‘𝐺)
4	3	gsumz 18713	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
5	1, 2, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
6	5	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ 0 )) = 0 )
7		gsumval3.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8	7	feqmptd 6957	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
10		gsumval3.h	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
11		f1f 6784	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → 𝐻:(1...𝑀)⟶𝐴)
14		f1f1orn 6841	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
17		f1ocnv 6842	. . . . . . . . . . . . . 14 ⊢ (𝐻:(1...𝑀)–1-1-onto→ran 𝐻 → ^◡𝐻:ran 𝐻–1-1-onto→(1...𝑀))
18		f1of 6830	. . . . . . . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (^◡𝐻‘𝑥) ∈ (1...𝑀))
21		fvco3 6987	. . . . . . . . . . . 12 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (^◡𝐻‘𝑥) ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘(^◡𝐻‘𝑥)) = (𝐹‘(𝐻‘(^◡𝐻‘𝑥))))
22	13, 20, 21	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → ((𝐹 ∘ 𝐻)‘(^◡𝐻‘𝑥)) = (𝐹‘(𝐻‘(^◡𝐻‘𝑥))))
23		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝑊 = ∅)
24	23	difeq2d 4121	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((1...𝑀) ∖ 𝑊) = ((1...𝑀) ∖ ∅))
25		dif0 4371	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∖ ∅) = (1...𝑀)
26	24, 25	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((1...𝑀) ∖ 𝑊) = (1...𝑀))
27	26	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → ((1...𝑀) ∖ 𝑊) = (1...𝑀))
28	20, 27	eleqtrrd 2836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (^◡𝐻‘𝑥) ∈ ((1...𝑀) ∖ 𝑊))
29		fco 6738	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
30	7, 12, 29	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
31	30	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
32		gsumval3.w	. . . . . . . . . . . . . . 15 ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )
33	32	eqimss2i 4042	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ 𝑊
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ 𝑊)
35		ovexd 7440	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (1...𝑀) ∈ V)
36	3	fvexi 6902	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 0 ∈ V)
38	31, 34, 35, 37	suppssr 8177	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (^◡𝐻‘𝑥) ∈ ((1...𝑀) ∖ 𝑊)) → ((𝐹 ∘ 𝐻)‘(^◡𝐻‘𝑥)) = 0 )
39	28, 38	syldan 591	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → ((𝐹 ∘ 𝐻)‘(^◡𝐻‘𝑥)) = 0 )
40		f1ocnvfv2 7271	. . . . . . . . . . . . 13 ⊢ ((𝐻:(1...𝑀)–1-1-onto→ran 𝐻 ∧ 𝑥 ∈ ran 𝐻) → (𝐻‘(^◡𝐻‘𝑥)) = 𝑥)
41	16, 40	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐻‘(^◡𝐻‘𝑥)) = 𝑥)
42	41	fveq2d 6892	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘(𝐻‘(^◡𝐻‘𝑥))) = (𝐹‘𝑥))
43	22, 39, 42	3eqtr3rd 2781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) = 0 )
44		fvex 6901	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
45	44	elsn 4642	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 )
46	43, 45	sylibr 233	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) ∈ { 0 })
47	46	adantlr 713	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) ∈ { 0 })
48		eldif 3957	. . . . . . . . . . 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 8177	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran 𝐻)) → (𝐹‘𝑥) = 0 )
52	51, 45	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran 𝐻)) → (𝐹‘𝑥) ∈ { 0 })
53	48, 52	sylan2br 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻)) → (𝐹‘𝑥) ∈ { 0 })
54	53	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐻)) → (𝐹‘𝑥) ∈ { 0 })
55	54	anassrs 468	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 ∈ ran 𝐻) → (𝐹‘𝑥) ∈ { 0 })
56	47, 55	pm2.61dan 811	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ { 0 })
57	56, 45	sylib 217	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 0 )
58	57	mpteq2dva 5247	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 0 ))
59	9, 58	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 0 ))
60	59	oveq2d 7421	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ 0 )))
61		gsumval3.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
62	61, 3	mndidcl 18636	. . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
63		gsumval3.p	. . . . . . 7 ⊢ + = (+_g‘𝐺)
64	61, 63, 3	mndlid 18641	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
65	1, 62, 64	syl2anc2 585	. . . . 5 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
66	65	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ( 0 + 0 ) = 0 )
67		gsumval3.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
68		nnuz 12861	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
69	67, 68	eleqtrdi 2843	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
70	69	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝑀 ∈ (ℤ_≥‘1))
71	26	eleq2d 2819	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ ((1...𝑀) ∖ 𝑊) ↔ 𝑥 ∈ (1...𝑀)))
72	71	biimpar 478	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (1...𝑀)) → 𝑥 ∈ ((1...𝑀) ∖ 𝑊))
73	31, 34, 35, 37	suppssr 8177	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ ((1...𝑀) ∖ 𝑊)) → ((𝐹 ∘ 𝐻)‘𝑥) = 0 )
74	72, 73	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑥) = 0 )
75	66, 70, 74	seqid3 14008	. . 3 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = 0 )
76	6, 60, 75	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
77		fzf 13484	. . . . 5 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
78		ffn 6714	. . . . 5 ⊢ (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
79		ovelrn 7579	. . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐺 ∈ Mnd)
82		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐴 = (𝑚...𝑛))
83		frel 6719	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
84		reldm0 5925	. . . . . . . . . . . . . . . . 17 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
85	7, 83, 84	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
86	7	fdmd 6725	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐹 = 𝐴)
87	86	eqeq1d 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (dom 𝐹 = ∅ ↔ 𝐴 = ∅))
88	85, 87	bitrd 278	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 = ∅ ↔ 𝐴 = ∅))
89		coeq1 5855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 = ∅ → (𝐹 ∘ 𝐻) = (∅ ∘ 𝐻))
90		co01 6257	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∘ 𝐻) = ∅
91	89, 90	eqtrdi 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = ∅ → (𝐹 ∘ 𝐻) = ∅)
92	91	oveq1d 7420	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = (∅ supp 0 ))
93		supp0 8147	. . . . . . . . . . . . . . . . . 18 ⊢ ( 0 ∈ V → (∅ supp 0 ) = ∅)
94	36, 93	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ supp 0 ) = ∅
95	92, 94	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅)
96	32, 95	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ (𝐹 = ∅ → 𝑊 = ∅)
97	88, 96	syl6bir 253	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 = ∅ → 𝑊 = ∅))
98	97	necon3d 2961	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ≠ ∅ → 𝐴 ≠ ∅))
99	98	imp 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → 𝐴 ≠ ∅)
100	99	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐴 ≠ ∅)
101	82, 100	eqnetrrd 3009	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝑚...𝑛) ≠ ∅)
102		fzn0 13511	. . . . . . . . . 10 ⊢ ((𝑚...𝑛) ≠ ∅ ↔ 𝑛 ∈ (ℤ_≥‘𝑚))
103	101, 102	sylib 217	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝑛 ∈ (ℤ_≥‘𝑚))
104	7	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐹:𝐴⟶𝐵)
105	82	feq2d 6700	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:(𝑚...𝑛)⟶𝐵))
106	104, 105	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → 𝐹:(𝑚...𝑛)⟶𝐵)
107	61, 63, 81, 103, 106	gsumval2 18601	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝐺 Σ_g 𝐹) = (seq𝑚( + , 𝐹)‘𝑛))
108		frn 6721	. . . . . . . . . . . . . . 15 ⊢ (𝐻:(1...𝑀)⟶𝐴 → ran 𝐻 ⊆ 𝐴)
109	10, 11, 108	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
110	109	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ⊆ 𝐴)
111	110, 82	sseqtrd 4021	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ⊆ (𝑚...𝑛))
112		fzssuz 13538	. . . . . . . . . . . . 13 ⊢ (𝑚...𝑛) ⊆ (ℤ_≥‘𝑚)
113		uzssz 12839	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑚) ⊆ ℤ
114		zssre 12561	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
115	113, 114	sstri 3990	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑚) ⊆ ℝ
116	112, 115	sstri 3990	. . . . . . . . . . . 12 ⊢ (𝑚...𝑛) ⊆ ℝ
117	111, 116	sstrdi 3993	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ⊆ ℝ)
118		ltso 11290	. . . . . . . . . . 11 ⊢ < Or ℝ
119		soss 5607	. . . . . . . . . . 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 13933	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ Fin
122	121	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
123	12, 122	fexd 7225	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 ∈ V)
124		f1oen3g 8958	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
125	123, 15, 124	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
126		enfi 9186	. . . . . . . . . . . . 13 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
127	125, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
128	121, 127	mpbii 232	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
129	128	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ran 𝐻 ∈ Fin)
130		fz1iso 14419	. . . . . . . . . 10 ⊢ (( < Or ran 𝐻 ∧ ran 𝐻 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))
131	120, 129, 130	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))
132	67	nnnn0d 12528	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
133		hashfz1 14302	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
134	132, 133	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
135	122, 15	hasheqf1od 14309	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(1...𝑀)) = (♯‘ran 𝐻))
136	134, 135	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 = (♯‘ran 𝐻))
137	136	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑀 = (♯‘ran 𝐻))
138	137	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq1( + , (𝐹 ∘ 𝑓))‘𝑀) = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘ran 𝐻)))
139	1	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐺 ∈ Mnd)
140	61, 63	mndcl 18629	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
141	140	3expb 1120	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
142	139, 141	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
143		gsumval3.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
144	143	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
145	144	sselda 3981	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
146		gsumval3.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (Cntz‘𝐺)
147	63, 146	cntzi 19187	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
148	145, 147	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
149	148	anasss 467	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
150	61, 63	mndass 18630	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
151	139, 150	sylan 580	. . . . . . . . . . . . 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)))
152	69	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑀 ∈ (ℤ_≥‘1))
153	7	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐹:𝐴⟶𝐵)
154	153	frnd 6722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐹 ⊆ 𝐵)
155		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))
156		isof1o 7316	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻) → 𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻)
157	155, 156	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻)
158	137	oveq2d 7421	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (1...𝑀) = (1...(♯‘ran 𝐻)))
159	158	f1oeq2d 6826	. . . . . . . . . . . . . . . 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))
160	157, 159	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...𝑀)–1-1-onto→ran 𝐻)
161		f1ocnv 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 → ^◡𝑓:ran 𝐻–1-1-onto→(1...𝑀))
162	160, 161	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ^◡𝑓:ran 𝐻–1-1-onto→(1...𝑀))
163	15	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
164		f1oco 6853	. . . . . . . . . . . . . 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))
165	162, 163, 164	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (^◡𝑓 ∘ 𝐻):(1...𝑀)–1-1-onto→(1...𝑀))
166		ffn 6714	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
167		dffn4 6808	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
168	166, 167	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
169		fof 6802	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐴–onto→ran 𝐹 → 𝐹:𝐴⟶ran 𝐹)
170	153, 168, 169	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐹:𝐴⟶ran 𝐹)
171		f1of 6830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 → 𝑓:(1...𝑀)⟶ran 𝐻)
172	160, 171	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...𝑀)⟶ran 𝐻)
173	109	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐻 ⊆ 𝐴)
174	172, 173	fssd 6732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...𝑀)⟶𝐴)
175		fco 6738	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ 𝑓:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝑓):(1...𝑀)⟶ran 𝐹)
176	170, 174, 175	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐹 ∘ 𝑓):(1...𝑀)⟶ran 𝐹)
177	176	ffvelcdmda 7083	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (1...𝑀)) → ((𝐹 ∘ 𝑓)‘𝑥) ∈ ran 𝐹)
178		f1ococnv2 6857	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:(1...𝑀)–1-1-onto→ran 𝐻 → (𝑓 ∘ ^◡𝑓) = ( I ↾ ran 𝐻))
179	160, 178	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑓 ∘ ^◡𝑓) = ( I ↾ ran 𝐻))
180	179	coeq1d 5859	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ((𝑓 ∘ ^◡𝑓) ∘ 𝐻) = (( I ↾ ran 𝐻) ∘ 𝐻))
181		f1of 6830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐻:(1...𝑀)–1-1-onto→ran 𝐻 → 𝐻:(1...𝑀)⟶ran 𝐻)
182		fcoi2 6763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐻:(1...𝑀)⟶ran 𝐻 → (( I ↾ ran 𝐻) ∘ 𝐻) = 𝐻)
183	163, 181, 182	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (( I ↾ ran 𝐻) ∘ 𝐻) = 𝐻)
184	180, 183	eqtr2d 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻 = ((𝑓 ∘ ^◡𝑓) ∘ 𝐻))
185		coass 6261	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∘ ^◡𝑓) ∘ 𝐻) = (𝑓 ∘ (^◡𝑓 ∘ 𝐻))
186	184, 185	eqtrdi 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻 = (𝑓 ∘ (^◡𝑓 ∘ 𝐻)))
187	186	coeq2d 5860	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐹 ∘ 𝐻) = (𝐹 ∘ (𝑓 ∘ (^◡𝑓 ∘ 𝐻))))
188		coass 6261	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∘ 𝑓) ∘ (^◡𝑓 ∘ 𝐻)) = (𝐹 ∘ (𝑓 ∘ (^◡𝑓 ∘ 𝐻)))
189	187, 188	eqtr4di 2790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐹 ∘ 𝐻) = ((𝐹 ∘ 𝑓) ∘ (^◡𝑓 ∘ 𝐻)))
190	189	fveq1d 6890	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ((𝐹 ∘ 𝐻)‘𝑘) = (((𝐹 ∘ 𝑓) ∘ (^◡𝑓 ∘ 𝐻))‘𝑘))
191	190	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑘) = (((𝐹 ∘ 𝑓) ∘ (^◡𝑓 ∘ 𝐻))‘𝑘))
192		f1of 6830	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝑓:ran 𝐻–1-1-onto→(1...𝑀) → ^◡𝑓:ran 𝐻⟶(1...𝑀))
193	160, 161, 192	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ^◡𝑓:ran 𝐻⟶(1...𝑀))
194	163, 181	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐻:(1...𝑀)⟶ran 𝐻)
195		fco 6738	. . . . . . . . . . . . . . . 16 ⊢ ((^◡𝑓:ran 𝐻⟶(1...𝑀) ∧ 𝐻:(1...𝑀)⟶ran 𝐻) → (^◡𝑓 ∘ 𝐻):(1...𝑀)⟶(1...𝑀))
196	193, 194, 195	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (^◡𝑓 ∘ 𝐻):(1...𝑀)⟶(1...𝑀))
197		fvco3 6987	. . . . . . . . . . . . . . 15 ⊢ (((^◡𝑓 ∘ 𝐻):(1...𝑀)⟶(1...𝑀) ∧ 𝑘 ∈ (1...𝑀)) → (((𝐹 ∘ 𝑓) ∘ (^◡𝑓 ∘ 𝐻))‘𝑘) = ((𝐹 ∘ 𝑓)‘((^◡𝑓 ∘ 𝐻)‘𝑘)))
198	196, 197	sylan 580	. . . . . . . . . . . . . 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)))
199	191, 198	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑘) = ((𝐹 ∘ 𝑓)‘((^◡𝑓 ∘ 𝐻)‘𝑘)))
200	142, 149, 151, 152, 154, 165, 177, 199	seqf1o 14005	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq1( + , (𝐹 ∘ 𝑓))‘𝑀))
201	61, 63, 3	mndlid 18641	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
202	139, 201	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
203	61, 63, 3	mndrid 18642	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
204	139, 203	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
205	139, 62	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 0 ∈ 𝐵)
206		fdm 6723	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:(1...𝑀)⟶𝐴 → dom 𝐻 = (1...𝑀))
207	10, 11, 206	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐻 = (1...𝑀))
208		eluzfz1 13504	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑀))
209		ne0i 4333	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ (1...𝑀) → (1...𝑀) ≠ ∅)
210	69, 208, 209	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑀) ≠ ∅)
211	207, 210	eqnetrd 3008	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐻 ≠ ∅)
212		dm0rn0 5922	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝐻 = ∅ ↔ ran 𝐻 = ∅)
213	212	necon3bii 2993	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐻 ≠ ∅ ↔ ran 𝐻 ≠ ∅)
214	211, 213	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐻 ≠ ∅)
215	214	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐻 ≠ ∅)
216	111	adantrr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → ran 𝐻 ⊆ (𝑚...𝑛))
217		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝐴 = (𝑚...𝑛))
218	217	eleq2d 2819	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝑚...𝑛)))
219	218	biimpar 478	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (𝑚...𝑛)) → 𝑥 ∈ 𝐴)
220	153	ffvelcdmda 7083	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
221	219, 220	syldan 591	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (𝑚...𝑛)) → (𝐹‘𝑥) ∈ 𝐵)
222	217	difeq1d 4120	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝐴 ∖ ran 𝐻) = ((𝑚...𝑛) ∖ ran 𝐻))
223	222	eleq2d 2819	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (𝑥 ∈ (𝐴 ∖ ran 𝐻) ↔ 𝑥 ∈ ((𝑚...𝑛) ∖ ran 𝐻)))
224	223	biimpar 478	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ((𝑚...𝑛) ∖ ran 𝐻)) → 𝑥 ∈ (𝐴 ∖ ran 𝐻))
225	51	ad4ant14 750	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝐻)) → (𝐹‘𝑥) = 0 )
226	224, 225	syldan 591	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑥 ∈ ((𝑚...𝑛) ∖ ran 𝐻)) → (𝐹‘𝑥) = 0 )
227		f1of 6830	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘ran 𝐻))–1-1-onto→ran 𝐻 → 𝑓:(1...(♯‘ran 𝐻))⟶ran 𝐻)
228	155, 156, 227	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → 𝑓:(1...(♯‘ran 𝐻))⟶ran 𝐻)
229		fvco3 6987	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘ran 𝐻))⟶ran 𝐻 ∧ 𝑦 ∈ (1...(♯‘ran 𝐻))) → ((𝐹 ∘ 𝑓)‘𝑦) = (𝐹‘(𝑓‘𝑦)))
230	228, 229	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) ∧ 𝑦 ∈ (1...(♯‘ran 𝐻))) → ((𝐹 ∘ 𝑓)‘𝑦) = (𝐹‘(𝑓‘𝑦)))
231	202, 204, 142, 205, 155, 215, 216, 221, 226, 230	seqcoll2 14422	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘ran 𝐻)))
232	138, 200, 231	3eqtr4d 2782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻))) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛))
233	232	expr 457	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛)))
234	233	exlimdv 1936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘ran 𝐻)), ran 𝐻) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛)))
235	131, 234	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq𝑚( + , 𝐹)‘𝑛))
236	107, 235	eqtr4d 2775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ 𝐴 = (𝑚...𝑛)) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
237	236	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐴 = (𝑚...𝑛) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
238	237	rexlimdvw 3160	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (∃𝑛 ∈ ℤ 𝐴 = (𝑚...𝑛) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
239	238	rexlimdvw 3160	. . . 4 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝐴 = (𝑚...𝑛) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
240	80, 239	biimtrid 241	. . 3 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐴 ∈ ran ... → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
241		suppssdm 8158	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
242	32, 241	eqsstri 4015	. . . . . . . . . 10 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
243	242, 30	fssdm 6734	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
244		fz1ssnn 13528	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
245		nnssre 12212	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ
246	244, 245	sstri 3990	. . . . . . . . 9 ⊢ (1...𝑀) ⊆ ℝ
247	243, 246	sstrdi 3993	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ ℝ)
248		soss 5607	. . . . . . . 8 ⊢ (𝑊 ⊆ ℝ → ( < Or ℝ → < Or 𝑊))
249	247, 118, 248	mpisyl 21	. . . . . . 7 ⊢ (𝜑 → < Or 𝑊)
250		ssfi 9169	. . . . . . . 8 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
251	121, 243, 250	sylancr 587	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
252		fz1iso 14419	. . . . . . 7 ⊢ (( < Or 𝑊 ∧ 𝑊 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
253	249, 251, 252	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
254	253	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
255	61, 3, 63, 146, 1, 2, 7, 143, 67, 10, 49, 32	gsumval3lem2 19768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))
256	1	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
257	256, 201	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
258	256, 203	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
259	256, 141	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
260	256, 62	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 0 ∈ 𝐵)
261		simprr 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))
262		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ≠ ∅)
263	243	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
264	30	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
265	264	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ (1...𝑀)) → ((𝐹 ∘ 𝐻)‘𝑥) ∈ 𝐵)
266	33	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ 𝑊)
267		ovexd 7440	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (1...𝑀) ∈ V)
268	36	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 0 ∈ V)
269	264, 266, 267, 268	suppssr 8177	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑥 ∈ ((1...𝑀) ∖ 𝑊)) → ((𝐹 ∘ 𝐻)‘𝑥) = 0 )
270		coass 6261	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐻) ∘ 𝑓) = (𝐹 ∘ (𝐻 ∘ 𝑓))
271	270	fveq1i 6889	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝐻) ∘ 𝑓)‘𝑦) = ((𝐹 ∘ (𝐻 ∘ 𝑓))‘𝑦)
272		isof1o 7316	. . . . . . . . . . . 12 ⊢ (𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
273		f1of 6830	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))⟶𝑊)
274	261, 272, 273	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
275		fvco3 6987	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑦 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘ 𝐻) ∘ 𝑓)‘𝑦) = ((𝐹 ∘ 𝐻)‘(𝑓‘𝑦)))
276	274, 275	sylan 580	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑦 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘ 𝐻) ∘ 𝑓)‘𝑦) = ((𝐹 ∘ 𝐻)‘(𝑓‘𝑦)))
277	271, 276	eqtr3id 2786	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) ∧ 𝑦 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ (𝐻 ∘ 𝑓))‘𝑦) = ((𝐹 ∘ 𝐻)‘(𝑓‘𝑦)))
278	257, 258, 259, 260, 261, 262, 263, 265, 269, 277	seqcoll2 14422	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (seq1( + , (𝐹 ∘ 𝐻))‘𝑀) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))
279	255, 278	eqtr4d 2775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
280	279	expr 457	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → (𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
281	280	exlimdv 1936	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
282	254, 281	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ ¬ 𝐴 ∈ ran ...) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
283	282	ex 413	. . 3 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (¬ 𝐴 ∈ ran ... → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)))
284	240, 283	pm2.61d 179	. 2 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
285	76, 284	pm2.61dane 3029	1 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 Vcvv 3474 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 I cid 5572 Or wor 5586 × cxp 5673 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ↾ cres 5677 ∘ ccom 5679 Rel wrel 5680 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 Isom wiso 6541 (class class class)co 7405 supp csupp 8142 ≈ cen 8932 Fincfn 8935 ℝcr 11105 1c1 11107 < clt 11244 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 seqcseq 13962 ♯chash 14286 Basecbs 17140 +_gcplusg 17193 0_gc0g 17381 Σ_g cgsu 17382 Mndcmnd 18621 Cntzccntz 19173

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-cntz 19175

This theorem is referenced by: gsumzres 19771 gsumzcl2 19772 gsumzf1o 19774 gsumzaddlem 19783 gsumconst 19796 gsumzmhm 19799 gsumzoppg 19806 gsumfsum 21004 wilthlem3 26563

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