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Theorem gsumval3lem2 19683

Description: Lemma 2 for gsumval3 19684. (Contributed 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
gsumval3lem2	⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))

Distinct variable groups: + ,𝑓 𝐴,𝑓 𝜑,𝑓 𝑓,𝐺 𝑓,𝑀 𝐵,𝑓 𝑓,𝐹 𝑓,𝐻 𝑓,𝑊 Allowed substitution hints: 𝑉(𝑓) 0 (𝑓) 𝑍(𝑓)

Proof of Theorem gsumval3lem2 Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3.h	. . . . . . 7 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
2		f1f 6738	. . . . . . 7 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
4		fzfid 13878	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
5	3, 4	fexd 7177	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ V)
6		vex 3449	. . . . 5 ⊢ 𝑓 ∈ V
7		coexg 7866	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑓 ∈ V) → (𝐻 ∘ 𝑓) ∈ V)
8	5, 6, 7	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑓) ∈ V)
9	8	ad2antrr 724	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓) ∈ V)
10		gsumval3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
11		gsumval3.0	. . . . 5 ⊢ 0 = (0_g‘𝐺)
12		gsumval3.p	. . . . 5 ⊢ + = (+_g‘𝐺)
13		gsumval3.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
14		gsumval3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
15		gsumval3.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16		gsumval3.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
17		gsumval3.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
18		gsumval3.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
19		gsumval3.n	. . . . 5 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
20		gsumval3.w	. . . . 5 ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )
21	10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 20	gsumval3lem1 19682	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
22		fzfi 13877	. . . . . . . 8 ⊢ (1...𝑀) ∈ Fin
23		suppssdm 8108	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
24	20, 23	eqsstri 3978	. . . . . . . . 9 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
25	16, 3	fcod 6694	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
26	24, 25	fssdm 6688	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
27		ssfi 9117	. . . . . . . 8 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
28	22, 26, 27	sylancr 587	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
29	28	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
30	1	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1→𝐴)
31	26	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
32		f1ores 6798	. . . . . . . 8 ⊢ ((𝐻:(1...𝑀)–1-1→𝐴 ∧ 𝑊 ⊆ (1...𝑀)) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
33	30, 31, 32	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
34	20	imaeq2i 6011	. . . . . . . . . 10 ⊢ (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 ))
35	16, 15	fexd 7177	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
36		ovex 7390	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
37		fex 7176	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
38	3, 36, 37	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ V)
39	35, 38	jca 512	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐻 ∈ V))
40		f1fun 6740	. . . . . . . . . . . . . 14 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → Fun 𝐻)
41	1, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐻)
42	41, 19	jca 512	. . . . . . . . . . . 12 ⊢ (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
43		imacosupp 8140	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
44	39, 42, 43	sylc 65	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
45	44	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
46	34, 45	eqtrid 2788	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
47	46	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
48	47	f1oeq3d 6781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊) ↔ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )))
49	33, 48	mpbid 231	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
50	29, 49	hasheqf1od 14253	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (♯‘𝑊) = (♯‘(𝐹 supp 0 )))
51	50	fveq2d 6846	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 ))))
52	21, 51	jca 512	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 )))))
53		f1oeq1 6772	. . . 4 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))
54		coeq2 5814	. . . . . . 7 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
55	54	seqeq3d 13914	. . . . . 6 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
56	55	fveq1d 6844	. . . . 5 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 ))))
57	56	eqeq2d 2747	. . . 4 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → ((seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))) ↔ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 )))))
58	53, 57	anbi12d 631	. . 3 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → ((𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ ((𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 ))))))
59	9, 52, 58	spcedv 3557	. 2 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))))
60	14	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
61	15	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐴 ∈ 𝑉)
62	16	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐹:𝐴⟶𝐵)
63	17	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
64		f1f1orn 6795	. . . . . . . . . . . . 13 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
65	1, 64	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
66		f1oen3g 8906	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
67	5, 65, 66	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
68		enfi 9134	. . . . . . . . . . 11 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
69	67, 68	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
70	22, 69	mpbii 232	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
71	70, 19	ssfid 9211	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
72	71	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
73	20	neeq1i 3008	. . . . . . . . . 10 ⊢ (𝑊 ≠ ∅ ↔ ((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅)
74		supp0cosupp0 8139	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((𝐹 supp 0 ) = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅))
75	74	necon3d 2964	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
76	35, 38, 75	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
77	73, 76	biimtrid 241	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
78	77	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐹 supp 0 ) ≠ ∅)
79	78	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ≠ ∅)
80	19	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ ran 𝐻)
81	3	frnd 6676	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
82	81	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝐻 ⊆ 𝐴)
83	80, 82	sstrd 3954	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ 𝐴)
84	10, 11, 12, 13, 60, 61, 62, 63, 72, 79, 83	gsumval3eu 19681	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ∃!𝑥∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))))
85		iota1 6473	. . . . . 6 ⊢ (∃!𝑥∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) → (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (℩𝑥∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))))) = 𝑥))
86	84, 85	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (℩𝑥∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))))) = 𝑥))
87		eqid 2736	. . . . . . 7 ⊢ (𝐹 supp 0 ) = (𝐹 supp 0 )
88		simprl 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ¬ 𝐴 ∈ ran ...)
89	10, 11, 12, 13, 60, 61, 62, 63, 72, 79, 87, 88	gsumval3a 19680	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (℩𝑥∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))))))
90	89	eqeq1d 2738	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐺 Σ_g 𝐹) = 𝑥 ↔ (℩𝑥∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))))) = 𝑥))
91	86, 90	bitr4d 281	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥))
92	91	alrimiv 1930	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ∀𝑥(∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥))
93		fvex 6855	. . . 4 ⊢ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) ∈ V
94		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) → (𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))) ↔ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))))
95	94	anbi2d 629	. . . . . 6 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) → ((𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))))))
96	95	exbidv 1924	. . . . 5 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) → (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ ∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))))))
97		eqeq2 2748	. . . . 5 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) → ((𝐺 Σ_g 𝐹) = 𝑥 ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊))))
98	96, 97	bibi12d 345	. . . 4 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) → ((∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥) ↔ (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))))
99	93, 98	spcv 3564	. . 3 ⊢ (∀𝑥(∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = 𝑥) → (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊))))
100	92, 99	syl 17	. 2 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (∃𝑔(𝑔:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 )))) ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊))))
101	59, 100	mpbid 231	1 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃!weu 2566 ≠ wne 2943 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 class class class wbr 5105 dom cdm 5633 ran crn 5634 ↾ cres 5635 “ cima 5636 ∘ ccom 5637 ℩cio 6446 Fun wfun 6490 ⟶wf 6492 –1-1→wf1 6493 –1-1-onto→wf1o 6495 ‘cfv 6496 Isom wiso 6497 (class class class)co 7357 supp csupp 8092 ≈ cen 8880 Fincfn 8883 1c1 11052 < clt 11189 ℕcn 12153 ...cfz 13424 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-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-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-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: gsumval3 19684

W3C validator