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Theorem gsumval3lem2 19835

Description: Lemma 2 for gsumval3 19836. (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 6730	. . . . . . 7 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
4		fzfid 13896	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
5	3, 4	fexd 7173	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ V)
6		vex 3444	. . . . 5 ⊢ 𝑓 ∈ V
7		coexg 7871	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑓 ∈ V) → (𝐻 ∘ 𝑓) ∈ V)
8	5, 6, 7	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑓) ∈ V)
9	8	ad2antrr 726	. . 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 19834	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
22		fzfi 13895	. . . . . . . 8 ⊢ (1...𝑀) ∈ Fin
23		suppssdm 8119	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
24	20, 23	eqsstri 3980	. . . . . . . . 9 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
25	16, 3	fcod 6687	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
26	24, 25	fssdm 6681	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
27		ssfi 9097	. . . . . . . 8 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
28	22, 26, 27	sylancr 587	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
29	28	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
30	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1→𝐴)
31	26	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
32		f1ores 6788	. . . . . . . 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 6017	. . . . . . . . . 10 ⊢ (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 ))
35	16, 15	fexd 7173	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
36		ovex 7391	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
37		fex 7172	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
38	3, 36, 37	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ V)
39	35, 38	jca 511	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐻 ∈ V))
40		f1fun 6732	. . . . . . . . . . . . . 14 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → Fun 𝐻)
41	1, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐻)
42	41, 19	jca 511	. . . . . . . . . . . 12 ⊢ (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
43		imacosupp 8151	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
46	34, 45	eqtrid 2783	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
47	46	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
48	47	f1oeq3d 6771	. . . . . . 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 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
50	29, 49	hasheqf1od 14276	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (♯‘𝑊) = (♯‘(𝐹 supp 0 )))
51	50	fveq2d 6838	. . . 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 511	. . 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 6762	. . . 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 5807	. . . . . . 7 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
55	54	seqeq3d 13932	. . . . . 6 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
56	55	fveq1d 6836	. . . . 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 632	. . 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 3552	. 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
61	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐴 ∈ 𝑉)
62	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐹:𝐴⟶𝐵)
63	17	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
64		f1f1orn 6785	. . . . . . . . . . . . 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 8903	. . . . . . . . . . . 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 9111	. . . . . . . . . . 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 233	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
71	70, 19	ssfid 9169	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
72	71	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
73	20	neeq1i 2996	. . . . . . . . . 10 ⊢ (𝑊 ≠ ∅ ↔ ((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅)
74		supp0cosupp0 8150	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((𝐹 supp 0 ) = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅))
75	74	necon3d 2953	. . . . . . . . . . 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 242	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
78	77	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐹 supp 0 ) ≠ ∅)
79	78	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ≠ ∅)
80	19	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ ran 𝐻)
81	3	frnd 6670	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
82	81	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝐻 ⊆ 𝐴)
83	80, 82	sstrd 3944	. . . . . . 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 19833	. . . . . 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 6471	. . . . . 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 770	. . . . . . 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 19832	. . . . . 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 282	. . . 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 1928	. . 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 6847	. . . 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 630	. . . . . 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 1922	. . . . 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 3559	. . 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 232	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 206 ∧ wa 395 ∀wal 1539 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∃!weu 2568 ≠ wne 2932 Vcvv 3440 ⊆ wss 3901 ∅c0 4285 class class class wbr 5098 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 ∘ ccom 5628 ℩cio 6446 Fun wfun 6486 ⟶wf 6488 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 Isom wiso 6493 (class class class)co 7358 supp csupp 8102 ≈ cen 8880 Fincfn 8883 1c1 11027 < clt 11166 ℕcn 12145 ...cfz 13423 seqcseq 13924 ♯chash 14253 Basecbs 17136 +_gcplusg 17177 0_gc0g 17359 Σ_g cgsu 17360 Mndcmnd 18659 Cntzccntz 19244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-0g 17361 df-gsum 17362 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-cntz 19246

This theorem is referenced by: gsumval3 19836

W3C validator