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Theorem gsumval3lem2 20129

Description: Lemma 2 for gsumval3 20130. (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 6783	. . . . . . 7 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
4		fzfid 14123	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
5	3, 4	fexd 7237	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ V)
6		vex 3454	. . . . 5 ⊢ 𝑓 ∈ V
7		coexg 7945	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑓 ∈ V) → (𝐻 ∘ 𝑓) ∈ V)
8	5, 6, 7	sylancl 598	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑓) ∈ V)
9	8	ad2antrr 739	. . 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 20128	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
22		fzfi 14122	. . . . . . . 8 ⊢ (1...𝑀) ∈ Fin
23		suppssdm 8201	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
24	20, 23	eqsstri 3955	. . . . . . . . 9 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
25	16, 3	fcod 6740	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
26	24, 25	fssdm 6734	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
27		ssfi 9195	. . . . . . . 8 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
28	22, 26, 27	sylancr 599	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
29	28	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
30	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1→𝐴)
31	26	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
32		f1ores 6844	. . . . . . . 8 ⊢ ((𝐻:(1...𝑀)–1-1→𝐴 ∧ 𝑊 ⊆ (1...𝑀)) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
33	30, 31, 32	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
34	20	imaeq2i 6052	. . . . . . . . . 10 ⊢ (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 ))
35	16, 15	fexd 7237	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
36		ovex 7457	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
37		fex 7236	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
38	3, 36, 37	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ V)
39	35, 38	jca 521	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐻 ∈ V))
40		f1fun 6785	. . . . . . . . . . . . . 14 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → Fun 𝐻)
41	1, 40	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐻)
42	41, 19	jca 521	. . . . . . . . . . . 12 ⊢ (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
43		imacosupp 8233	. . . . . . . . . . . 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 66	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
45	44	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
46	34, 45	eqtrid 2807	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
47	46	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
48	47	f1oeq3d 6826	. . . . . . 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 235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
50	29, 49	hasheqf1od 14504	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (♯‘𝑊) = (♯‘(𝐹 supp 0 )))
51	50	fveq2d 6894	. . . 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 521	. . 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 6817	. . . 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 5837	. . . . . . 7 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
55	54	seqeq3d 14159	. . . . . 6 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
56	55	fveq1d 6892	. . . . 5 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘(𝐹 supp 0 ))) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 ))))
57	56	eqeq2d 2771	. . . 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 644	. . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
61	15	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐴 ∈ 𝑉)
62	16	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐹:𝐴⟶𝐵)
63	17	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
64		f1f1orn 6841	. . . . . . . . . . . . 13 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
65	1, 64	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
66		f1oen3g 9000	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
67	5, 65, 66	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
68		enfi 9209	. . . . . . . . . . 11 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
69	67, 68	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
70	22, 69	mpbii 236	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
71	70, 19	ssfid 9267	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
72	71	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
73	20	neeq1i 3019	. . . . . . . . . 10 ⊢ (𝑊 ≠ ∅ ↔ ((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅)
74		supp0cosupp0 8232	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((𝐹 supp 0 ) = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅))
75	74	necon3d 2976	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
76	35, 38, 75	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
77	73, 76	biimtrid 245	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
78	77	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐹 supp 0 ) ≠ ∅)
79	78	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ≠ ∅)
80	19	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ ran 𝐻)
81	3	frnd 6723	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
82	81	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝐻 ⊆ 𝐴)
83	80, 82	sstrd 3919	. . . . . . 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 20127	. . . . . 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 6521	. . . . . 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 18	. . . . 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 2760	. . . . . . 7 ⊢ (𝐹 supp 0 ) = (𝐹 supp 0 )
88		simprl 783	. . . . . . 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 20126	. . . . . 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 2762	. . . . 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 285	. . . 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 1959	. . 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 6903	. . . 4 ⊢ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)) ∈ V
94		eqeq1 2764	. . . . . . 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 642	. . . . . 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 1953	. . . . 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 2772	. . . . 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 348	. . . 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 18	. 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 235	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1567 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ∃!weu 2593 ≠ wne 2955 Vcvv 3450 ⊆ wss 3896 ∅c0 4277 class class class wbr 5102 dom cdm 5651 ran crn 5652 ↾ cres 5653 “ cima 5654 ∘ ccom 5655 ℩cio 6495 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7424 supp csupp 8183 ≈ cen 8977 Fincfn 8980 1c1 11208 < clt 11350 ℕcn 12342 ...cfz 13646 seqcseq 14151 ♯chash 14481 Basecbs 17396 +_gcplusg 17437 0_gc0g 17619 Σ_g cgsu 17620 Mndcmnd 18932 Cntzccntz 19538

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8184 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-n0 12614 df-z 12701 df-uz 12973 df-fz 13647 df-fzo 13796 df-seq 14152 df-hash 14482 df-0g 17621 df-gsum 17622 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-cntz 19540

This theorem is used by: gsumval3 20130

W3C validator