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Theorem gsumval3lem2 18307

Description: Lemma 2 for gsumval3 18308. (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 6101	. . . . . . 7 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
4		fzfid 12772	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
5		gsumval3.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		fex2 7121	. . . . . 6 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
7	3, 4, 5, 6	syl3anc 1326	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ V)
8		vex 3203	. . . . 5 ⊢ 𝑓 ∈ V
9		coexg 7117	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑓 ∈ V) → (𝐻 ∘ 𝑓) ∈ V)
10	7, 8, 9	sylancl 694	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑓) ∈ V)
11	10	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓) ∈ V)
12		gsumval3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
13		gsumval3.0	. . . . 5 ⊢ 0 = (0_g‘𝐺)
14		gsumval3.p	. . . . 5 ⊢ + = (+_g‘𝐺)
15		gsumval3.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
16		gsumval3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
17		gsumval3.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
18		gsumval3.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
19		gsumval3.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
20		gsumval3.n	. . . . 5 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
21		gsumval3.w	. . . . 5 ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )
22	12, 13, 14, 15, 16, 5, 17, 18, 19, 1, 20, 21	gsumval3lem1 18306	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
23		resexg 5442	. . . . . . . . 9 ⊢ (𝐻 ∈ V → (𝐻 ↾ 𝑊) ∈ V)
24	7, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ 𝑊) ∈ V)
25	24	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊) ∈ V)
26	1	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1→𝐴)
27		suppssdm 7308	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
28	21, 27	eqsstri 3635	. . . . . . . . . . 11 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
29		fco 6058	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
30	17, 3, 29	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
31		fdm 6051	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵 → dom (𝐹 ∘ 𝐻) = (1...𝑀))
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝐹 ∘ 𝐻) = (1...𝑀))
33	28, 32	syl5sseq 3653	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
34	33	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
35		f1ores 6151	. . . . . . . . 9 ⊢ ((𝐻:(1...𝑀)–1-1→𝐴 ∧ 𝑊 ⊆ (1...𝑀)) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
36	26, 34, 35	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
37	21	imaeq2i 5464	. . . . . . . . . . 11 ⊢ (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 ))
38		fex 6490	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
39	17, 5, 38	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
40		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ V
41		fex 6490	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
42	3, 40, 41	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 ∈ V)
43	39, 42	jca 554	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐻 ∈ V))
44		f1fun 6103	. . . . . . . . . . . . . . 15 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → Fun 𝐻)
45	1, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun 𝐻)
46	45, 20	jca 554	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
47		imacosupp 7335	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
48	43, 46, 47	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
49	48	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
50	37, 49	syl5eq 2668	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
51	50	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
52		f1oeq3 6129	. . . . . . . . 9 ⊢ ((𝐻 “ 𝑊) = (𝐹 supp 0 ) → ((𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊) ↔ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )))
53	51, 52	syl 17	. . . . . . . 8 ⊢ (((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 )))
54	36, 53	mpbid 222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
55		f1oen3g 7971	. . . . . . 7 ⊢ (((𝐻 ↾ 𝑊) ∈ V ∧ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )) → 𝑊 ≈ (𝐹 supp 0 ))
56	25, 54, 55	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ≈ (𝐹 supp 0 ))
57		fzfi 12771	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
58		ssfi 8180	. . . . . . . . 9 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
59	57, 33, 58	sylancr 695	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Fin)
60	59	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
61		f1f1orn 6148	. . . . . . . . . . . . 13 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
62	1, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
63		f1oen3g 7971	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
64	7, 62, 63	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
65		enfi 8176	. . . . . . . . . . 11 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
67	57, 66	mpbii 223	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
68		ssfi 8180	. . . . . . . . 9 ⊢ ((ran 𝐻 ∈ Fin ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐹 supp 0 ) ∈ Fin)
69	67, 20, 68	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
70	69	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
71		hashen 13135	. . . . . . 7 ⊢ ((𝑊 ∈ Fin ∧ (𝐹 supp 0 ) ∈ Fin) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
72	60, 70, 71	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
73	56, 72	mpbird 247	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (#‘𝑊) = (#‘(𝐹 supp 0 )))
74	73	fveq2d 6195	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 ))))
75	22, 74	jca 554	. . 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 )))))
76		f1oeq1 6127	. . . . 5 ⊢ (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 )))
77		coeq2 5280	. . . . . . . 8 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
78	77	seqeq3d 12809	. . . . . . 7 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
79	78	fveq1d 6193	. . . . . 6 ⊢ (𝑔 = (𝐻 ∘ 𝑓) → (seq1( + , (𝐹 ∘ 𝑔))‘(#‘(𝐹 supp 0 ))) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 ))))
80	79	eqeq2d 2632	. . . . 5 ⊢ (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 )))))
81	76, 80	anbi12d 747	. . . 4 ⊢ (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 ))))))
82	81	spcegv 3294	. . 3 ⊢ ((𝐻 ∘ 𝑓) ∈ V → (((𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘(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 ))))))
83	11, 75, 82	sylc 65	. 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 )))))
84	16	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐺 ∈ Mnd)
85	5	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐴 ∈ 𝑉)
86	17	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐹:𝐴⟶𝐵)
87	18	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
88	21	neeq1i 2858	. . . . . . . . . 10 ⊢ (𝑊 ≠ ∅ ↔ ((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅)
89		supp0cosupp0 7334	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((𝐹 supp 0 ) = ∅ → ((𝐹 ∘ 𝐻) supp 0 ) = ∅))
90	89	necon3d 2815	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
91	39, 42, 90	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
92	88, 91	syl5bi 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ≠ ∅ → (𝐹 supp 0 ) ≠ ∅))
93	92	imp 445	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 ≠ ∅) → (𝐹 supp 0 ) ≠ ∅)
94	93	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ≠ ∅)
95	20	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ ran 𝐻)
96		frn 6053	. . . . . . . . . 10 ⊢ (𝐻:(1...𝑀)⟶𝐴 → ran 𝐻 ⊆ 𝐴)
97	3, 96	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐴)
98	97	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ran 𝐻 ⊆ 𝐴)
99	95, 98	sstrd 3613	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ⊆ 𝐴)
100	12, 13, 14, 15, 84, 85, 86, 87, 70, 94, 99	gsumval3eu 18305	. . . . . 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 )))))
101		iota1 5865	. . . . . 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))
102	100, 101	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))
103		eqid 2622	. . . . . . 7 ⊢ (𝐹 supp 0 ) = (𝐹 supp 0 )
104		simprl 794	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ¬ 𝐴 ∈ ran ...)
105	12, 13, 14, 15, 84, 85, 86, 87, 70, 94, 103, 104	gsumval3a 18304	. . . . . 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 ))))))
106	105	eqeq1d 2624	. . . . 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))
107	102, 106	bitr4d 271	. . . 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))
108	107	alrimiv 1855	. . 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))
109		fvex 6201	. . . 4 ⊢ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) ∈ V
110		eqeq1 2626	. . . . . . 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 )))))
111	110	anbi2d 740	. . . . . 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 ))))))
112	111	exbidv 1850	. . . . 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 ))))))
113		eqeq2 2633	. . . . 5 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊)) → ((𝐺 Σ_g 𝐹) = 𝑥 ↔ (𝐺 Σ_g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(#‘𝑊))))
114	112, 113	bibi12d 335	. . . 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)))))
115	109, 114	spcv 3299	. . 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))))
116	108, 115	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))))
117	83, 116	mpbid 222	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 196 ∧ wa 384 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ ccom 5118 ℩cio 5849 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 supp csupp 7295 ≈ cen 7952 Fincfn 7955 1c1 9937 < clt 10074 ℕcn 11020 ...cfz 12326 seqcseq 12801 #chash 13117 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Σ_g cgsu 16101 Mndcmnd 17294 Cntzccntz 17748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-cntz 17750

This theorem is referenced by: gsumval3 18308

W3C validator