Theorem gsumval3lem1 18306

Description: Lemma 1 for gsumval3 18308. (Contributed by AV, 31-May-2019.)

Hypotheses

Ref	Expression
gsumval3.b	⊢ 𝐵 = (Base‘𝐺)
gsumval3.0	⊢ 0 = (0g‘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
gsumval3lem1	⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))

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 gsumval3lem1

Step	Hyp	Ref	Expression
1		gsumval3.h	. . . . . . 7 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)
2	1	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1→𝐴)
3		gsumval3.w	. . . . . . . . 9 ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )
4		suppssdm 7308	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) supp 0 ) ⊆ dom (𝐹 ∘ 𝐻)
5	3, 4	eqsstri 3635	. . . . . . . 8 ⊢ 𝑊 ⊆ dom (𝐹 ∘ 𝐻)
6		gsumval3.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7		f1f 6101	. . . . . . . . . . 11 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)⟶𝐴)
8	1, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:(1...𝑀)⟶𝐴)
9		fco 6058	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:(1...𝑀)⟶𝐴) → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
10	6, 8, 9	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵)
11		fdm 6051	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻):(1...𝑀)⟶𝐵 → dom (𝐹 ∘ 𝐻) = (1...𝑀))
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom (𝐹 ∘ 𝐻) = (1...𝑀))
13	5, 12	syl5sseq 3653	. . . . . . 7 ⊢ (𝜑 → 𝑊 ⊆ (1...𝑀))
14	13	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
15		f1ores 6151	. . . . . 6 ⊢ ((𝐻:(1...𝑀)–1-1→𝐴 ∧ 𝑊 ⊆ (1...𝑀)) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
16	2, 14, 15	syl2anc 693	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊))
17	3	imaeq2i 5464	. . . . . . 7 ⊢ (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 ))
18		gsumval3.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
19		fex 6490	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
20	6, 18, 19	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
21		ovex 6678	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ V
22		fex 6490	. . . . . . . . . . . 12 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
23	7, 21, 22	sylancl 694	. . . . . . . . . . 11 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻 ∈ V)
24	1, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ V)
25		f1fun 6103	. . . . . . . . . . . 12 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → Fun 𝐻)
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐻)
27		gsumval3.n	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
28	26, 27	jca 554	. . . . . . . . . 10 ⊢ (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
29	20, 24, 28	jca31 557	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
30	29	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
31		imacosupp 7335	. . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
32	31	imp 445	. . . . . . . 8 ⊢ (((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
33	30, 32	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
34	17, 33	syl5eq 2668	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
35		f1oeq3 6129	. . . . . 6 ⊢ ((𝐻 “ 𝑊) = (𝐹 supp 0 ) → ((𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐻 “ 𝑊) ↔ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )))
36	34, 35	syl 17	. . . . 5 ⊢ (((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 )))
37	16, 36	mpbid 222	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
38		isof1o 6573	. . . . 5 ⊢ (𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊) → 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊)
39	38	ad2antll 765	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊)
40		f1oco 6159	. . . 4 ⊢ (((𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ) ∧ 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊) → ((𝐻 ↾ 𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ))
41	37, 39, 40	syl2anc 693	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻 ↾ 𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ))
42		f1of 6137	. . . . 5 ⊢ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(#‘𝑊))⟶𝑊)
43		frn 6053	. . . . 5 ⊢ (𝑓:(1...(#‘𝑊))⟶𝑊 → ran 𝑓 ⊆ 𝑊)
44	39, 42, 43	3syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ran 𝑓 ⊆ 𝑊)
45		cores 5638	. . . 4 ⊢ (ran 𝑓 ⊆ 𝑊 → ((𝐻 ↾ 𝑊) ∘ 𝑓) = (𝐻 ∘ 𝑓))
46		f1oeq1 6127	. . . 4 ⊢ (((𝐻 ↾ 𝑊) ∘ 𝑓) = (𝐻 ∘ 𝑓) → (((𝐻 ↾ 𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 )))
47	44, 45, 46	3syl 18	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (((𝐻 ↾ 𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 )))
48	41, 47	mpbid 222	. 2 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ))
49		fzfi 12771	. . . . . . . . . 10 ⊢ (1...𝑀) ∈ Fin
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
51		fex2 7121	. . . . . . . . 9 ⊢ ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
52	8, 50, 18, 51	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ V)
53		resexg 5442	. . . . . . . 8 ⊢ (𝐻 ∈ V → (𝐻 ↾ 𝑊) ∈ V)
54	52, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ 𝑊) ∈ V)
55	54	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊) ∈ V)
56	3	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 = ((𝐹 ∘ 𝐻) supp 0 ))
57	56	imaeq2d 5466	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )))
58	20, 52, 28	jca31 557	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
59	58	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
60	59, 32	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
61	57, 60	eqtrd 2656	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ 𝑊) = (𝐹 supp 0 ))
62	61, 35	syl 17	. . . . . . 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 )))
63	16, 62	mpbid 222	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 ))
64		f1oen3g 7971	. . . . . 6 ⊢ (((𝐻 ↾ 𝑊) ∈ V ∧ (𝐻 ↾ 𝑊):𝑊–1-1-onto→(𝐹 supp 0 )) → 𝑊 ≈ (𝐹 supp 0 ))
65	55, 63, 64	syl2anc 693	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ≈ (𝐹 supp 0 ))
66		ssfi 8180	. . . . . . . 8 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
67	49, 13, 66	sylancr 695	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
68	67	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
69		f1f1orn 6148	. . . . . . . . . . . 12 ⊢ (𝐻:(1...𝑀)–1-1→𝐴 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
70	1, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
71		f1oen3g 7971	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
72	52, 70, 71	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ≈ ran 𝐻)
73		enfi 8176	. . . . . . . . . 10 ⊢ ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
74	72, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
75	49, 74	mpbii 223	. . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ∈ Fin)
76		ssfi 8180	. . . . . . . 8 ⊢ ((ran 𝐻 ∈ Fin ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐹 supp 0 ) ∈ Fin)
77	75, 27, 76	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
78	77	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
79		hashen 13135	. . . . . 6 ⊢ ((𝑊 ∈ Fin ∧ (𝐹 supp 0 ) ∈ Fin) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
80	68, 78, 79	syl2anc 693	. . . . 5 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
81	65, 80	mpbird 247	. . . 4 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (#‘𝑊) = (#‘(𝐹 supp 0 )))
82	81	oveq2d 6666	. . 3 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (1...(#‘𝑊)) = (1...(#‘(𝐹 supp 0 ))))
83		f1oeq2 6128	. . 3 ⊢ ((1...(#‘𝑊)) = (1...(#‘(𝐹 supp 0 ))) → ((𝐻 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))
84	82, 83	syl 17	. 2 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))
85	48, 84	mpbid 222	1 ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ 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  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  #chash 13117  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  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-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-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-hash 13118

This theorem is referenced by:  gsumval3lem2  18307