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Theorem gsumval3lem1 18230
Description: Lemma 1 for gsumval3 18232. (Contributed by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumval3.b 𝐵 = (Base‘𝐺)
gsumval3.0 0 = (0g𝐺)
gsumval3.p + = (+g𝐺)
gsumval3.z 𝑍 = (Cntz‘𝐺)
gsumval3.g (𝜑𝐺 ∈ Mnd)
gsumval3.a (𝜑𝐴𝑉)
gsumval3.f (𝜑𝐹:𝐴𝐵)
gsumval3.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3.m (𝜑𝑀 ∈ ℕ)
gsumval3.h (𝜑𝐻:(1...𝑀)–1-1𝐴)
gsumval3.n (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
gsumval3.w 𝑊 = ((𝐹𝐻) supp 0 )
Assertion
Ref Expression
gsumval3lem1 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑓):(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
StepHypRef Expression
1 gsumval3.h . . . . . . 7 (𝜑𝐻:(1...𝑀)–1-1𝐴)
21ad2antrr 761 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1𝐴)
3 gsumval3.w . . . . . . . . 9 𝑊 = ((𝐹𝐻) supp 0 )
4 suppssdm 7256 . . . . . . . . 9 ((𝐹𝐻) supp 0 ) ⊆ dom (𝐹𝐻)
53, 4eqsstri 3616 . . . . . . . 8 𝑊 ⊆ dom (𝐹𝐻)
6 gsumval3.f . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
7 f1f 6060 . . . . . . . . . . 11 (𝐻:(1...𝑀)–1-1𝐴𝐻:(1...𝑀)⟶𝐴)
81, 7syl 17 . . . . . . . . . 10 (𝜑𝐻:(1...𝑀)⟶𝐴)
9 fco 6017 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐻:(1...𝑀)⟶𝐴) → (𝐹𝐻):(1...𝑀)⟶𝐵)
106, 8, 9syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐹𝐻):(1...𝑀)⟶𝐵)
11 fdm 6010 . . . . . . . . 9 ((𝐹𝐻):(1...𝑀)⟶𝐵 → dom (𝐹𝐻) = (1...𝑀))
1210, 11syl 17 . . . . . . . 8 (𝜑 → dom (𝐹𝐻) = (1...𝑀))
135, 12syl5sseq 3634 . . . . . . 7 (𝜑𝑊 ⊆ (1...𝑀))
1413ad2antrr 761 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
15 f1ores 6110 . . . . . 6 ((𝐻:(1...𝑀)–1-1𝐴𝑊 ⊆ (1...𝑀)) → (𝐻𝑊):𝑊1-1-onto→(𝐻𝑊))
162, 14, 15syl2anc 692 . . . . 5 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊):𝑊1-1-onto→(𝐻𝑊))
173imaeq2i 5425 . . . . . . 7 (𝐻𝑊) = (𝐻 “ ((𝐹𝐻) supp 0 ))
18 gsumval3.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
19 fex 6447 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
206, 18, 19syl2anc 692 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
21 ovex 6635 . . . . . . . . . . . 12 (1...𝑀) ∈ V
22 fex 6447 . . . . . . . . . . . 12 ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
237, 21, 22sylancl 693 . . . . . . . . . . 11 (𝐻:(1...𝑀)–1-1𝐴𝐻 ∈ V)
241, 23syl 17 . . . . . . . . . 10 (𝜑𝐻 ∈ V)
25 f1fun 6062 . . . . . . . . . . . 12 (𝐻:(1...𝑀)–1-1𝐴 → Fun 𝐻)
261, 25syl 17 . . . . . . . . . . 11 (𝜑 → Fun 𝐻)
27 gsumval3.n . . . . . . . . . . 11 (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)
2826, 27jca 554 . . . . . . . . . 10 (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
2920, 24, 28jca31 556 . . . . . . . . 9 (𝜑 → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
3029ad2antrr 761 . . . . . . . 8 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
31 imacosupp 7283 . . . . . . . . 9 ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐻 “ ((𝐹𝐻) supp 0 )) = (𝐹 supp 0 )))
3231imp 445 . . . . . . . 8 (((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)) → (𝐻 “ ((𝐹𝐻) supp 0 )) = (𝐹 supp 0 ))
3330, 32syl 17 . . . . . . 7 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ ((𝐹𝐻) supp 0 )) = (𝐹 supp 0 ))
3417, 33syl5eq 2667 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊) = (𝐹 supp 0 ))
35 f1oeq3 6088 . . . . . 6 ((𝐻𝑊) = (𝐹 supp 0 ) → ((𝐻𝑊):𝑊1-1-onto→(𝐻𝑊) ↔ (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 )))
3634, 35syl 17 . . . . 5 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻𝑊):𝑊1-1-onto→(𝐻𝑊) ↔ (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 )))
3716, 36mpbid 222 . . . 4 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 ))
38 isof1o 6530 . . . . 5 (𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊) → 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
3938ad2antll 764 . . . 4 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
40 f1oco 6118 . . . 4 (((𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 ) ∧ 𝑓:(1...(#‘𝑊))–1-1-onto𝑊) → ((𝐻𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ))
4137, 39, 40syl2anc 692 . . 3 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ))
42 f1of 6096 . . . . 5 (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑓:(1...(#‘𝑊))⟶𝑊)
43 frn 6012 . . . . 5 (𝑓:(1...(#‘𝑊))⟶𝑊 → ran 𝑓𝑊)
4439, 42, 433syl 18 . . . 4 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ran 𝑓𝑊)
45 cores 5599 . . . 4 (ran 𝑓𝑊 → ((𝐻𝑊) ∘ 𝑓) = (𝐻𝑓))
46 f1oeq1 6086 . . . 4 (((𝐻𝑊) ∘ 𝑓) = (𝐻𝑓) → (((𝐻𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 )))
4744, 45, 463syl 18 . . 3 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (((𝐻𝑊) ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 )))
4841, 47mpbid 222 . 2 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ))
49 fzfi 12714 . . . . . . . . . 10 (1...𝑀) ∈ Fin
5049a1i 11 . . . . . . . . 9 (𝜑 → (1...𝑀) ∈ Fin)
51 fex2 7071 . . . . . . . . 9 ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin ∧ 𝐴𝑉) → 𝐻 ∈ V)
528, 50, 18, 51syl3anc 1323 . . . . . . . 8 (𝜑𝐻 ∈ V)
53 resexg 5403 . . . . . . . 8 (𝐻 ∈ V → (𝐻𝑊) ∈ V)
5452, 53syl 17 . . . . . . 7 (𝜑 → (𝐻𝑊) ∈ V)
5554ad2antrr 761 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊) ∈ V)
563a1i 11 . . . . . . . . . 10 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 = ((𝐹𝐻) supp 0 ))
5756imaeq2d 5427 . . . . . . . . 9 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊) = (𝐻 “ ((𝐹𝐻) supp 0 )))
5820, 52, 28jca31 556 . . . . . . . . . . 11 (𝜑 → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
5958ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
6059, 32syl 17 . . . . . . . . 9 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻 “ ((𝐹𝐻) supp 0 )) = (𝐹 supp 0 ))
6157, 60eqtrd 2655 . . . . . . . 8 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊) = (𝐹 supp 0 ))
6261, 35syl 17 . . . . . . 7 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻𝑊):𝑊1-1-onto→(𝐻𝑊) ↔ (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 )))
6316, 62mpbid 222 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 ))
64 f1oen3g 7918 . . . . . 6 (((𝐻𝑊) ∈ V ∧ (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 )) → 𝑊 ≈ (𝐹 supp 0 ))
6555, 63, 64syl2anc 692 . . . . 5 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ≈ (𝐹 supp 0 ))
66 ssfi 8127 . . . . . . . 8 (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
6749, 13, 66sylancr 694 . . . . . . 7 (𝜑𝑊 ∈ Fin)
6867ad2antrr 761 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
69 f1f1orn 6107 . . . . . . . . . . . 12 (𝐻:(1...𝑀)–1-1𝐴𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
701, 69syl 17 . . . . . . . . . . 11 (𝜑𝐻:(1...𝑀)–1-1-onto→ran 𝐻)
71 f1oen3g 7918 . . . . . . . . . . 11 ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
7252, 70, 71syl2anc 692 . . . . . . . . . 10 (𝜑 → (1...𝑀) ≈ ran 𝐻)
73 enfi 8123 . . . . . . . . . 10 ((1...𝑀) ≈ ran 𝐻 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
7472, 73syl 17 . . . . . . . . 9 (𝜑 → ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
7549, 74mpbii 223 . . . . . . . 8 (𝜑 → ran 𝐻 ∈ Fin)
76 ssfi 8127 . . . . . . . 8 ((ran 𝐻 ∈ Fin ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐹 supp 0 ) ∈ Fin)
7775, 27, 76syl2anc 692 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
7877ad2antrr 761 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
79 hashen 13078 . . . . . 6 ((𝑊 ∈ Fin ∧ (𝐹 supp 0 ) ∈ Fin) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
8068, 78, 79syl2anc 692 . . . . 5 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((#‘𝑊) = (#‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
8165, 80mpbird 247 . . . 4 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (#‘𝑊) = (#‘(𝐹 supp 0 )))
8281oveq2d 6623 . . 3 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (1...(#‘𝑊)) = (1...(#‘(𝐹 supp 0 ))))
83 f1oeq2 6087 . . 3 ((1...(#‘𝑊)) = (1...(#‘(𝐹 supp 0 ))) → ((𝐻𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))
8482, 83syl 17 . 2 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → ((𝐻𝑓):(1...(#‘𝑊))–1-1-onto→(𝐹 supp 0 ) ↔ (𝐻𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )))
8548, 84mpbid 222 1 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(#‘𝑊)), 𝑊))) → (𝐻𝑓):(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  wss 3556  c0 3893   class class class wbr 4615  dom cdm 5076  ran crn 5077  cres 5078  cima 5079  ccom 5080  Fun wfun 5843  wf 5845  1-1wf1 5846  1-1-ontowf1o 5848  cfv 5849   Isom wiso 5850  (class class class)co 6607   supp csupp 7243  cen 7899  Fincfn 7902  1c1 9884   < clt 10021  cn 10967  ...cfz 12271  #chash 13060  Basecbs 15784  +gcplusg 15865  0gc0g 16024  Mndcmnd 17218  Cntzccntz 17672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-hash 13061
This theorem is referenced by:  gsumval3lem2  18231
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