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Theorem gsumval3lem1 18513
 Description: Lemma 1 for gsumval3 18515. (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 705 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝐻:(1...𝑀)–1-1𝐴)
3 gsumval3.w . . . . . . . . 9 𝑊 = ((𝐹𝐻) supp 0 )
4 suppssdm 7459 . . . . . . . . 9 ((𝐹𝐻) supp 0 ) ⊆ dom (𝐹𝐻)
53, 4eqsstri 3784 . . . . . . . 8 𝑊 ⊆ dom (𝐹𝐻)
6 gsumval3.f . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
7 f1f 6241 . . . . . . . . . . 11 (𝐻:(1...𝑀)–1-1𝐴𝐻:(1...𝑀)⟶𝐴)
81, 7syl 17 . . . . . . . . . 10 (𝜑𝐻:(1...𝑀)⟶𝐴)
9 fco 6198 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐻:(1...𝑀)⟶𝐴) → (𝐹𝐻):(1...𝑀)⟶𝐵)
106, 8, 9syl2anc 573 . . . . . . . . 9 (𝜑 → (𝐹𝐻):(1...𝑀)⟶𝐵)
11 fdm 6191 . . . . . . . . 9 ((𝐹𝐻):(1...𝑀)⟶𝐵 → dom (𝐹𝐻) = (1...𝑀))
1210, 11syl 17 . . . . . . . 8 (𝜑 → dom (𝐹𝐻) = (1...𝑀))
135, 12syl5sseq 3802 . . . . . . 7 (𝜑𝑊 ⊆ (1...𝑀))
1413ad2antrr 705 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ⊆ (1...𝑀))
15 f1ores 6292 . . . . . 6 ((𝐻:(1...𝑀)–1-1𝐴𝑊 ⊆ (1...𝑀)) → (𝐻𝑊):𝑊1-1-onto→(𝐻𝑊))
162, 14, 15syl2anc 573 . . . . 5 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻𝑊):𝑊1-1-onto→(𝐻𝑊))
173imaeq2i 5605 . . . . . . 7 (𝐻𝑊) = (𝐻 “ ((𝐹𝐻) supp 0 ))
18 gsumval3.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
19 fex 6633 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
206, 18, 19syl2anc 573 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
21 ovex 6823 . . . . . . . . . . . 12 (1...𝑀) ∈ V
22 fex 6633 . . . . . . . . . . . 12 ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐻 ∈ V)
237, 21, 22sylancl 574 . . . . . . . . . . 11 (𝐻:(1...𝑀)–1-1𝐴𝐻 ∈ V)
241, 23syl 17 . . . . . . . . . 10 (𝜑𝐻 ∈ V)
25 f1fun 6243 . . . . . . . . . . . 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 501 . . . . . . . . . 10 (𝜑 → (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻))
2920, 24, 28jca31 504 . . . . . . . . 9 (𝜑 → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
3029ad2antrr 705 . . . . . . . 8 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
31 imacosupp 7487 . . . . . . . . 9 ((𝐹 ∈ V ∧ 𝐻 ∈ V) → ((Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐻 “ ((𝐹𝐻) supp 0 )) = (𝐹 supp 0 )))
3231imp 393 . . . . . . . 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 2817 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻𝑊) = (𝐹 supp 0 ))
35 f1oeq3 6270 . . . . . 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 6716 . . . . 5 (𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊) → 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
3938ad2antll 708 . . . 4 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
40 f1oco 6300 . . . 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 573 . . 3 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ((𝐻𝑊) ∘ 𝑓):(1...(♯‘𝑊))–1-1-onto→(𝐹 supp 0 ))
42 f1of 6278 . . . . 5 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))⟶𝑊)
43 frn 6193 . . . . 5 (𝑓:(1...(♯‘𝑊))⟶𝑊 → ran 𝑓𝑊)
4439, 42, 433syl 18 . . . 4 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → ran 𝑓𝑊)
45 cores 5782 . . . 4 (ran 𝑓𝑊 → ((𝐻𝑊) ∘ 𝑓) = (𝐻𝑓))
46 f1oeq1 6268 . . . 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 12979 . . . . . . . . . 10 (1...𝑀) ∈ Fin
5049a1i 11 . . . . . . . . 9 (𝜑 → (1...𝑀) ∈ Fin)
51 fex2 7268 . . . . . . . . 9 ((𝐻:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ Fin ∧ 𝐴𝑉) → 𝐻 ∈ V)
528, 50, 18, 51syl3anc 1476 . . . . . . . 8 (𝜑𝐻 ∈ V)
53 resexg 5583 . . . . . . . 8 (𝐻 ∈ V → (𝐻𝑊) ∈ V)
5452, 53syl 17 . . . . . . 7 (𝜑 → (𝐻𝑊) ∈ V)
5554ad2antrr 705 . . . . . 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 5607 . . . . . . . . 9 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻𝑊) = (𝐻 “ ((𝐹𝐻) supp 0 )))
5820, 52, 28jca31 504 . . . . . . . . . . 11 (𝜑 → ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) ⊆ ran 𝐻)))
5958ad2antrr 705 . . . . . . . . . 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 2805 . . . . . . . 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 8125 . . . . . 6 (((𝐻𝑊) ∈ V ∧ (𝐻𝑊):𝑊1-1-onto→(𝐹 supp 0 )) → 𝑊 ≈ (𝐹 supp 0 ))
6555, 63, 64syl2anc 573 . . . . 5 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ≈ (𝐹 supp 0 ))
66 ssfi 8336 . . . . . . . 8 (((1...𝑀) ∈ Fin ∧ 𝑊 ⊆ (1...𝑀)) → 𝑊 ∈ Fin)
6749, 13, 66sylancr 575 . . . . . . 7 (𝜑𝑊 ∈ Fin)
6867ad2antrr 705 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → 𝑊 ∈ Fin)
69 f1f1orn 6289 . . . . . . . . . . . 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 8125 . . . . . . . . . . 11 ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-onto→ran 𝐻) → (1...𝑀) ≈ ran 𝐻)
7252, 70, 71syl2anc 573 . . . . . . . . . 10 (𝜑 → (1...𝑀) ≈ ran 𝐻)
73 enfi 8332 . . . . . . . . . 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 8336 . . . . . . . 8 ((ran 𝐻 ∈ Fin ∧ (𝐹 supp 0 ) ⊆ ran 𝐻) → (𝐹 supp 0 ) ∈ Fin)
7775, 27, 76syl2anc 573 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
7877ad2antrr 705 . . . . . 6 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐹 supp 0 ) ∈ Fin)
79 hashen 13339 . . . . . 6 ((𝑊 ∈ Fin ∧ (𝐹 supp 0 ) ∈ Fin) → ((♯‘𝑊) = (♯‘(𝐹 supp 0 )) ↔ 𝑊 ≈ (𝐹 supp 0 )))
8068, 78, 79syl2anc 573 . . . . 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 6809 . . 3 (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (1...(♯‘𝑊)) = (1...(♯‘(𝐹 supp 0 ))))
83 f1oeq2 6269 . . 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 382   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  Vcvv 3351   ⊆ wss 3723  ∅c0 4063   class class class wbr 4786  dom cdm 5249  ran crn 5250   ↾ cres 5251   “ cima 5252   ∘ ccom 5253  Fun wfun 6025  ⟶wf 6027  –1-1→wf1 6028  –1-1-onto→wf1o 6030  ‘cfv 6031   Isom wiso 6032  (class class class)co 6793   supp csupp 7446   ≈ cen 8106  Fincfn 8109  1c1 10139   < clt 10276  ℕcn 11222  ...cfz 12533  ♯chash 13321  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502  Cntzccntz 17955 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-hash 13322 This theorem is referenced by:  gsumval3lem2  18514
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