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Theorem gsumfzval 12964
Description: An expression for Σg when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)
Hypotheses
Ref Expression
gsumval.b 𝐵 = (Base‘𝐺)
gsumval.z 0 = (0g𝐺)
gsumval.p + = (+g𝐺)
gsumval.g (𝜑𝐺𝑉)
gsumfzval.m (𝜑𝑀 ∈ ℤ)
gsumfzval.n (𝜑𝑁 ∈ ℤ)
gsumfzval.f (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
Assertion
Ref Expression
gsumfzval (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))

Proof of Theorem gsumfzval
Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval.b . . 3 𝐵 = (Base‘𝐺)
2 gsumval.z . . 3 0 = (0g𝐺)
3 gsumval.p . . 3 + = (+g𝐺)
4 gsumval.g . . 3 (𝜑𝐺𝑉)
5 gsumfzval.m . . . 4 (𝜑𝑀 ∈ ℤ)
6 gsumfzval.n . . . 4 (𝜑𝑁 ∈ ℤ)
75, 6fzfigd 10492 . . 3 (𝜑 → (𝑀...𝑁) ∈ Fin)
8 gsumfzval.f . . 3 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
91, 2, 3, 4, 7, 8igsumval 12963 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
10 fn0g 12948 . . . . . 6 0g Fn V
114elexd 2773 . . . . . 6 (𝜑𝐺 ∈ V)
12 funfvex 5563 . . . . . . 7 ((Fun 0g𝐺 ∈ dom 0g) → (0g𝐺) ∈ V)
1312funfni 5346 . . . . . 6 ((0g Fn V ∧ 𝐺 ∈ V) → (0g𝐺) ∈ V)
1410, 11, 13sylancr 414 . . . . 5 (𝜑 → (0g𝐺) ∈ V)
152, 14eqeltrid 2280 . . . 4 (𝜑0 ∈ V)
16 seqex 10510 . . . . 5 seq𝑀( + , 𝐹) ∈ V
17 fvexg 5565 . . . . 5 ((seq𝑀( + , 𝐹) ∈ V ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
1816, 6, 17sylancr 414 . . . 4 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
1915, 18ifexd 4513 . . 3 (𝜑 → if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V)
20 zdclt 9384 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
216, 5, 20syl2anc 411 . . . . . . 7 (𝜑DECID 𝑁 < 𝑀)
22 eqifdc 3592 . . . . . . 7 (DECID 𝑁 < 𝑀 → (𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ↔ ((𝑁 < 𝑀𝑥 = 0 ) ∨ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))))
2321, 22syl 14 . . . . . 6 (𝜑 → (𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ↔ ((𝑁 < 𝑀𝑥 = 0 ) ∨ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))))
24 fzn 10098 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
255, 6, 24syl2anc 411 . . . . . . . 8 (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
2625anbi1d 465 . . . . . . 7 (𝜑 → ((𝑁 < 𝑀𝑥 = 0 ) ↔ ((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 )))
275adantr 276 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ∈ ℤ)
2827zred 9429 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ∈ ℝ)
296adantr 276 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ ℤ)
3029zred 9429 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ ℝ)
31 simprl 529 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ¬ 𝑁 < 𝑀)
3228, 30, 31nltled 8130 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀𝑁)
33 eluz 9595 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
3427, 29, 33syl2anc 411 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
3532, 34mpbird 167 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ (ℤ𝑀))
36 oveq2 5918 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
3736eqeq2d 2205 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
38 fveq2 5546 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
3938eqeq2d 2205 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
4037, 39anbi12d 473 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
4140adantl 277 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) ∧ 𝑛 = 𝑁) → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
42 eqidd 2194 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → (𝑀...𝑁) = (𝑀...𝑁))
43 simprr 531 . . . . . . . . . . . 12 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
4442, 43jca 306 . . . . . . . . . . 11 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
4535, 41, 44rspcedvd 2870 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
46 fveq2 5546 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
47 oveq1 5917 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
4847eqeq2d 2205 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
49 seqeq1 10511 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
5049fveq1d 5548 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
5150eqeq2d 2205 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
5248, 51anbi12d 473 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
5346, 52rexeqbidv 2707 . . . . . . . . . . 11 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
5453spcegv 2848 . . . . . . . . . 10 (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
5527, 45, 54sylc 62 . . . . . . . . 9 ((𝜑 ∧ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
5655ex 115 . . . . . . . 8 (𝜑 → ((¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
57 eluzel2 9587 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑚) → 𝑚 ∈ ℤ)
5857ad2antlr 489 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℤ)
5958zred 9429 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℝ)
60 eluzelre 9592 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑚) → 𝑛 ∈ ℝ)
6160ad2antlr 489 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑛 ∈ ℝ)
62 eluzle 9594 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑚) → 𝑚𝑛)
6362ad2antlr 489 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚𝑛)
6459, 61, 63lensymd 8131 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ¬ 𝑛 < 𝑚)
65 simprl 529 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
6665eqcomd 2199 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚...𝑛) = (𝑀...𝑁))
67 fzopth 10117 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝑚) → ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑚 = 𝑀𝑛 = 𝑁)))
6867ad2antlr 489 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑚 = 𝑀𝑛 = 𝑁)))
6966, 68mpbid 147 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚 = 𝑀𝑛 = 𝑁))
7069simprd 114 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑛 = 𝑁)
7169simpld 112 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 = 𝑀)
7270, 71breq12d 4042 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑛 < 𝑚𝑁 < 𝑀))
7364, 72mtbid 673 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ¬ 𝑁 < 𝑀)
74 simprr 531 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
7571seqeq1d 10514 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
7675, 70fveq12d 5553 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
7774, 76eqtrd 2226 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
7873, 77jca 306 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
7978rexlimdva2 2614 . . . . . . . . 9 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
8079exlimdv 1830 . . . . . . . 8 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
8156, 80impbid 129 . . . . . . 7 (𝜑 → ((¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
8226, 81orbi12d 794 . . . . . 6 (𝜑 → (((𝑁 < 𝑀𝑥 = 0 ) ∨ (¬ 𝑁 < 𝑀𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
8323, 82bitr2d 189 . . . . 5 (𝜑 → ((((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) ↔ 𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁))))
8483adantr 276 . . . 4 ((𝜑 ∧ if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V) → ((((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) ↔ 𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁))))
8584iota5 5228 . . 3 ((𝜑 ∧ if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V) → (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
8619, 85mpdan 421 . 2 (𝜑 → (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
879, 86eqtrd 2226 1 (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1364  wex 1503  wcel 2164  wrex 2473  Vcvv 2760  c0 3446  ifcif 3557   class class class wbr 4029  cio 5205   Fn wfn 5241  wf 5242  cfv 5246  (class class class)co 5910  Fincfn 6785  cr 7861   < clt 8044  cle 8045  cz 9307  cuz 9582  ...cfz 10064  seqcseq 10508  Basecbs 12608  +gcplusg 12685  0gc0g 12857   Σg cgsu 12858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-frec 6435  df-1o 6460  df-er 6578  df-en 6786  df-fin 6788  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-seqfrec 10509  df-ndx 12611  df-slot 12612  df-base 12614  df-0g 12859  df-igsum 12860
This theorem is referenced by:  gsumfzz  13057  gsumfzcl  13061  gsumfzreidx  13396  gsumfzsubmcl  13397  gsumfzmptfidmadd  13398
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