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Theorem isummolem2a 10735
 Description: Lemma for isummo 10737. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 3-Sep-2022.)
Hypotheses
Ref Expression
isummo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
isummo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
isummolem2a.dc ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)
isummolem2a.g 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
isummolem2a.h 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))
summolem2.5 (𝜑𝑁 ∈ ℕ)
summolem2.6 (𝜑𝑀 ∈ ℤ)
summolem2.7 (𝜑𝐴 ⊆ (ℤ𝑀))
summolem2.8 (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)
summolem2.9 (𝜑𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))
Assertion
Ref Expression
isummolem2a (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑁))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝑘,𝑀,𝑛   𝐵,𝑛   𝑘,𝐹   𝑘,𝐾,𝑛   𝑓,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑓,𝑘)   𝐹(𝑓)   𝐺(𝑓,𝑘,𝑛)   𝐻(𝑓,𝑘,𝑛)   𝐾(𝑓)   𝑀(𝑓)   𝑁(𝑓)

Proof of Theorem isummolem2a
Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isummo.1 . . 3 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
2 isummo.2 . . 3 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3 isummolem2a.dc . . 3 ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)
4 summolem2.7 . . . 4 (𝜑𝐴 ⊆ (ℤ𝑀))
5 summolem2.9 . . . . . . . 8 (𝜑𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))
6 1zzd 8747 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
7 summolem2.5 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℕ)
87nnzd 8837 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℤ)
96, 8fzfigd 9803 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) ∈ Fin)
10 summolem2.8 . . . . . . . . . . . 12 (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)
119, 10fihasheqf1od 10163 . . . . . . . . . . 11 (𝜑 → (♯‘(1...𝑁)) = (♯‘𝐴))
12 nnnn0 8650 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
13 hashfz1 10156 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
147, 12, 133syl 17 . . . . . . . . . . 11 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
1511, 14eqtr3d 2122 . . . . . . . . . 10 (𝜑 → (♯‘𝐴) = 𝑁)
1615oveq2d 5650 . . . . . . . . 9 (𝜑 → (1...(♯‘𝐴)) = (1...𝑁))
17 isoeq4 5565 . . . . . . . . 9 ((1...(♯‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
1816, 17syl 14 . . . . . . . 8 (𝜑 → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
195, 18mpbid 145 . . . . . . 7 (𝜑𝐾 Isom < , < ((1...𝑁), 𝐴))
20 isof1o 5568 . . . . . . 7 (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto𝐴)
2119, 20syl 14 . . . . . 6 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
22 f1of 5237 . . . . . 6 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
2321, 22syl 14 . . . . 5 (𝜑𝐾:(1...𝑁)⟶𝐴)
24 nnuz 9023 . . . . . . 7 ℕ = (ℤ‘1)
257, 24syl6eleq 2180 . . . . . 6 (𝜑𝑁 ∈ (ℤ‘1))
26 eluzfz2 9415 . . . . . 6 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
2725, 26syl 14 . . . . 5 (𝜑𝑁 ∈ (1...𝑁))
2823, 27ffvelrnd 5419 . . . 4 (𝜑 → (𝐾𝑁) ∈ 𝐴)
294, 28sseldd 3024 . . 3 (𝜑 → (𝐾𝑁) ∈ (ℤ𝑀))
304sselda 3023 . . . . . 6 ((𝜑𝑛𝐴) → 𝑛 ∈ (ℤ𝑀))
31 f1ocnvfv2 5539 . . . . . . . . 9 ((𝐾:(1...𝑁)–1-1-onto𝐴𝑛𝐴) → (𝐾‘(𝐾𝑛)) = 𝑛)
3221, 31sylan 277 . . . . . . . 8 ((𝜑𝑛𝐴) → (𝐾‘(𝐾𝑛)) = 𝑛)
33 f1ocnv 5250 . . . . . . . . . . . 12 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:𝐴1-1-onto→(1...𝑁))
34 f1of 5237 . . . . . . . . . . . 12 (𝐾:𝐴1-1-onto→(1...𝑁) → 𝐾:𝐴⟶(1...𝑁))
3521, 33, 343syl 17 . . . . . . . . . . 11 (𝜑𝐾:𝐴⟶(1...𝑁))
3635ffvelrnda 5418 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (𝐾𝑛) ∈ (1...𝑁))
37 elfzle2 9411 . . . . . . . . . 10 ((𝐾𝑛) ∈ (1...𝑁) → (𝐾𝑛) ≤ 𝑁)
3836, 37syl 14 . . . . . . . . 9 ((𝜑𝑛𝐴) → (𝐾𝑛) ≤ 𝑁)
3919adantr 270 . . . . . . . . . 10 ((𝜑𝑛𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
40 fzssuz 9447 . . . . . . . . . . . . 13 (1...𝑁) ⊆ (ℤ‘1)
41 uzssz 9007 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
42 zssre 8727 . . . . . . . . . . . . . 14 ℤ ⊆ ℝ
4341, 42sstri 3032 . . . . . . . . . . . . 13 (ℤ‘1) ⊆ ℝ
4440, 43sstri 3032 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℝ
45 ressxr 7510 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
4644, 45sstri 3032 . . . . . . . . . . 11 (1...𝑁) ⊆ ℝ*
4746a1i 9 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (1...𝑁) ⊆ ℝ*)
484adantr 270 . . . . . . . . . . . 12 ((𝜑𝑛𝐴) → 𝐴 ⊆ (ℤ𝑀))
49 uzssz 9007 . . . . . . . . . . . . 13 (ℤ𝑀) ⊆ ℤ
5049, 42sstri 3032 . . . . . . . . . . . 12 (ℤ𝑀) ⊆ ℝ
5148, 50syl6ss 3035 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → 𝐴 ⊆ ℝ)
5251, 45syl6ss 3035 . . . . . . . . . 10 ((𝜑𝑛𝐴) → 𝐴 ⊆ ℝ*)
5327adantr 270 . . . . . . . . . 10 ((𝜑𝑛𝐴) → 𝑁 ∈ (1...𝑁))
54 leisorel 10207 . . . . . . . . . 10 ((𝐾 Isom < , < ((1...𝑁), 𝐴) ∧ ((1...𝑁) ⊆ ℝ*𝐴 ⊆ ℝ*) ∧ ((𝐾𝑛) ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → ((𝐾𝑛) ≤ 𝑁 ↔ (𝐾‘(𝐾𝑛)) ≤ (𝐾𝑁)))
5539, 47, 52, 36, 53, 54syl122anc 1183 . . . . . . . . 9 ((𝜑𝑛𝐴) → ((𝐾𝑛) ≤ 𝑁 ↔ (𝐾‘(𝐾𝑛)) ≤ (𝐾𝑁)))
5638, 55mpbid 145 . . . . . . . 8 ((𝜑𝑛𝐴) → (𝐾‘(𝐾𝑛)) ≤ (𝐾𝑁))
5732, 56eqbrtrrd 3859 . . . . . . 7 ((𝜑𝑛𝐴) → 𝑛 ≤ (𝐾𝑁))
58 eluzelz 8997 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
5930, 58syl 14 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝑛 ∈ ℤ)
60 eluzelz 8997 . . . . . . . . . 10 ((𝐾𝑁) ∈ (ℤ𝑀) → (𝐾𝑁) ∈ ℤ)
6129, 60syl 14 . . . . . . . . 9 (𝜑 → (𝐾𝑁) ∈ ℤ)
6261adantr 270 . . . . . . . 8 ((𝜑𝑛𝐴) → (𝐾𝑁) ∈ ℤ)
63 eluz 9001 . . . . . . . 8 ((𝑛 ∈ ℤ ∧ (𝐾𝑁) ∈ ℤ) → ((𝐾𝑁) ∈ (ℤ𝑛) ↔ 𝑛 ≤ (𝐾𝑁)))
6459, 62, 63syl2anc 403 . . . . . . 7 ((𝜑𝑛𝐴) → ((𝐾𝑁) ∈ (ℤ𝑛) ↔ 𝑛 ≤ (𝐾𝑁)))
6557, 64mpbird 165 . . . . . 6 ((𝜑𝑛𝐴) → (𝐾𝑁) ∈ (ℤ𝑛))
66 elfzuzb 9403 . . . . . 6 (𝑛 ∈ (𝑀...(𝐾𝑁)) ↔ (𝑛 ∈ (ℤ𝑀) ∧ (𝐾𝑁) ∈ (ℤ𝑛)))
6730, 65, 66sylanbrc 408 . . . . 5 ((𝜑𝑛𝐴) → 𝑛 ∈ (𝑀...(𝐾𝑁)))
6867ex 113 . . . 4 (𝜑 → (𝑛𝐴𝑛 ∈ (𝑀...(𝐾𝑁))))
6968ssrdv 3029 . . 3 (𝜑𝐴 ⊆ (𝑀...(𝐾𝑁)))
701, 2, 3, 29, 69fisumcvg 10730 . 2 (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (seq𝑀( + , 𝐹, ℂ)‘(𝐾𝑁)))
71 addid2 7600 . . . . 5 (𝑚 ∈ ℂ → (0 + 𝑚) = 𝑚)
7271adantl 271 . . . 4 ((𝜑𝑚 ∈ ℂ) → (0 + 𝑚) = 𝑚)
73 addid1 7599 . . . . 5 (𝑚 ∈ ℂ → (𝑚 + 0) = 𝑚)
7473adantl 271 . . . 4 ((𝜑𝑚 ∈ ℂ) → (𝑚 + 0) = 𝑚)
75 addcl 7446 . . . . 5 ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 + 𝑥) ∈ ℂ)
7675adantl 271 . . . 4 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 + 𝑥) ∈ ℂ)
77 0cnd 7460 . . . 4 (𝜑 → 0 ∈ ℂ)
7827, 16eleqtrrd 2167 . . . 4 (𝜑𝑁 ∈ (1...(♯‘𝐴)))
79 iftrue 3394 . . . . . . . . . . . 12 (𝑘𝐴 → if(𝑘𝐴, 𝐵, 0) = 𝐵)
8079adantl 271 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 0) = 𝐵)
8180, 2eqeltrd 2164 . . . . . . . . . 10 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
8281adantlr 461 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ 𝑘𝐴) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
8382adantlr 461 . . . . . . . 8 ((((𝜑𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) ∧ 𝑘𝐴) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
84 iffalse 3397 . . . . . . . . . 10 𝑘𝐴 → if(𝑘𝐴, 𝐵, 0) = 0)
85 0cn 7459 . . . . . . . . . 10 0 ∈ ℂ
8684, 85syl6eqel 2178 . . . . . . . . 9 𝑘𝐴 → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
8786adantl 271 . . . . . . . 8 ((((𝜑𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) ∧ ¬ 𝑘𝐴) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
883adantlr 461 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)
89 exmiddc 782 . . . . . . . . 9 (DECID 𝑘𝐴 → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
9088, 89syl 14 . . . . . . . 8 (((𝜑𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
9183, 87, 90mpjaodan 747 . . . . . . 7 (((𝜑𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
92 simpll 496 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ𝑀)) → 𝜑)
93 simpr 108 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ𝑀)) → ¬ 𝑘 ∈ (ℤ𝑀))
944ssneld 3025 . . . . . . . . 9 (𝜑 → (¬ 𝑘 ∈ (ℤ𝑀) → ¬ 𝑘𝐴))
9592, 93, 94sylc 61 . . . . . . . 8 (((𝜑𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ𝑀)) → ¬ 𝑘𝐴)
9695, 86syl 14 . . . . . . 7 (((𝜑𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ𝑀)) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
97 summolem2.6 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
98 eluzdc 9066 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → DECID 𝑘 ∈ (ℤ𝑀))
9997, 98sylan 277 . . . . . . . 8 ((𝜑𝑘 ∈ ℤ) → DECID 𝑘 ∈ (ℤ𝑀))
100 exmiddc 782 . . . . . . . 8 (DECID 𝑘 ∈ (ℤ𝑀) → (𝑘 ∈ (ℤ𝑀) ∨ ¬ 𝑘 ∈ (ℤ𝑀)))
10199, 100syl 14 . . . . . . 7 ((𝜑𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ𝑀) ∨ ¬ 𝑘 ∈ (ℤ𝑀)))
10291, 96, 101mpjaodan 747 . . . . . 6 ((𝜑𝑘 ∈ ℤ) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
103102, 1fmptd 5436 . . . . 5 (𝜑𝐹:ℤ⟶ℂ)
104 eluzelz 8997 . . . . 5 (𝑚 ∈ (ℤ𝑀) → 𝑚 ∈ ℤ)
105 ffvelrn 5416 . . . . 5 ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹𝑚) ∈ ℂ)
106103, 104, 105syl2an 283 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑀)) → (𝐹𝑚) ∈ ℂ)
107 elnnuz 9024 . . . . . . . 8 (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ‘1))
108107biimpri 131 . . . . . . 7 (𝑚 ∈ (ℤ‘1) → 𝑚 ∈ ℕ)
109108adantl 271 . . . . . 6 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑚 ∈ ℕ)
110 isof1o 5568 . . . . . . . . . . . 12 (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐾:(1...(♯‘𝐴))–1-1-onto𝐴)
111 f1of 5237 . . . . . . . . . . . 12 (𝐾:(1...(♯‘𝐴))–1-1-onto𝐴𝐾:(1...(♯‘𝐴))⟶𝐴)
1125, 110, 1113syl 17 . . . . . . . . . . 11 (𝜑𝐾:(1...(♯‘𝐴))⟶𝐴)
113112ad2antrr 472 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝐾:(1...(♯‘𝐴))⟶𝐴)
114 1zzd 8747 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 1 ∈ ℤ)
11515, 8eqeltrd 2164 . . . . . . . . . . . . 13 (𝜑 → (♯‘𝐴) ∈ ℤ)
116115ad2antrr 472 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (♯‘𝐴) ∈ ℤ)
117 eluzelz 8997 . . . . . . . . . . . . 13 (𝑚 ∈ (ℤ‘1) → 𝑚 ∈ ℤ)
118117ad2antlr 473 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝑚 ∈ ℤ)
119114, 116, 1183jca 1123 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑚 ∈ ℤ))
120 eluzle 9000 . . . . . . . . . . . . 13 (𝑚 ∈ (ℤ‘1) → 1 ≤ 𝑚)
121120ad2antlr 473 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 1 ≤ 𝑚)
122 simpr 108 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝑚𝑁)
12315breq2d 3849 . . . . . . . . . . . . . 14 (𝜑 → (𝑚 ≤ (♯‘𝐴) ↔ 𝑚𝑁))
124123ad2antrr 472 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝑚 ≤ (♯‘𝐴) ↔ 𝑚𝑁))
125122, 124mpbird 165 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝑚 ≤ (♯‘𝐴))
126121, 125jca 300 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (1 ≤ 𝑚𝑚 ≤ (♯‘𝐴)))
127 elfz2 9400 . . . . . . . . . . 11 (𝑚 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (1 ≤ 𝑚𝑚 ≤ (♯‘𝐴))))
128119, 126, 127sylanbrc 408 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝑚 ∈ (1...(♯‘𝐴)))
129113, 128ffvelrnd 5419 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝐾𝑚) ∈ 𝐴)
130129iftrued 3396 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0) = (𝐾𝑚) / 𝑘𝐵)
1314ad2antrr 472 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝐴 ⊆ (ℤ𝑀))
13223ad2antrr 472 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝐾:(1...𝑁)⟶𝐴)
13316eleq2d 2157 . . . . . . . . . . . . . 14 (𝜑 → (𝑚 ∈ (1...(♯‘𝐴)) ↔ 𝑚 ∈ (1...𝑁)))
134133ad2antrr 472 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝑚 ∈ (1...(♯‘𝐴)) ↔ 𝑚 ∈ (1...𝑁)))
135128, 134mpbid 145 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → 𝑚 ∈ (1...𝑁))
136132, 135ffvelrnd 5419 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝐾𝑚) ∈ 𝐴)
137131, 136sseldd 3024 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝐾𝑚) ∈ (ℤ𝑀))
13849, 137sseldi 3021 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝐾𝑚) ∈ ℤ)
139102ralrimiva 2446 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℤ if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
140139ad2antrr 472 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → ∀𝑘 ∈ ℤ if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
141 nfv 1466 . . . . . . . . . . . 12 𝑘(𝐾𝑚) ∈ 𝐴
142 nfcsb1v 2961 . . . . . . . . . . . 12 𝑘(𝐾𝑚) / 𝑘𝐵
143 nfcv 2228 . . . . . . . . . . . 12 𝑘0
144141, 142, 143nfif 3415 . . . . . . . . . . 11 𝑘if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0)
145144nfel1 2239 . . . . . . . . . 10 𝑘if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ
146 eleq1 2150 . . . . . . . . . . . 12 (𝑘 = (𝐾𝑚) → (𝑘𝐴 ↔ (𝐾𝑚) ∈ 𝐴))
147 csbeq1a 2939 . . . . . . . . . . . 12 (𝑘 = (𝐾𝑚) → 𝐵 = (𝐾𝑚) / 𝑘𝐵)
148146, 147ifbieq1d 3409 . . . . . . . . . . 11 (𝑘 = (𝐾𝑚) → if(𝑘𝐴, 𝐵, 0) = if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0))
149148eleq1d 2156 . . . . . . . . . 10 (𝑘 = (𝐾𝑚) → (if(𝑘𝐴, 𝐵, 0) ∈ ℂ ↔ if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ))
150145, 149rspc 2716 . . . . . . . . 9 ((𝐾𝑚) ∈ ℤ → (∀𝑘 ∈ ℤ if(𝑘𝐴, 𝐵, 0) ∈ ℂ → if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ))
151138, 140, 150sylc 61 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → if((𝐾𝑚) ∈ 𝐴, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
152130, 151eqeltrrd 2165 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚𝑁) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
153 0cnd 7460 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ ¬ 𝑚𝑁) → 0 ∈ ℂ)
154109nnzd 8837 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑚 ∈ ℤ)
1558adantr 270 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑁 ∈ ℤ)
156 zdcle 8793 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑚𝑁)
157154, 155, 156syl2anc 403 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ‘1)) → DECID 𝑚𝑁)
158152, 153, 157ifcldadc 3416 . . . . . 6 ((𝜑𝑚 ∈ (ℤ‘1)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
159 breq1 3840 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛𝑁𝑚𝑁))
160 fveq2 5289 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐾𝑛) = (𝐾𝑚))
161160csbeq1d 2937 . . . . . . . 8 (𝑛 = 𝑚(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑚) / 𝑘𝐵)
162159, 161ifbieq1d 3409 . . . . . . 7 (𝑛 = 𝑚 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
163 isummolem2a.h . . . . . . 7 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))
164162, 163fvmptg 5364 . . . . . 6 ((𝑚 ∈ ℕ ∧ if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
165109, 158, 164syl2anc 403 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
166165, 158eqeltrd 2164 . . . 4 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) ∈ ℂ)
167 fveqeq2 5298 . . . . . 6 (𝑘 = 𝑚 → ((𝐹𝑘) = 0 ↔ (𝐹𝑚) = 0))
168 eldifi 3120 . . . . . . . . 9 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))))
169 elfzelz 9409 . . . . . . . . 9 (𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))) → 𝑘 ∈ ℤ)
170168, 169syl 14 . . . . . . . 8 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
171 eldifn 3121 . . . . . . . . . 10 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → ¬ 𝑘𝐴)
172171, 84syl 14 . . . . . . . . 9 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘𝐴, 𝐵, 0) = 0)
173172, 85syl6eqel 2178 . . . . . . . 8 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
1741fvmpt2 5370 . . . . . . . 8 ((𝑘 ∈ ℤ ∧ if(𝑘𝐴, 𝐵, 0) ∈ ℂ) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
175170, 173, 174syl2anc 403 . . . . . . 7 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
176175, 172eqtrd 2120 . . . . . 6 (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹𝑘) = 0)
177167, 176vtoclga 2685 . . . . 5 (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹𝑚) = 0)
178177adantl 271 . . . 4 ((𝜑𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹𝑚) = 0)
179112ffvelrnda 5418 . . . . . 6 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐾𝑥) ∈ 𝐴)
180179iftrued 3396 . . . . 5 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) = (𝐾𝑥) / 𝑘𝐵)
1814adantr 270 . . . . . . . 8 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ (ℤ𝑀))
182181, 179sseldd 3024 . . . . . . 7 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐾𝑥) ∈ (ℤ𝑀))
183 eluzelz 8997 . . . . . . 7 ((𝐾𝑥) ∈ (ℤ𝑀) → (𝐾𝑥) ∈ ℤ)
184182, 183syl 14 . . . . . 6 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐾𝑥) ∈ ℤ)
185 simpl 107 . . . . . . . 8 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → 𝜑)
186185, 184jca 300 . . . . . . 7 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝜑 ∧ (𝐾𝑥) ∈ ℤ))
187 nfcv 2228 . . . . . . . 8 𝑘(𝐾𝑥)
188 nfv 1466 . . . . . . . . 9 𝑘(𝜑 ∧ (𝐾𝑥) ∈ ℤ)
189 nfv 1466 . . . . . . . . . . 11 𝑘(𝐾𝑥) ∈ 𝐴
190 nfcsb1v 2961 . . . . . . . . . . 11 𝑘(𝐾𝑥) / 𝑘𝐵
191189, 190, 143nfif 3415 . . . . . . . . . 10 𝑘if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0)
192191nfel1 2239 . . . . . . . . 9 𝑘if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ
193188, 192nfim 1509 . . . . . . . 8 𝑘((𝜑 ∧ (𝐾𝑥) ∈ ℤ) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ)
194 eleq1 2150 . . . . . . . . . 10 (𝑘 = (𝐾𝑥) → (𝑘 ∈ ℤ ↔ (𝐾𝑥) ∈ ℤ))
195194anbi2d 452 . . . . . . . . 9 (𝑘 = (𝐾𝑥) → ((𝜑𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝐾𝑥) ∈ ℤ)))
196 eleq1 2150 . . . . . . . . . . 11 (𝑘 = (𝐾𝑥) → (𝑘𝐴 ↔ (𝐾𝑥) ∈ 𝐴))
197 csbeq1a 2939 . . . . . . . . . . 11 (𝑘 = (𝐾𝑥) → 𝐵 = (𝐾𝑥) / 𝑘𝐵)
198196, 197ifbieq1d 3409 . . . . . . . . . 10 (𝑘 = (𝐾𝑥) → if(𝑘𝐴, 𝐵, 0) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0))
199198eleq1d 2156 . . . . . . . . 9 (𝑘 = (𝐾𝑥) → (if(𝑘𝐴, 𝐵, 0) ∈ ℂ ↔ if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ))
200195, 199imbi12d 232 . . . . . . . 8 (𝑘 = (𝐾𝑥) → (((𝜑𝑘 ∈ ℤ) → if(𝑘𝐴, 𝐵, 0) ∈ ℂ) ↔ ((𝜑 ∧ (𝐾𝑥) ∈ ℤ) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ)))
201187, 193, 200, 102vtoclgf 2677 . . . . . . 7 ((𝐾𝑥) ∈ 𝐴 → ((𝜑 ∧ (𝐾𝑥) ∈ ℤ) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ))
202179, 186, 201sylc 61 . . . . . 6 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ)
203 eleq1 2150 . . . . . . . 8 (𝑛 = (𝐾𝑥) → (𝑛𝐴 ↔ (𝐾𝑥) ∈ 𝐴))
204 csbeq1 2934 . . . . . . . 8 (𝑛 = (𝐾𝑥) → 𝑛 / 𝑘𝐵 = (𝐾𝑥) / 𝑘𝐵)
205203, 204ifbieq1d 3409 . . . . . . 7 (𝑛 = (𝐾𝑥) → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0))
206 nfcv 2228 . . . . . . . . 9 𝑛if(𝑘𝐴, 𝐵, 0)
207 nfv 1466 . . . . . . . . . 10 𝑘 𝑛𝐴
208 nfcsb1v 2961 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
209207, 208, 143nfif 3415 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
210 eleq1 2150 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
211 csbeq1a 2939 . . . . . . . . . 10 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
212210, 211ifbieq1d 3409 . . . . . . . . 9 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
213206, 209, 212cbvmpt 3925 . . . . . . . 8 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
2141, 213eqtri 2108 . . . . . . 7 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
215205, 214fvmptg 5364 . . . . . 6 (((𝐾𝑥) ∈ ℤ ∧ if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ) → (𝐹‘(𝐾𝑥)) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0))
216184, 202, 215syl2anc 403 . . . . 5 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾𝑥)) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 0))
217 elfznn 9437 . . . . . . . 8 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
218217adantl 271 . . . . . . 7 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ ℕ)
219 elfzle2 9411 . . . . . . . . . . 11 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
220219adantl 271 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴))
22115breq2d 3849 . . . . . . . . . . 11 (𝜑 → (𝑥 ≤ (♯‘𝐴) ↔ 𝑥𝑁))
222221adantr 270 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝑥 ≤ (♯‘𝐴) ↔ 𝑥𝑁))
223220, 222mpbid 145 . . . . . . . . 9 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → 𝑥𝑁)
224223iftrued 3396 . . . . . . . 8 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥𝑁, (𝐾𝑥) / 𝑘𝐵, 0) = (𝐾𝑥) / 𝑘𝐵)
225180, 202eqeltrrd 2165 . . . . . . . 8 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐾𝑥) / 𝑘𝐵 ∈ ℂ)
226224, 225eqeltrd 2164 . . . . . . 7 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥𝑁, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ)
227 breq1 3840 . . . . . . . . 9 (𝑛 = 𝑥 → (𝑛𝑁𝑥𝑁))
228 fveq2 5289 . . . . . . . . . 10 (𝑛 = 𝑥 → (𝐾𝑛) = (𝐾𝑥))
229228csbeq1d 2937 . . . . . . . . 9 (𝑛 = 𝑥(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑥) / 𝑘𝐵)
230227, 229ifbieq1d 3409 . . . . . . . 8 (𝑛 = 𝑥 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑥𝑁, (𝐾𝑥) / 𝑘𝐵, 0))
231230, 163fvmptg 5364 . . . . . . 7 ((𝑥 ∈ ℕ ∧ if(𝑥𝑁, (𝐾𝑥) / 𝑘𝐵, 0) ∈ ℂ) → (𝐻𝑥) = if(𝑥𝑁, (𝐾𝑥) / 𝑘𝐵, 0))
232218, 226, 231syl2anc 403 . . . . . 6 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐻𝑥) = if(𝑥𝑁, (𝐾𝑥) / 𝑘𝐵, 0))
233232, 224eqtrd 2120 . . . . 5 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐻𝑥) = (𝐾𝑥) / 𝑘𝐵)
234180, 216, 2333eqtr4rd 2131 . . . 4 ((𝜑𝑥 ∈ (1...(♯‘𝐴))) → (𝐻𝑥) = (𝐹‘(𝐾𝑥)))
23572, 74, 76, 77, 5, 78, 4, 106, 166, 178, 234iseqcoll 10212 . . 3 (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘(𝐾𝑁)) = (seq1( + , 𝐻, ℂ)‘𝑁))
23615, 7eqeltrd 2164 . . . . 5 (𝜑 → (♯‘𝐴) ∈ ℕ)
237236, 7jca 300 . . . 4 (𝜑 → ((♯‘𝐴) ∈ ℕ ∧ 𝑁 ∈ ℕ))
23816eqcomd 2093 . . . . . 6 (𝜑 → (1...𝑁) = (1...(♯‘𝐴)))
239 f1oeq2 5229 . . . . . 6 ((1...𝑁) = (1...(♯‘𝐴)) → (𝑓:(1...𝑁)–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1-onto𝐴))
240238, 239syl 14 . . . . 5 (𝜑 → (𝑓:(1...𝑁)–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1-onto𝐴))
24110, 240mpbid 145 . . . 4 (𝜑𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
242 isummolem2a.g . . . 4 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
2431, 2, 237, 241, 21, 242, 163isummolem3 10734 . . 3 (𝜑 → (seq1( + , 𝐺, ℂ)‘(♯‘𝐴)) = (seq1( + , 𝐻, ℂ)‘𝑁))
24415fveq2d 5293 . . 3 (𝜑 → (seq1( + , 𝐺, ℂ)‘(♯‘𝐴)) = (seq1( + , 𝐺, ℂ)‘𝑁))
245235, 243, 2443eqtr2d 2126 . 2 (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘(𝐾𝑁)) = (seq1( + , 𝐺, ℂ)‘𝑁))
24670, 245breqtrd 3861 1 (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑁))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 664  DECID wdc 780   ∧ w3a 924   = wceq 1289   ∈ wcel 1438  ∀wral 2359  ⦋csb 2931   ∖ cdif 2994   ⊆ wss 2997  ifcif 3389   class class class wbr 3837   ↦ cmpt 3891  ◡ccnv 4427  ⟶wf 4998  –1-1-onto→wf1o 5001  ‘cfv 5002   Isom wiso 5003  (class class class)co 5634  ℂcc 7327  ℝcr 7328  0cc0 7329  1c1 7330   + caddc 7332  ℝ*cxr 7500   < clt 7501   ≤ cle 7502  ℕcn 8394  ℕ0cn0 8643  ℤcz 8720  ℤ≥cuz 8988  ...cfz 9393  seqcseq4 9816  ♯chash 10148   ⇝ cli 10630 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442 This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9920  df-ihash 10149  df-cj 10241  df-rsqrt 10396  df-abs 10397  df-clim 10631 This theorem is referenced by:  isummolem2  10736  zisum  10738
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