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Theorem isercolllem3 14567
Description: Lemma for isercoll 14568. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
isercoll.z 𝑍 = (ℤ𝑀)
isercoll.m (𝜑𝑀 ∈ ℤ)
isercoll.g (𝜑𝐺:ℕ⟶𝑍)
isercoll.i ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
isercoll.0 ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
isercoll.f ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)
isercoll.h ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
Assertion
Ref Expression
isercolllem3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))))
Distinct variable groups:   𝑘,𝑛,𝐹   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝐻,𝑛   𝑘,𝑀,𝑛   𝑛,𝑍
Allowed substitution hint:   𝑍(𝑘)

Proof of Theorem isercolllem3
StepHypRef Expression
1 addid2 10382 . . 3 (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
21adantl 473 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3 addid1 10379 . . 3 (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
43adantl 473 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5 addcl 10181 . . 3 ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑛 + 𝑘) ∈ ℂ)
65adantl 473 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ (𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ)) → (𝑛 + 𝑘) ∈ ℂ)
7 0cnd 10196 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 0 ∈ ℂ)
8 cnvimass 5631 . . . . 5 (𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
9 isercoll.g . . . . . . 7 (𝜑𝐺:ℕ⟶𝑍)
109adantr 472 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
11 fdm 6200 . . . . . 6 (𝐺:ℕ⟶𝑍 → dom 𝐺 = ℕ)
1210, 11syl 17 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → dom 𝐺 = ℕ)
138, 12syl5sseq 3782 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
14 isercoll.z . . . . 5 𝑍 = (ℤ𝑀)
15 isercoll.m . . . . 5 (𝜑𝑀 ∈ ℤ)
16 isercoll.i . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
1714, 15, 9, 16isercolllem1 14565 . . . 4 ((𝜑 ∧ (𝐺 “ (𝑀...𝑁)) ⊆ ℕ) → (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((𝐺 “ (𝑀...𝑁)), (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
1813, 17syldan 488 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((𝐺 “ (𝑀...𝑁)), (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
1914, 15, 9, 16isercolllem2 14566 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) = (𝐺 “ (𝑀...𝑁)))
20 isoeq4 6721 . . . 4 ((1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) = (𝐺 “ (𝑀...𝑁)) → ((𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))), (𝐺 “ (𝐺 “ (𝑀...𝑁)))) ↔ (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((𝐺 “ (𝑀...𝑁)), (𝐺 “ (𝐺 “ (𝑀...𝑁))))))
2119, 20syl 17 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))), (𝐺 “ (𝐺 “ (𝑀...𝑁)))) ↔ (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((𝐺 “ (𝑀...𝑁)), (𝐺 “ (𝐺 “ (𝑀...𝑁))))))
2218, 21mpbird 247 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))), (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
238a1i 11 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺)
24 sseqin2 3948 . . . . 5 ((𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺 ↔ (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) = (𝐺 “ (𝑀...𝑁)))
2523, 24sylib 208 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) = (𝐺 “ (𝑀...𝑁)))
26 1nn 11194 . . . . . . 7 1 ∈ ℕ
2726a1i 11 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 1 ∈ ℕ)
28 ffvelrn 6508 . . . . . . . . . 10 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
299, 26, 28sylancl 697 . . . . . . . . 9 (𝜑 → (𝐺‘1) ∈ 𝑍)
3029, 14syl6eleq 2837 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
3130adantr 472 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ𝑀))
32 simpr 479 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝑁 ∈ (ℤ‘(𝐺‘1)))
33 elfzuzb 12500 . . . . . . 7 ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ‘(𝐺‘1))))
3431, 32, 33sylanbrc 701 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
35 ffn 6194 . . . . . . 7 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
36 elpreima 6488 . . . . . . 7 (𝐺 Fn ℕ → (1 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
3710, 35, 363syl 18 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1 ∈ (𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
3827, 34, 37mpbir2and 995 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 1 ∈ (𝐺 “ (𝑀...𝑁)))
39 ne0i 4052 . . . . 5 (1 ∈ (𝐺 “ (𝑀...𝑁)) → (𝐺 “ (𝑀...𝑁)) ≠ ∅)
4038, 39syl 17 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ≠ ∅)
4125, 40eqnetrd 2987 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) ≠ ∅)
42 imadisj 5630 . . . 4 ((𝐺 “ (𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) = ∅)
4342necon3bii 2972 . . 3 ((𝐺 “ (𝐺 “ (𝑀...𝑁))) ≠ ∅ ↔ (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) ≠ ∅)
4441, 43sylibr 224 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ≠ ∅)
45 ffun 6197 . . . 4 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
46 funimacnv 6119 . . . 4 (Fun 𝐺 → (𝐺 “ (𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
4710, 45, 463syl 18 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
48 inss1 3964 . . . 4 ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
4948a1i 11 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁))
5047, 49eqsstrd 3768 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
51 simpl 474 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝜑)
52 elfzuz 12502 . . . 4 (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ𝑀))
5352, 14syl6eleqr 2838 . . 3 (𝑛 ∈ (𝑀...𝑁) → 𝑛𝑍)
54 isercoll.f . . 3 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)
5551, 53, 54syl2an 495 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑀...𝑁)) → (𝐹𝑛) ∈ ℂ)
5647difeq2d 3859 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
57 difin 3992 . . . . . 6 ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
5856, 57syl6eq 2798 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
5953ssriv 3736 . . . . . 6 (𝑀...𝑁) ⊆ 𝑍
60 ssdif 3876 . . . . . 6 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑀...𝑁) ∖ ran 𝐺) ⊆ (𝑍 ∖ ran 𝐺))
6159, 60mp1i 13 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ ran 𝐺) ⊆ (𝑍 ∖ ran 𝐺))
6258, 61eqsstrd 3768 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
6362sselda 3732 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁))))) → 𝑛 ∈ (𝑍 ∖ ran 𝐺))
64 isercoll.0 . . . 4 ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
6564adantlr 753 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
6663, 65syldan 488 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁))))) → (𝐹𝑛) = 0)
67 elfznn 12534 . . . 4 (𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) → 𝑘 ∈ ℕ)
68 isercoll.h . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
6951, 67, 68syl2an 495 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
7019eleq2d 2813 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) ↔ 𝑘 ∈ (𝐺 “ (𝑀...𝑁))))
7170biimpa 502 . . . . 5 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → 𝑘 ∈ (𝐺 “ (𝑀...𝑁)))
72 fvres 6356 . . . . 5 (𝑘 ∈ (𝐺 “ (𝑀...𝑁)) → ((𝐺 ↾ (𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺𝑘))
7371, 72syl 17 . . . 4 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → ((𝐺 ↾ (𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺𝑘))
7473fveq2d 6344 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → (𝐹‘((𝐺 ↾ (𝐺 “ (𝑀...𝑁)))‘𝑘)) = (𝐹‘(𝐺𝑘)))
7569, 74eqtr4d 2785 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → (𝐻𝑘) = (𝐹‘((𝐺 ↾ (𝐺 “ (𝑀...𝑁)))‘𝑘)))
762, 4, 6, 7, 22, 44, 50, 55, 66, 75seqcoll2 13412 1 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wne 2920  cdif 3700  cin 3702  wss 3703  c0 4046   class class class wbr 4792  ccnv 5253  dom cdm 5254  ran crn 5255  cres 5256  cima 5257  Fun wfun 6031   Fn wfn 6032  wf 6033  cfv 6037   Isom wiso 6038  (class class class)co 6801  cc 10097  0cc0 10099  1c1 10100   + caddc 10102   < clt 10237  cn 11183  cz 11540  cuz 11850  ...cfz 12490  seqcseq 12966  chash 13282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8501  df-card 8926  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-n0 11456  df-z 11541  df-uz 11851  df-fz 12491  df-seq 12967  df-hash 13283
This theorem is referenced by:  isercoll  14568
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