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Theorem isercolllem3 14331
 Description: Lemma for isercoll 14332. (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 10163 . . 3 (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
21adantl 482 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3 addid1 10160 . . 3 (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
43adantl 482 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5 addcl 9962 . . 3 ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑛 + 𝑘) ∈ ℂ)
65adantl 482 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ (𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ)) → (𝑛 + 𝑘) ∈ ℂ)
7 0cnd 9977 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 0 ∈ ℂ)
8 cnvimass 5444 . . . . 5 (𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
9 isercoll.g . . . . . . 7 (𝜑𝐺:ℕ⟶𝑍)
109adantr 481 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
11 fdm 6008 . . . . . 6 (𝐺:ℕ⟶𝑍 → dom 𝐺 = ℕ)
1210, 11syl 17 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → dom 𝐺 = ℕ)
138, 12syl5sseq 3632 . . . 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 14329 . . . 4 ((𝜑 ∧ (𝐺 “ (𝑀...𝑁)) ⊆ ℕ) → (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((𝐺 “ (𝑀...𝑁)), (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
1813, 17syldan 487 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 ↾ (𝐺 “ (𝑀...𝑁))) Isom < , < ((𝐺 “ (𝑀...𝑁)), (𝐺 “ (𝐺 “ (𝑀...𝑁)))))
1914, 15, 9, 16isercolllem2 14330 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) = (𝐺 “ (𝑀...𝑁)))
20 isoeq4 6524 . . . 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 3795 . . . . 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 10975 . . . . . . 7 1 ∈ ℕ
2726a1i 11 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 1 ∈ ℕ)
28 ffvelrn 6313 . . . . . . . . . 10 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
299, 26, 28sylancl 693 . . . . . . . . 9 (𝜑 → (𝐺‘1) ∈ 𝑍)
3029, 14syl6eleq 2708 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
3130adantr 481 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ𝑀))
32 simpr 477 . . . . . . 7 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝑁 ∈ (ℤ‘(𝐺‘1)))
33 elfzuzb 12278 . . . . . . 7 ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ‘(𝐺‘1))))
3431, 32, 33sylanbrc 697 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
35 ffn 6002 . . . . . . 7 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
36 elpreima 6293 . . . . . . 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 956 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 1 ∈ (𝐺 “ (𝑀...𝑁)))
39 ne0i 3897 . . . . 5 (1 ∈ (𝐺 “ (𝑀...𝑁)) → (𝐺 “ (𝑀...𝑁)) ≠ ∅)
4038, 39syl 17 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝑀...𝑁)) ≠ ∅)
4125, 40eqnetrd 2857 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) ≠ ∅)
42 imadisj 5443 . . . 4 ((𝐺 “ (𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) = ∅)
4342necon3bii 2842 . . 3 ((𝐺 “ (𝐺 “ (𝑀...𝑁))) ≠ ∅ ↔ (dom 𝐺 ∩ (𝐺 “ (𝑀...𝑁))) ≠ ∅)
4441, 43sylibr 224 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ≠ ∅)
45 ffun 6005 . . . 4 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
46 funimacnv 5928 . . . 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 3811 . . . 4 ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
4948a1i 11 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁))
5047, 49eqsstrd 3618 . 2 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
51 simpl 473 . . 3 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → 𝜑)
52 elfzuz 12280 . . . 4 (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ𝑀))
5352, 14syl6eleqr 2709 . . 3 (𝑛 ∈ (𝑀...𝑁) → 𝑛𝑍)
54 isercoll.f . . 3 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)
5551, 53, 54syl2an 494 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑀...𝑁)) → (𝐹𝑛) ∈ ℂ)
5647difeq2d 3706 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
57 difin 3839 . . . . . 6 ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
5856, 57syl6eq 2671 . . . . 5 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
5953ssriv 3587 . . . . . 6 (𝑀...𝑁) ⊆ 𝑍
60 ssdif 3723 . . . . . 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 3618 . . . 4 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
6362sselda 3583 . . 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 750 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
6663, 65syldan 487 . 2 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (𝐺 “ (𝑀...𝑁))))) → (𝐹𝑛) = 0)
67 elfznn 12312 . . . 4 (𝑘 ∈ (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) → 𝑘 ∈ ℕ)
68 isercoll.h . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
6951, 67, 68syl2an 494 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
7019eleq2d 2684 . . . . . 6 ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (𝑘 ∈ (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) ↔ 𝑘 ∈ (𝐺 “ (𝑀...𝑁))))
7170biimpa 501 . . . . 5 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → 𝑘 ∈ (𝐺 “ (𝑀...𝑁)))
72 fvres 6164 . . . . 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 6152 . . 3 (((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁)))))) → (𝐹‘((𝐺 ↾ (𝐺 “ (𝑀...𝑁)))‘𝑘)) = (𝐹‘(𝐺𝑘)))
7569, 74eqtr4d 2658 . 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 13187 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 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   ∖ cdif 3552   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891   class class class wbr 4613  ◡ccnv 5073  dom cdm 5074  ran crn 5075   ↾ cres 5076   “ cima 5077  Fun wfun 5841   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847   Isom wiso 5848  (class class class)co 6604  ℂcc 9878  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018  ℕcn 10964  ℤcz 11321  ℤ≥cuz 11631  ...cfz 12268  seqcseq 12741  #chash 13057 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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 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-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-seq 12742  df-hash 13058 This theorem is referenced by:  isercoll  14332
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