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Theorem seqof 12795
Description: Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.)
Hypotheses
Ref Expression
seqof.1 (𝜑𝐴𝑉)
seqof.2 (𝜑𝑁 ∈ (ℤ𝑀))
seqof.3 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
Assertion
Ref Expression
seqof (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝐹,𝑧   𝑥,𝐺   𝑥,𝑀,𝑧   𝑥,𝑁,𝑧   𝑥, + ,𝑧   𝜑,𝑥,𝑧
Allowed substitution hints:   𝐺(𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem seqof
Dummy variables 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqof.2 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
2 fvex 6160 . . . . . . . . 9 (𝐺𝑥) ∈ V
32rgenw 2924 . . . . . . . 8 𝑧𝐴 (𝐺𝑥) ∈ V
4 eqid 2626 . . . . . . . . 9 (𝑧𝐴 ↦ (𝐺𝑥)) = (𝑧𝐴 ↦ (𝐺𝑥))
54fnmpt 5979 . . . . . . . 8 (∀𝑧𝐴 (𝐺𝑥) ∈ V → (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴)
63, 5mp1i 13 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴)
7 seqof.3 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
87fneq1d 5941 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥) Fn 𝐴 ↔ (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴))
96, 8mpbird 247 . . . . . 6 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) Fn 𝐴)
10 fvex 6160 . . . . . . 7 (𝐹𝑥) ∈ V
11 fneq1 5939 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹𝑥) Fn 𝐴))
1210, 11elab 3338 . . . . . 6 ((𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝐹𝑥) Fn 𝐴)
139, 12sylibr 224 . . . . 5 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
14 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16 seqof.1 . . . . . . . . . 10 (𝜑𝐴𝑉)
1716adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝐴𝑉)
18 inidm 3805 . . . . . . . . 9 (𝐴𝐴) = 𝐴
1914, 15, 17, 17, 18offn 6862 . . . . . . . 8 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → (𝑥𝑓 + 𝑦) Fn 𝐴)
2019ex 450 . . . . . . 7 (𝜑 → ((𝑥 Fn 𝐴𝑦 Fn 𝐴) → (𝑥𝑓 + 𝑦) Fn 𝐴))
21 vex 3194 . . . . . . . . 9 𝑥 ∈ V
22 fneq1 5939 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 Fn 𝐴𝑥 Fn 𝐴))
2321, 22elab 3338 . . . . . . . 8 (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24 vex 3194 . . . . . . . . 9 𝑦 ∈ V
25 fneq1 5939 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 Fn 𝐴𝑦 Fn 𝐴))
2624, 25elab 3338 . . . . . . . 8 (𝑦 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
2723, 26anbi12i 732 . . . . . . 7 ((𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴𝑦 Fn 𝐴))
28 ovex 6633 . . . . . . . 8 (𝑥𝑓 + 𝑦) ∈ V
29 fneq1 5939 . . . . . . . 8 (𝑧 = (𝑥𝑓 + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥𝑓 + 𝑦) Fn 𝐴))
3028, 29elab 3338 . . . . . . 7 ((𝑥𝑓 + 𝑦) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝑥𝑓 + 𝑦) Fn 𝐴)
3120, 27, 303imtr4g 285 . . . . . 6 (𝜑 → ((𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴}) → (𝑥𝑓 + 𝑦) ∈ {𝑧𝑧 Fn 𝐴}))
3231imp 445 . . . . 5 ((𝜑 ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥𝑓 + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
331, 13, 32seqcl 12758 . . . 4 (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴})
34 fvex 6160 . . . . 5 (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ V
35 fneq1 5939 . . . . 5 (𝑧 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴))
3634, 35elab 3338 . . . 4 ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴} ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴)
3733, 36sylib 208 . . 3 (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴)
38 dffn5 6199 . . 3 ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)))
3937, 38sylib 208 . 2 (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)))
40 fveq1 6149 . . . . . 6 (𝑤 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑤𝑧) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
41 eqid 2626 . . . . . 6 (𝑤 ∈ V ↦ (𝑤𝑧)) = (𝑤 ∈ V ↦ (𝑤𝑧))
42 fvex 6160 . . . . . 6 ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) ∈ V
4340, 41, 42fvmpt 6240 . . . . 5 ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
4434, 43mp1i 13 . . . 4 ((𝜑𝑧𝐴) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
4532adantlr 750 . . . . 5 (((𝜑𝑧𝐴) ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥𝑓 + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
4613adantlr 750 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
471adantr 481 . . . . 5 ((𝜑𝑧𝐴) → 𝑁 ∈ (ℤ𝑀))
48 eqidd 2627 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑥𝑧) = (𝑥𝑧))
49 eqidd 2627 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑦𝑧) = (𝑦𝑧))
5014, 15, 17, 17, 18, 48, 49ofval 6860 . . . . . . . 8 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → ((𝑥𝑓 + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
5150an32s 845 . . . . . . 7 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑥𝑓 + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
52 fveq1 6149 . . . . . . . . 9 (𝑤 = (𝑥𝑓 + 𝑦) → (𝑤𝑧) = ((𝑥𝑓 + 𝑦)‘𝑧))
53 fvex 6160 . . . . . . . . 9 ((𝑥𝑓 + 𝑦)‘𝑧) ∈ V
5452, 41, 53fvmpt 6240 . . . . . . . 8 ((𝑥𝑓 + 𝑦) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥𝑓 + 𝑦)) = ((𝑥𝑓 + 𝑦)‘𝑧))
5528, 54ax-mp 5 . . . . . . 7 ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥𝑓 + 𝑦)) = ((𝑥𝑓 + 𝑦)‘𝑧)
56 fveq1 6149 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑤𝑧) = (𝑥𝑧))
57 fvex 6160 . . . . . . . . . 10 (𝑥𝑧) ∈ V
5856, 41, 57fvmpt 6240 . . . . . . . . 9 (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧))
5921, 58ax-mp 5 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧)
60 fveq1 6149 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤𝑧) = (𝑦𝑧))
61 fvex 6160 . . . . . . . . . 10 (𝑦𝑧) ∈ V
6260, 41, 61fvmpt 6240 . . . . . . . . 9 (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧))
6324, 62ax-mp 5 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧)
6459, 63oveq12i 6617 . . . . . . 7 (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)) = ((𝑥𝑧) + (𝑦𝑧))
6551, 55, 643eqtr4g 2685 . . . . . 6 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥𝑓 + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)))
6627, 65sylan2b 492 . . . . 5 (((𝜑𝑧𝐴) ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥𝑓 + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)))
67 fveq1 6149 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝑤𝑧) = ((𝐹𝑥)‘𝑧))
68 fvex 6160 . . . . . . . 8 ((𝐹𝑥)‘𝑧) ∈ V
6967, 41, 68fvmpt 6240 . . . . . . 7 ((𝐹𝑥) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧))
7010, 69ax-mp 5 . . . . . 6 ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧)
717adantlr 750 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
7271fveq1d 6152 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧))
73 simplr 791 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧𝐴)
744fvmpt2 6249 . . . . . . . 8 ((𝑧𝐴 ∧ (𝐺𝑥) ∈ V) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7573, 2, 74sylancl 693 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7672, 75eqtrd 2660 . . . . . 6 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = (𝐺𝑥))
7770, 76syl5eq 2672 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = (𝐺𝑥))
7845, 46, 47, 66, 77seqhomo 12785 . . . 4 ((𝜑𝑧𝐴) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
7944, 78eqtr3d 2662 . . 3 ((𝜑𝑧𝐴) → ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
8079mpteq2dva 4709 . 2 (𝜑 → (𝑧𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
8139, 80eqtrd 2660 1 (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  Vcvv 3191  cmpt 4678   Fn wfn 5845  cfv 5850  (class class class)co 6605  𝑓 cof 6849  cuz 11631  ...cfz 12265  seqcseq 12738
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-seq 12739
This theorem is referenced by:  seqof2  12796  mtest  24057  pserulm  24075  knoppcnlem7  32123
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