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Theorem seqof 13044
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 6354 . . . . . . . . 9 (𝐺𝑥) ∈ V
32rgenw 3054 . . . . . . . 8 𝑧𝐴 (𝐺𝑥) ∈ V
4 eqid 2752 . . . . . . . . 9 (𝑧𝐴 ↦ (𝐺𝑥)) = (𝑧𝐴 ↦ (𝐺𝑥))
54fnmpt 6173 . . . . . . . 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 6134 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥) Fn 𝐴 ↔ (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴))
96, 8mpbird 247 . . . . . 6 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) Fn 𝐴)
10 fvex 6354 . . . . . . 7 (𝐹𝑥) ∈ V
11 fneq1 6132 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹𝑥) Fn 𝐴))
1210, 11elab 3482 . . . . . 6 ((𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝐹𝑥) Fn 𝐴)
139, 12sylibr 224 . . . . 5 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
14 simprl 811 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16 seqof.1 . . . . . . . . . 10 (𝜑𝐴𝑉)
1716adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝐴𝑉)
18 inidm 3957 . . . . . . . . 9 (𝐴𝐴) = 𝐴
1914, 15, 17, 17, 18offn 7065 . . . . . . . 8 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → (𝑥𝑓 + 𝑦) Fn 𝐴)
2019ex 449 . . . . . . 7 (𝜑 → ((𝑥 Fn 𝐴𝑦 Fn 𝐴) → (𝑥𝑓 + 𝑦) Fn 𝐴))
21 vex 3335 . . . . . . . . 9 𝑥 ∈ V
22 fneq1 6132 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 Fn 𝐴𝑥 Fn 𝐴))
2321, 22elab 3482 . . . . . . . 8 (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24 vex 3335 . . . . . . . . 9 𝑦 ∈ V
25 fneq1 6132 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 Fn 𝐴𝑦 Fn 𝐴))
2624, 25elab 3482 . . . . . . . 8 (𝑦 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
2723, 26anbi12i 735 . . . . . . 7 ((𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴𝑦 Fn 𝐴))
28 ovex 6833 . . . . . . . 8 (𝑥𝑓 + 𝑦) ∈ V
29 fneq1 6132 . . . . . . . 8 (𝑧 = (𝑥𝑓 + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥𝑓 + 𝑦) Fn 𝐴))
3028, 29elab 3482 . . . . . . 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 444 . . . . 5 ((𝜑 ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥𝑓 + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
331, 13, 32seqcl 13007 . . . 4 (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴})
34 fvex 6354 . . . . 5 (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ V
35 fneq1 6132 . . . . 5 (𝑧 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴))
3634, 35elab 3482 . . . 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 6395 . . 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 6343 . . . . . 6 (𝑤 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑤𝑧) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
41 eqid 2752 . . . . . 6 (𝑤 ∈ V ↦ (𝑤𝑧)) = (𝑤 ∈ V ↦ (𝑤𝑧))
42 fvex 6354 . . . . . 6 ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) ∈ V
4340, 41, 42fvmpt 6436 . . . . 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 753 . . . . 5 (((𝜑𝑧𝐴) ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥𝑓 + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
4613adantlr 753 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
471adantr 472 . . . . 5 ((𝜑𝑧𝐴) → 𝑁 ∈ (ℤ𝑀))
48 eqidd 2753 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑥𝑧) = (𝑥𝑧))
49 eqidd 2753 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑦𝑧) = (𝑦𝑧))
5014, 15, 17, 17, 18, 48, 49ofval 7063 . . . . . . . 8 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → ((𝑥𝑓 + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
5150an32s 881 . . . . . . 7 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑥𝑓 + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
52 fveq1 6343 . . . . . . . . 9 (𝑤 = (𝑥𝑓 + 𝑦) → (𝑤𝑧) = ((𝑥𝑓 + 𝑦)‘𝑧))
53 fvex 6354 . . . . . . . . 9 ((𝑥𝑓 + 𝑦)‘𝑧) ∈ V
5452, 41, 53fvmpt 6436 . . . . . . . 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 6343 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑤𝑧) = (𝑥𝑧))
57 fvex 6354 . . . . . . . . . 10 (𝑥𝑧) ∈ V
5856, 41, 57fvmpt 6436 . . . . . . . . 9 (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧))
5921, 58ax-mp 5 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧)
60 fveq1 6343 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤𝑧) = (𝑦𝑧))
61 fvex 6354 . . . . . . . . . 10 (𝑦𝑧) ∈ V
6260, 41, 61fvmpt 6436 . . . . . . . . 9 (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧))
6324, 62ax-mp 5 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧)
6459, 63oveq12i 6817 . . . . . . 7 (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)) = ((𝑥𝑧) + (𝑦𝑧))
6551, 55, 643eqtr4g 2811 . . . . . 6 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥𝑓 + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)))
6627, 65sylan2b 493 . . . . 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 6343 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝑤𝑧) = ((𝐹𝑥)‘𝑧))
68 fvex 6354 . . . . . . . 8 ((𝐹𝑥)‘𝑧) ∈ V
6967, 41, 68fvmpt 6436 . . . . . . 7 ((𝐹𝑥) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧))
7010, 69ax-mp 5 . . . . . 6 ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧)
717adantlr 753 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
7271fveq1d 6346 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧))
73 simplr 809 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧𝐴)
744fvmpt2 6445 . . . . . . . 8 ((𝑧𝐴 ∧ (𝐺𝑥) ∈ V) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7573, 2, 74sylancl 697 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7672, 75eqtrd 2786 . . . . . 6 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = (𝐺𝑥))
7770, 76syl5eq 2798 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = (𝐺𝑥))
7845, 46, 47, 66, 77seqhomo 13034 . . . 4 ((𝜑𝑧𝐴) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
7944, 78eqtr3d 2788 . . 3 ((𝜑𝑧𝐴) → ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
8079mpteq2dva 4888 . 2 (𝜑 → (𝑧𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
8139, 80eqtrd 2786 1 (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  {cab 2738  wral 3042  Vcvv 3332  cmpt 4873   Fn wfn 6036  cfv 6041  (class class class)co 6805  𝑓 cof 7052  cuz 11871  ...cfz 12511  seqcseq 12987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-seq 12988
This theorem is referenced by:  seqof2  13045  mtest  24349  pserulm  24367  knoppcnlem7  32787
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