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Theorem seqof 13974
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 6859 . . . . . . . . 9 (𝐺𝑥) ∈ V
32rgenw 3065 . . . . . . . 8 𝑧𝐴 (𝐺𝑥) ∈ V
4 eqid 2733 . . . . . . . . 9 (𝑧𝐴 ↦ (𝐺𝑥)) = (𝑧𝐴 ↦ (𝐺𝑥))
54fnmpt 6645 . . . . . . . 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 6599 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥) Fn 𝐴 ↔ (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴))
96, 8mpbird 257 . . . . . 6 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) Fn 𝐴)
10 fvex 6859 . . . . . . 7 (𝐹𝑥) ∈ V
11 fneq1 6597 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹𝑥) Fn 𝐴))
1210, 11elab 3634 . . . . . 6 ((𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝐹𝑥) Fn 𝐴)
139, 12sylibr 233 . . . . 5 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
14 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16 seqof.1 . . . . . . . . . 10 (𝜑𝐴𝑉)
1716adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝐴𝑉)
18 inidm 4182 . . . . . . . . 9 (𝐴𝐴) = 𝐴
1914, 15, 17, 17, 18offn 7634 . . . . . . . 8 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → (𝑥f + 𝑦) Fn 𝐴)
2019ex 414 . . . . . . 7 (𝜑 → ((𝑥 Fn 𝐴𝑦 Fn 𝐴) → (𝑥f + 𝑦) Fn 𝐴))
21 vex 3451 . . . . . . . . 9 𝑥 ∈ V
22 fneq1 6597 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 Fn 𝐴𝑥 Fn 𝐴))
2321, 22elab 3634 . . . . . . . 8 (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24 vex 3451 . . . . . . . . 9 𝑦 ∈ V
25 fneq1 6597 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 Fn 𝐴𝑦 Fn 𝐴))
2624, 25elab 3634 . . . . . . . 8 (𝑦 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
2723, 26anbi12i 628 . . . . . . 7 ((𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴𝑦 Fn 𝐴))
28 ovex 7394 . . . . . . . 8 (𝑥f + 𝑦) ∈ V
29 fneq1 6597 . . . . . . . 8 (𝑧 = (𝑥f + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥f + 𝑦) Fn 𝐴))
3028, 29elab 3634 . . . . . . 7 ((𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝑥f + 𝑦) Fn 𝐴)
3120, 27, 303imtr4g 296 . . . . . 6 (𝜑 → ((𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴}) → (𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴}))
3231imp 408 . . . . 5 ((𝜑 ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
331, 13, 32seqcl 13937 . . . 4 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴})
34 fvex 6859 . . . . 5 (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ V
35 fneq1 6597 . . . . 5 (𝑧 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴))
3634, 35elab 3634 . . . 4 ((seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴} ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
3733, 36sylib 217 . . 3 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
38 dffn5 6905 . . 3 ((seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
3937, 38sylib 217 . 2 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
40 fveq1 6845 . . . . . 6 (𝑤 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑤𝑧) = ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧))
41 eqid 2733 . . . . . 6 (𝑤 ∈ V ↦ (𝑤𝑧)) = (𝑤 ∈ V ↦ (𝑤𝑧))
42 fvex 6859 . . . . . 6 ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) ∈ V
4340, 41, 42fvmpt 6952 . . . . 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 714 . . . . 5 (((𝜑𝑧𝐴) ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
4613adantlr 714 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
471adantr 482 . . . . 5 ((𝜑𝑧𝐴) → 𝑁 ∈ (ℤ𝑀))
48 eqidd 2734 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑥𝑧) = (𝑥𝑧))
49 eqidd 2734 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑦𝑧) = (𝑦𝑧))
5014, 15, 17, 17, 18, 48, 49ofval 7632 . . . . . . . 8 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → ((𝑥f + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
5150an32s 651 . . . . . . 7 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑥f + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
52 fveq1 6845 . . . . . . . . 9 (𝑤 = (𝑥f + 𝑦) → (𝑤𝑧) = ((𝑥f + 𝑦)‘𝑧))
53 fvex 6859 . . . . . . . . 9 ((𝑥f + 𝑦)‘𝑧) ∈ V
5452, 41, 53fvmpt 6952 . . . . . . . 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 6845 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑤𝑧) = (𝑥𝑧))
57 fvex 6859 . . . . . . . . . 10 (𝑥𝑧) ∈ V
5856, 41, 57fvmpt 6952 . . . . . . . . 9 (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧))
5958elv 3453 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧)
60 fveq1 6845 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤𝑧) = (𝑦𝑧))
61 fvex 6859 . . . . . . . . . 10 (𝑦𝑧) ∈ V
6260, 41, 61fvmpt 6952 . . . . . . . . 9 (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧))
6362elv 3453 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧)
6459, 63oveq12i 7373 . . . . . . 7 (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)) = ((𝑥𝑧) + (𝑦𝑧))
6551, 55, 643eqtr4g 2798 . . . . . 6 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥f + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)))
6627, 65sylan2b 595 . . . . 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 6845 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝑤𝑧) = ((𝐹𝑥)‘𝑧))
68 fvex 6859 . . . . . . . 8 ((𝐹𝑥)‘𝑧) ∈ V
6967, 41, 68fvmpt 6952 . . . . . . 7 ((𝐹𝑥) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧))
7010, 69ax-mp 5 . . . . . 6 ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧)
717adantlr 714 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
7271fveq1d 6848 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧))
73 simplr 768 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧𝐴)
744fvmpt2 6963 . . . . . . . 8 ((𝑧𝐴 ∧ (𝐺𝑥) ∈ V) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7573, 2, 74sylancl 587 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7672, 75eqtrd 2773 . . . . . 6 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = (𝐺𝑥))
7770, 76eqtrid 2785 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = (𝐺𝑥))
7845, 46, 47, 66, 77seqhomo 13964 . . . 4 ((𝜑𝑧𝐴) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘f + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
7944, 78eqtr3d 2775 . . 3 ((𝜑𝑧𝐴) → ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
8079mpteq2dva 5209 . 2 (𝜑 → (𝑧𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
8139, 80eqtrd 2773 1 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3061  Vcvv 3447  cmpt 5192   Fn wfn 6495  cfv 6500  (class class class)co 7361  f cof 7619  cuz 12771  ...cfz 13433  seqcseq 13915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-seq 13916
This theorem is referenced by:  seqof2  13975  mtest  25786  pserulm  25804  knoppcnlem7  35015
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