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Theorem seqof 14032
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 6904 . . . . . . . . 9 (𝐺𝑥) ∈ V
32rgenw 3064 . . . . . . . 8 𝑧𝐴 (𝐺𝑥) ∈ V
4 eqid 2731 . . . . . . . . 9 (𝑧𝐴 ↦ (𝐺𝑥)) = (𝑧𝐴 ↦ (𝐺𝑥))
54fnmpt 6690 . . . . . . . 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 6642 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥) Fn 𝐴 ↔ (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴))
96, 8mpbird 257 . . . . . 6 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) Fn 𝐴)
10 fvex 6904 . . . . . . 7 (𝐹𝑥) ∈ V
11 fneq1 6640 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹𝑥) Fn 𝐴))
1210, 11elab 3668 . . . . . 6 ((𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝐹𝑥) Fn 𝐴)
139, 12sylibr 233 . . . . 5 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
14 simprl 768 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15 simprr 770 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16 seqof.1 . . . . . . . . . 10 (𝜑𝐴𝑉)
1716adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝐴𝑉)
18 inidm 4218 . . . . . . . . 9 (𝐴𝐴) = 𝐴
1914, 15, 17, 17, 18offn 7687 . . . . . . . 8 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → (𝑥f + 𝑦) Fn 𝐴)
2019ex 412 . . . . . . 7 (𝜑 → ((𝑥 Fn 𝐴𝑦 Fn 𝐴) → (𝑥f + 𝑦) Fn 𝐴))
21 vex 3477 . . . . . . . . 9 𝑥 ∈ V
22 fneq1 6640 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 Fn 𝐴𝑥 Fn 𝐴))
2321, 22elab 3668 . . . . . . . 8 (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24 vex 3477 . . . . . . . . 9 𝑦 ∈ V
25 fneq1 6640 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 Fn 𝐴𝑦 Fn 𝐴))
2624, 25elab 3668 . . . . . . . 8 (𝑦 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
2723, 26anbi12i 626 . . . . . . 7 ((𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴𝑦 Fn 𝐴))
28 ovex 7445 . . . . . . . 8 (𝑥f + 𝑦) ∈ V
29 fneq1 6640 . . . . . . . 8 (𝑧 = (𝑥f + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥f + 𝑦) Fn 𝐴))
3028, 29elab 3668 . . . . . . 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 406 . . . . 5 ((𝜑 ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
331, 13, 32seqcl 13995 . . . 4 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴})
34 fvex 6904 . . . . 5 (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ V
35 fneq1 6640 . . . . 5 (𝑧 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴))
3634, 35elab 3668 . . . 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 6950 . . 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 6890 . . . . . 6 (𝑤 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑤𝑧) = ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧))
41 eqid 2731 . . . . . 6 (𝑤 ∈ V ↦ (𝑤𝑧)) = (𝑤 ∈ V ↦ (𝑤𝑧))
42 fvex 6904 . . . . . 6 ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) ∈ V
4340, 41, 42fvmpt 6998 . . . . 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 712 . . . . 5 (((𝜑𝑧𝐴) ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
4613adantlr 712 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
471adantr 480 . . . . 5 ((𝜑𝑧𝐴) → 𝑁 ∈ (ℤ𝑀))
48 eqidd 2732 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑥𝑧) = (𝑥𝑧))
49 eqidd 2732 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑦𝑧) = (𝑦𝑧))
5014, 15, 17, 17, 18, 48, 49ofval 7685 . . . . . . . 8 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → ((𝑥f + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
5150an32s 649 . . . . . . 7 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑥f + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
52 fveq1 6890 . . . . . . . . 9 (𝑤 = (𝑥f + 𝑦) → (𝑤𝑧) = ((𝑥f + 𝑦)‘𝑧))
53 fvex 6904 . . . . . . . . 9 ((𝑥f + 𝑦)‘𝑧) ∈ V
5452, 41, 53fvmpt 6998 . . . . . . . 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 6890 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑤𝑧) = (𝑥𝑧))
57 fvex 6904 . . . . . . . . . 10 (𝑥𝑧) ∈ V
5856, 41, 57fvmpt 6998 . . . . . . . . 9 (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧))
5958elv 3479 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧)
60 fveq1 6890 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤𝑧) = (𝑦𝑧))
61 fvex 6904 . . . . . . . . . 10 (𝑦𝑧) ∈ V
6260, 41, 61fvmpt 6998 . . . . . . . . 9 (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧))
6362elv 3479 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧)
6459, 63oveq12i 7424 . . . . . . 7 (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)) = ((𝑥𝑧) + (𝑦𝑧))
6551, 55, 643eqtr4g 2796 . . . . . 6 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥f + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)))
6627, 65sylan2b 593 . . . . 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 6890 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝑤𝑧) = ((𝐹𝑥)‘𝑧))
68 fvex 6904 . . . . . . . 8 ((𝐹𝑥)‘𝑧) ∈ V
6967, 41, 68fvmpt 6998 . . . . . . 7 ((𝐹𝑥) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧))
7010, 69ax-mp 5 . . . . . 6 ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧)
717adantlr 712 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
7271fveq1d 6893 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧))
73 simplr 766 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧𝐴)
744fvmpt2 7009 . . . . . . . 8 ((𝑧𝐴 ∧ (𝐺𝑥) ∈ V) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7573, 2, 74sylancl 585 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7672, 75eqtrd 2771 . . . . . 6 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = (𝐺𝑥))
7770, 76eqtrid 2783 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = (𝐺𝑥))
7845, 46, 47, 66, 77seqhomo 14022 . . . 4 ((𝜑𝑧𝐴) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘f + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
7944, 78eqtr3d 2773 . . 3 ((𝜑𝑧𝐴) → ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
8079mpteq2dva 5248 . 2 (𝜑 → (𝑧𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
8139, 80eqtrd 2771 1 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  Vcvv 3473  cmpt 5231   Fn wfn 6538  cfv 6543  (class class class)co 7412  f cof 7672  cuz 12829  ...cfz 13491  seqcseq 13973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-seq 13974
This theorem is referenced by:  seqof2  14033  mtest  26256  pserulm  26274  knoppcnlem7  35842
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