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Theorem seqof 14097
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 6920 . . . . . . . . 9 (𝐺𝑥) ∈ V
32rgenw 3063 . . . . . . . 8 𝑧𝐴 (𝐺𝑥) ∈ V
4 eqid 2735 . . . . . . . . 9 (𝑧𝐴 ↦ (𝐺𝑥)) = (𝑧𝐴 ↦ (𝐺𝑥))
54fnmpt 6709 . . . . . . . 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 6662 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥) Fn 𝐴 ↔ (𝑧𝐴 ↦ (𝐺𝑥)) Fn 𝐴))
96, 8mpbird 257 . . . . . 6 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) Fn 𝐴)
10 fvex 6920 . . . . . . 7 (𝐹𝑥) ∈ V
11 fneq1 6660 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹𝑥) Fn 𝐴))
1210, 11elab 3681 . . . . . 6 ((𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴} ↔ (𝐹𝑥) Fn 𝐴)
139, 12sylibr 234 . . . . 5 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
14 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15 simprr 773 . . . . . . . . 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 4235 . . . . . . . . 9 (𝐴𝐴) = 𝐴
1914, 15, 17, 17, 18offn 7710 . . . . . . . 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 3482 . . . . . . . . 9 𝑥 ∈ V
22 fneq1 6660 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 Fn 𝐴𝑥 Fn 𝐴))
2321, 22elab 3681 . . . . . . . 8 (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24 vex 3482 . . . . . . . . 9 𝑦 ∈ V
25 fneq1 6660 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 Fn 𝐴𝑦 Fn 𝐴))
2624, 25elab 3681 . . . . . . . 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 7464 . . . . . . . 8 (𝑥f + 𝑦) ∈ V
29 fneq1 6660 . . . . . . . 8 (𝑧 = (𝑥f + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥f + 𝑦) Fn 𝐴))
3028, 29elab 3681 . . . . . . 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 14060 . . . 4 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴})
34 fvex 6920 . . . . 5 (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ V
35 fneq1 6660 . . . . 5 (𝑧 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴))
3634, 35elab 3681 . . . 4 ((seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧𝑧 Fn 𝐴} ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
3733, 36sylib 218 . . 3 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
38 dffn5 6967 . . 3 ((seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
3937, 38sylib 218 . 2 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
40 fveq1 6906 . . . . . 6 (𝑤 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑤𝑧) = ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧))
41 eqid 2735 . . . . . 6 (𝑤 ∈ V ↦ (𝑤𝑧)) = (𝑤 ∈ V ↦ (𝑤𝑧))
42 fvex 6920 . . . . . 6 ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) ∈ V
4340, 41, 42fvmpt 7016 . . . . 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 715 . . . . 5 (((𝜑𝑧𝐴) ∧ (𝑥 ∈ {𝑧𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧𝑧 Fn 𝐴})) → (𝑥f + 𝑦) ∈ {𝑧𝑧 Fn 𝐴})
4613adantlr 715 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ {𝑧𝑧 Fn 𝐴})
471adantr 480 . . . . 5 ((𝜑𝑧𝐴) → 𝑁 ∈ (ℤ𝑀))
48 eqidd 2736 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑥𝑧) = (𝑥𝑧))
49 eqidd 2736 . . . . . . . . 9 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → (𝑦𝑧) = (𝑦𝑧))
5014, 15, 17, 17, 18, 48, 49ofval 7708 . . . . . . . 8 (((𝜑 ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) ∧ 𝑧𝐴) → ((𝑥f + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
5150an32s 652 . . . . . . 7 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑥f + 𝑦)‘𝑧) = ((𝑥𝑧) + (𝑦𝑧)))
52 fveq1 6906 . . . . . . . . 9 (𝑤 = (𝑥f + 𝑦) → (𝑤𝑧) = ((𝑥f + 𝑦)‘𝑧))
53 fvex 6920 . . . . . . . . 9 ((𝑥f + 𝑦)‘𝑧) ∈ V
5452, 41, 53fvmpt 7016 . . . . . . . 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 6906 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑤𝑧) = (𝑥𝑧))
57 fvex 6920 . . . . . . . . . 10 (𝑥𝑧) ∈ V
5856, 41, 57fvmpt 7016 . . . . . . . . 9 (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧))
5958elv 3483 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) = (𝑥𝑧)
60 fveq1 6906 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤𝑧) = (𝑦𝑧))
61 fvex 6920 . . . . . . . . . 10 (𝑦𝑧) ∈ V
6260, 41, 61fvmpt 7016 . . . . . . . . 9 (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧))
6362elv 3483 . . . . . . . 8 ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦) = (𝑦𝑧)
6459, 63oveq12i 7443 . . . . . . 7 (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)) = ((𝑥𝑧) + (𝑦𝑧))
6551, 55, 643eqtr4g 2800 . . . . . 6 (((𝜑𝑧𝐴) ∧ (𝑥 Fn 𝐴𝑦 Fn 𝐴)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝑥f + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤𝑧))‘𝑦)))
6627, 65sylan2b 594 . . . . 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 6906 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝑤𝑧) = ((𝐹𝑥)‘𝑧))
68 fvex 6920 . . . . . . . 8 ((𝐹𝑥)‘𝑧) ∈ V
6967, 41, 68fvmpt 7016 . . . . . . 7 ((𝐹𝑥) ∈ V → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧))
7010, 69ax-mp 5 . . . . . 6 ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = ((𝐹𝑥)‘𝑧)
717adantlr 715 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = (𝑧𝐴 ↦ (𝐺𝑥)))
7271fveq1d 6909 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧))
73 simplr 769 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧𝐴)
744fvmpt2 7027 . . . . . . . 8 ((𝑧𝐴 ∧ (𝐺𝑥) ∈ V) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7573, 2, 74sylancl 586 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧𝐴 ↦ (𝐺𝑥))‘𝑧) = (𝐺𝑥))
7672, 75eqtrd 2775 . . . . . 6 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹𝑥)‘𝑧) = (𝐺𝑥))
7770, 76eqtrid 2787 . . . . 5 (((𝜑𝑧𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(𝐹𝑥)) = (𝐺𝑥))
7845, 46, 47, 66, 77seqhomo 14087 . . . 4 ((𝜑𝑧𝐴) → ((𝑤 ∈ V ↦ (𝑤𝑧))‘(seq𝑀( ∘f + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
7944, 78eqtr3d 2777 . . 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 2775 1 (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  cmpt 5231   Fn wfn 6558  cfv 6563  (class class class)co 7431  f cof 7695  cuz 12876  ...cfz 13544  seqcseq 14039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040
This theorem is referenced by:  seqof2  14098  mtest  26462  pserulm  26480  knoppcnlem7  36482
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