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Theorem itcoval1 45030
Description: A function iterated once. (Contributed by AV, 2-May-2024.)
Assertion
Ref Expression
itcoval1 ((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)

Proof of Theorem itcoval1
Dummy variables 𝑔 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itcoval 45028 . . . 4 (𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
21fveq1d 6665 . . 3 (𝐹𝑉 → ((IterComp‘𝐹)‘1) = (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1))
32adantl 485 . 2 ((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1))
4 nn0uz 12279 . . . 4 0 = (ℤ‘0)
5 0nn0 11911 . . . . 5 0 ∈ ℕ0
65a1i 11 . . . 4 ((Rel 𝐹𝐹𝑉) → 0 ∈ ℕ0)
7 1e0p1 12139 . . . 4 1 = (0 + 1)
81eqcomd 2830 . . . . . . 7 (𝐹𝑉 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) = (IterComp‘𝐹))
98fveq1d 6665 . . . . . 6 (𝐹𝑉 → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ((IterComp‘𝐹)‘0))
10 itcoval0 45029 . . . . . 6 (𝐹𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
119, 10eqtrd 2859 . . . . 5 (𝐹𝑉 → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ( I ↾ dom 𝐹))
1211adantl 485 . . . 4 ((Rel 𝐹𝐹𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ( I ↾ dom 𝐹))
13 eqidd 2825 . . . . 5 ((Rel 𝐹𝐹𝑉) → (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))
14 ax-1ne0 10606 . . . . . . . . 9 1 ≠ 0
1514neii 3016 . . . . . . . 8 ¬ 1 = 0
16 eqeq1 2828 . . . . . . . 8 (𝑖 = 1 → (𝑖 = 0 ↔ 1 = 0))
1715, 16mtbiri 330 . . . . . . 7 (𝑖 = 1 → ¬ 𝑖 = 0)
1817iffalsed 4461 . . . . . 6 (𝑖 = 1 → if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹) = 𝐹)
1918adantl 485 . . . . 5 (((Rel 𝐹𝐹𝑉) ∧ 𝑖 = 1) → if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹) = 𝐹)
20 1nn0 11912 . . . . . 6 1 ∈ ℕ0
2120a1i 11 . . . . 5 ((Rel 𝐹𝐹𝑉) → 1 ∈ ℕ0)
22 simpr 488 . . . . 5 ((Rel 𝐹𝐹𝑉) → 𝐹𝑉)
2313, 19, 21, 22fvmptd 6768 . . . 4 ((Rel 𝐹𝐹𝑉) → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))‘1) = 𝐹)
244, 6, 7, 12, 23seqp1d 13392 . . 3 ((Rel 𝐹𝐹𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1) = (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹))
25 eqidd 2825 . . . . . 6 (𝐹𝑉 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)))
26 coeq2 5717 . . . . . . 7 (𝑔 = ( I ↾ dom 𝐹) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ dom 𝐹)))
2726ad2antrl 727 . . . . . 6 ((𝐹𝑉 ∧ (𝑔 = ( I ↾ dom 𝐹) ∧ 𝑗 = 𝐹)) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ dom 𝐹)))
28 dmexg 7610 . . . . . . 7 (𝐹𝑉 → dom 𝐹 ∈ V)
2928resiexd 6972 . . . . . 6 (𝐹𝑉 → ( I ↾ dom 𝐹) ∈ V)
30 elex 3498 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 coexg 7631 . . . . . . 7 ((𝐹𝑉 ∧ ( I ↾ dom 𝐹) ∈ V) → (𝐹 ∘ ( I ↾ dom 𝐹)) ∈ V)
3229, 31mpdan 686 . . . . . 6 (𝐹𝑉 → (𝐹 ∘ ( I ↾ dom 𝐹)) ∈ V)
3325, 27, 29, 30, 32ovmpod 7297 . . . . 5 (𝐹𝑉 → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹) = (𝐹 ∘ ( I ↾ dom 𝐹)))
3433adantl 485 . . . 4 ((Rel 𝐹𝐹𝑉) → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹) = (𝐹 ∘ ( I ↾ dom 𝐹)))
35 coires1 6106 . . . . 5 (𝐹 ∘ ( I ↾ dom 𝐹)) = (𝐹 ↾ dom 𝐹)
36 resdm 5886 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
3736adantr 484 . . . . 5 ((Rel 𝐹𝐹𝑉) → (𝐹 ↾ dom 𝐹) = 𝐹)
3835, 37syl5eq 2871 . . . 4 ((Rel 𝐹𝐹𝑉) → (𝐹 ∘ ( I ↾ dom 𝐹)) = 𝐹)
3934, 38eqtrd 2859 . . 3 ((Rel 𝐹𝐹𝑉) → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹) = 𝐹)
4024, 39eqtrd 2859 . 2 ((Rel 𝐹𝐹𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1) = 𝐹)
413, 40eqtrd 2859 1 ((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  ifcif 4450  cmpt 5133   I cid 5447  dom cdm 5543  cres 5545  ccom 5547  Rel wrel 5548  cfv 6345  (class class class)co 7151  cmpo 7153  0cc0 10537  1c1 10538  0cn0 11896  seqcseq 13375  IterCompcitco 45024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-inf2 9103  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-n0 11897  df-z 11981  df-uz 12243  df-seq 13376  df-itco 45026
This theorem is referenced by:  itcoval2  45031  ackvalsuc0val  45054
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