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Theorem itcovalpclem2 45012
Description: Lemma 2 for itcovalpc 45013: induction step. (Contributed by AV, 4-May-2024.)
Hypothesis
Ref Expression
itcovalpc.f 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))
Assertion
Ref Expression
itcovalpclem2 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
Distinct variable groups:   𝐶,𝑛   𝑦,𝑛
Allowed substitution hints:   𝐶(𝑦)   𝐹(𝑦,𝑛)

Proof of Theorem itcovalpclem2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 itcovalpc.f . . . . 5 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))
2 nn0ex 11903 . . . . . 6 0 ∈ V
32mptex 6978 . . . . 5 (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ∈ V
41, 3eqeltri 2912 . . . 4 𝐹 ∈ V
5 simpl 486 . . . 4 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝑦 ∈ ℕ0)
6 simpr 488 . . . 4 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
7 itcovalsucov 45009 . . . 4 ((𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
84, 5, 6, 7mp3an2ani 1465 . . 3 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
9 simpr 488 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10 simplr 768 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℕ0)
115adantr 484 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑦 ∈ ℕ0)
1210, 11nn0mulcld 11960 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℕ0)
139, 12nn0addcld 11959 . . . . . 6 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + (𝐶 · 𝑦)) ∈ ℕ0)
14 eqidd 2825 . . . . . 6 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
15 oveq1 7157 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 + 𝐶) = (𝑚 + 𝐶))
1615cbvmptv 5156 . . . . . . . 8 (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
171, 16eqtri 2847 . . . . . . 7 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
1817a1i 11 . . . . . 6 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶)))
19 oveq1 7157 . . . . . 6 (𝑚 = (𝑛 + (𝐶 · 𝑦)) → (𝑚 + 𝐶) = ((𝑛 + (𝐶 · 𝑦)) + 𝐶))
2013, 14, 18, 19fmptco 6883 . . . . 5 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)))
219nn0cnd 11957 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
2212nn0cnd 11957 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℂ)
2310nn0cnd 11957 . . . . . . . 8 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℂ)
2421, 22, 23addassd 10662 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + ((𝐶 · 𝑦) + 𝐶)))
25 nn0cn 11907 . . . . . . . . . . . . . 14 (𝐶 ∈ ℕ0𝐶 ∈ ℂ)
2625mulid1d 10657 . . . . . . . . . . . . 13 (𝐶 ∈ ℕ0 → (𝐶 · 1) = 𝐶)
2726adantl 485 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐶 · 1) = 𝐶)
2827eqcomd 2830 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 = (𝐶 · 1))
2928oveq2d 7166 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = ((𝐶 · 𝑦) + (𝐶 · 1)))
30 simpr 488 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 ∈ ℕ0)
3130nn0cnd 11957 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
325nn0cnd 11957 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 𝑦 ∈ ℂ)
33 1cnd 10635 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → 1 ∈ ℂ)
3431, 32, 33adddid 10664 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐶 · (𝑦 + 1)) = ((𝐶 · 𝑦) + (𝐶 · 1)))
3529, 34eqtr4d 2862 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = (𝐶 · (𝑦 + 1)))
3635oveq2d 7166 . . . . . . . 8 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
3736adantr 484 . . . . . . 7 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
3824, 37eqtrd 2859 . . . . . 6 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + (𝐶 · (𝑦 + 1))))
3938mpteq2dva 5148 . . . . 5 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
4020, 39eqtrd 2859 . . . 4 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
4140adantr 484 . . 3 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
428, 41eqtrd 2859 . 2 (((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
4342ex 416 1 ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3481  cmpt 5133  ccom 5547  cfv 6344  (class class class)co 7150  1c1 10537   + caddc 10539   · cmul 10541  0cn0 11897  IterCompcitco 44998
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 7456  ax-inf2 9102  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
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 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-iun 4908  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 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-2nd 7686  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-er 8286  df-en 8507  df-dom 8508  df-sdom 8509  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-n0 11898  df-z 11982  df-uz 12244  df-seq 13377  df-itco 45000
This theorem is referenced by:  itcovalpc  45013
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